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GraphiXT User`s Manual

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1. DLL functions x Edit New Insen before Duplicate Delete Delete All No Name 1 DLL1 TEST 3 C TestDLL Test dll 2 DLL2 TEST1 1 C TestDLL Test dll 3 DLL3 TEST 3 C TestDLL Test dll 4 DLL4 TEST2 2 C TestDLL Test dll Copy Copy All Cut Paste Paste before OK Apply l Cancel Fig 4 10 An example of a list of user defined DLL functions DLL function definition rx Function name in programs DLLT Number of arguments 3 Copies of argument values will be passed to the function C function prototype double func long ier double arg double arg2 double arg3 In order to pass addresses of arguments insert the minus sign before the number of arguments An example of the function call DLL1 a b c Function name in the DLL file TEST DLL file name C TestDLL T est dll Browse Cancel Fig 4 11 An example of the dialog window DLL function definition 21 The mentioned 32 bit integer variable is used as the error code Prior to calling the DLL function GraphiXT sets that variable equal to the number of all other arguments if copies of argument values are passed to the function or opposite to the number of the other arguments if argument addresses are passed to the function If the DLL function modifies that variable this is interpreted as a runtime error for more information about runtime error handling see Section 4 7 If the pr
2. Fig 5 1 An example of the dialog window X datasets 27 28 Formulas can be unlinked and re linked using the controls of the programming tools dialog window see Fig 4 1 whereas free curves can not be re linked after unlinking The IDs of formulas and free curves are constructed as follows Fn formula No n for example F10 FCn free curve No n for example FC10 Notes 1 The numbers in curve IDs reflects the order in which the formulas and free curves were created in the graph window 2 If the selected X set is a set of model time or coordinate values i e if its ID is t or XL7 then the list of formulas linked to it is sorted in the order of increasing formula IDs If the selected X set is independent i e if its ID is of the type Xn then the row order of formulas and free curves in the bottom grid control is the order in which they were linked to the selected X set After selecting a cell or a group of cells in the curve list the two buttons in the bottom right corner of this dialog window become active The button Unlink the selected curves causes removal of the selected curves from the list of curves linked to the selected X set and creation of the same number of new X datasets which are exact copies of the selected X set the X set list is modified accordingly Note If the selected X set is independent i e if its ID is of the type Xn and i
3. x um Electron Hole concentration concentration 1 cm 3 1 cm 3 3x104 162 8 05 1 390828e 014 6 878913e 011 163 81 1371585e 014 6 830527e 011 2x10 164 815 1352396e 014 6 188076e 011 165 5 742185e 011 166 825 1 314962e 014 4 678605e 011 167 83 1 296921e 014 6 71193e 011 4 168 8 35 1 278878e 014 6 962918e 011 1 261168e 014 7 098352e 011 4x1014 1x1014 E Data 1 Free surface potential IZ 1 Free surface potential S E Is Potential of the left edge of the system V 1 2 3 4 A o Measurement data t FT16 XI r S 800 4 ts Potential of the left t s Measurement 7 edge of the system data an v 1 0 878 15281500766 0 878 15 soo 4 2 0 0002001464606771 582 03890541054 0 01 134 82 3 0 0010675893303196 364 22676335466 0 02 93 28 4 0 0035126822880201 251 58734449054 0 03 81 43 ann I etl 200 a O 5 0 0075113975865957 200 03231823104 004 7136 Sa ndonag al ie 6 0 01 184 03897658102 0 05 64 28 0 005 01 015 02 25 03 035 04 045 05 7 0 016076755036054 160 97657678768 0 06 58 95 ts 8 0 02 151 65601057075 0 07 54 76 m ae EE Speed J WlAuto E lt gt 0 109523561 0 11 Vit syne V t sync all E x spne x sync all gt E3 ir Language Fig 16 1 An example of the GraphiXT main window with two data tables open 65
4. Convert to a function graph from the line s context menu An opposite conversion is also possible i e any function curve can be converted to a free form line Point coordinates of function curves in axis units are stored in memory as 64 bit floating point variables this corresponds to precision of approximately 15 significant digits Point coordinates of free form lines in axis units are not stored in memory and when they need to be displayed the program calculates them from the vertex pixel coordinates 1 e coordinates expressed in terms of number of pixels from the left and top edges of the graph window Thus vertex coordinates of free form lines in axis units depend on the screen resolution and those coordinates are much less precise than point coordinates of function graphs This fact should be kept in mind when converting a function graph to a free form line that conversion causes loss of precision Notes 1 If x coordinates of vertices of a free form line are not sorted in ascending or descending order then that line can not be converted to a function graph In this case after an attempt to convert the line to a function graph a warning message will appear 2 After converting a free form line to a function graph the user has an option to keep the original line or to delete it After an opposite conversion when a function graph is converted to a free form line there is no such option the original curve is always
5. Nonlinear least squares fitting a Used formulas and model functions Fitted datasets corresponding to the selected fitting function No 2 No Graph Function di Free surface pot Potential of the lv 3 Graph4 F1 Add formula model function Remove Add dataset Dataset options Remove dataset O E Li U UL E r Varied parameters Current model parameter No 4 A Name al 1 Graph 2 b G 10 Formula 3 ja L 4F1 2 Current value 9 65369e 009 v vary 4 BS TE I D Initial value 1 8e 009 normalize 5 jbl M 0 0100783 v 0 000171572 Upper bound Lower bound 0 New Copy to the Into window flemove No 4 Renovel Initial values of all parameters curent values ko Number of fitted points n 17 Number of varied parameters m 2 Number of degrees of freedom df n m 15 Total number of iterations 29 Chi 2 14 0153 Chi 2 df 0 934356 P Chi 2ldf 0 475633 Fitting stopped because the minimum value of the sum of squared errors has been reached Total fitting time 0 06 03 015 Maximum number of iterations LM method 20 r Simplex method 200 General options Start fitting Fig 14 1 An example of the nonlinear fitting dialog window Prior to instructions on using GraphiXT for nonlinear fitting an introduction to the theory of nonlinear fitting is presented 14 1 Elements of the theory of nonlinear fitting 14 1 1 The used terminology and formulation of the
6. in this example B lt 0 44 Fig 13 4 Linear interpolation thin solid line and exponential interpolation dashed line Since the integrand function is positive and sublinear i e its first derivative decreases with increasing x a more accurate estimate of the integral is obtained by applying the method of linear interpolation In the case of exponential interpolation the error grows with increasing ratio of connected y values for example see the dashed line connecting the last two points Interpolated values are computed using the points belonging to the interval of integration and the closest points outside that interval if available one point near each limit of integration In the case of exponential interpolation only positive or only negative values of y are used In the latter case if the interval of integration contains both positive and negative y values then GraphiXT computes both the positive and the negative integral 13 3 Statistical analysis Let us assume that a data set consists of the argument values x1 X2 Xn 1 Xn and corresponding values of the function y x which will be denoted y yo Yn During statistical analysis the program computes the following statistical parameters e sum and average Ves 13 10 Il s eR S e variance 1 I 1 n D v 77 Xy ny 13 11 n li n li e standard deviation of a single point o JD 13 1
7. A group of three checkboxes that are to the right of the text Apply these t and x limits to are used when time or coordinate limits of two or more fitted datasets of the given type i e f t model functions or free curves f x model functions or f x free curves must be computed according to the same rule specified by the previous controls Different checkboxes correspond to different subsets of fitted datasets of the given type the subset is indicated near each checkbox The controls that belong to the group Error factors of the selected dataset define the method of computing the error factors which are denoted w in the SSE expression 14 1 3 The error factors either have the same value for all points of the fitted dataset or they are calculated using a pre defined formula or they are extracted from a special dataset Four choices are available 1 constant error factors 1 w 14 2 1 e where e is a positive constant 2 error factors that are inversely proportional to the absolute value of the corresponding measured response v 7 1 J v 2 where f and e are positive constants the constant e may be zero 14 2 2 We 3 error factors that are inversely proportional to the square root of the absolute value of the corresponding measured response v 1 fly ite where f and e are positive constants one of them may be zero 14 2 3 W 4 error factors that are equal to inve
8. Available physical memory is self explanatory The button Delete model data is used to delete all stored model time values and associated model function values Depending on the state of the checkbox Don t delete model data corresponding to the initial time the first stored time value together with the corresponding set of function values may be deleted or not see below Note The menu command Simulation options Delete model data serves the same purpose as this button If the checkbox Don t delete model data corresponding to the initial time is not checked then by clicking the button Delete model data or by selecting the menu command Simulation options Delete model data all model data will be deleted the text box Number of time values in memory will contain 0 If that checkbox is checked then the mentioned action will cause deletion of all data except the data corresponding to the first stored model time value the text box Number of time values in memory will contain 1 This option is useful when simulation is intended to be done several times starting from the same initial state corresponding to the minimum stored time value for example the initial state could be the state of thermodynamic equilibrium The static text field Computation time contains the total processing time of the simulation Its format is H MM SS SSS where H is the number
9. GraphiXT uses one of the variants of the simplex algorithm which is described below 1 Sort the values of the SSE corresponding to each vertex of the simplex in the ascending order Fain F D lt F D3 SS F Din Ex 14 1 24 where the subscripts at p are the numbers of the vertices 2 Compute the centroid p i e the arithmetic average of all vertices except p m 1 m P D 14 1 25 m 1 3 Reflect the vertex pm in the opposite face of the simplex along the line p p see Fig 14 3a P P 0 P Pau 14 1 26 GraphiXT uses the reflection coefficient value a 1 If F p SF p lt F pn 14 1 27 then obtain a new simplex by replacing the worst point pn with the reflected point p and go to step 1 Otherwise if F p lt F p 14 1 28 then go to step 4 If this inequality is false then go to step 5 4 Compute the expanded point see Fig 14 3b P P 7Y P Pau 14 1 29 This simplex algorithm is described in the Wikipedia article Nelder Mead method web page http en wikipedia org wiki Nelder Mead method 52 GraphiXT uses the expansion coefficient value y 2 If F p lt F 14 1 30 then obtain a new simplex by replacing the worst point pmii with the expanded point pe Otherwise obtain a new simplex by replacing the worst point p with the reflected point p Go to Step 1 5 Compute the contracted point see Fig 14 3c Pe Pins P Pma 14 1 31
10. addition subtraction gj multiplication division A raising to a power assignment of a value For example the statement a a 1 increases the value of the variable a by 1 Comparison operators lt less than lt less than or equal to gt greater than gt greater than or equal to equal to not equal to lt not equal to HB EEB MV x TI K RT 3x101 Fi 5x10 Vr 4098 15x10 10 2x10 10 Insert reference to a function Built in function Function argument values Argument values correspond to the X axis range Number of points N Independent set of values tMin 0 Max 1 98e 010 N 102 Use this set of argument values Model time values Arrays Global parameters Local parameters Curve format and name Compile Apply E auto 1 5 0 0000e 000 2 0000e 010 7 tsyne T tsyne all xsync all gt FI ilevy x TI A sz pr Insert reference to a function ft fx free curve subroutine DLL Built in function e A Error in this formula the identifier X can not appear on the left hand side of an assignment statement Total current A cm 2 Function argument values Argument values correspond to the X axis range Number of points N 101
11. n NLFT n the number of ft functions in the layer No n NLFX n the number of f x t functions in the layer No n All built in arrays are read only i e user programs can not modify their elements those arrays are automatically modified by GraphiXT when the user changes the relevant data The smallest allowed value of array indices is 1 An exception to that rule is the index that means the sequence number of a layer of the simulated system i e the first index of the arrays XL I2FT T2FX NLFT and NLFX the smallest allowed value of that index is 0 and the largest value is one less than the number of layers However if any of the just mentioned five arrays is used as an argument actual parameter of a subroutine then the smallest allowed value of the index of the corresponding formal parameter is still equal to 1 GraphiXT includes an array viewer The array viewer allows direct access and modification of element values of user defined arrays The array viewer can be opened either by clicking the button View of the array list dialog window see Fig 4 4 or by selecting the command Open array viewer from the array context menu or by double clicking an item of the array list box Only allocated arrays can be viewed using the array viewer if an array is not allocated then the double click opens the array properties dialog An example of an array view is shown in Fig 4 6 V
12. Arrays in GraphiXT are global objects which exist independently of user defined programs A new array can be only created in the list of arrays That list is opened using the menu bar command Programming Arrays or the context menu of the program editor or the array editor or by clicking 15 r 1 Arrays a View Properties New Insen before Duplicate Delete Delete All No Name and dimensions 1 Array1 100 2 Arayan E 1 4 Ap10000000 r Copy Copy All Cut Paste Paste before Deallocate Deallocate All Fig 4 4 An example of a list of arrays the button Arrays of the programming tools dialog see Fig 4 1 The list of arrays can also be opened by clicking the toolbar button 53 see Fig 4 1 An example of a dialog window with such a list is shown in Fig 4 4 Array properties i e name dimensions data type and initialization expression can be modified in the array properties dialog window which is opened by clicking the button Properties or by selecting the command Array properties from the array context menu An example of the array properties dialog window is shown in Fig 4 5 The smallest allowed value of each index of a user defined array is 1 and the largest allowed value of each index is defined in the text box Dimensions of the array properties dialog see Fig 4 5 The array data type is defined using
13. GraphiXT uses the contraction coefficient value 7 0 5 If 20 Aa 14 1 32 then obtain a new simplex by replacing the worst point Pm with the contracted point p and go to Step 1 Otherwise go to Step 6 6 Shrink the entire simplex see Fig 14 3d by replacing all points except the best point p with P P o p P 2 3 m 1 14 1 33 GraphiXT uses the proportional contraction coefficient value o 0 5 Go to Step 1 Fitting is stopped when the simplex dimensions become so small that the differences of SSE values corresponding to different vertices of the simplex become less than some small predefined value or when the differences of parameter values corresponding to different vertices of the simplex become less than some small predefined value 14 2 Nonlinear fitting with GraphiXT 14 2 1 Definition of fitting functions The nonlinear fitting dialog window is opened by selecting the menu command Data analysis Nonlinear fitting An example of that window is shown in Fig 14 1 When that window is opened for the first time all its input and information fields are empty Then the user must define the fitting functions whose parameters will be optimized This is done by clicking the button Add formula if parameters of a user defined formula will be optimized or Add model function if parameters of a physical system simulated by the loaded plug in will be optimized and if the fitted datasets contain values of
14. and y are variables parameters z is a two dimensional array and g is a three argument subroutine whose formal parameters conform to the requirements stated in the fourth comment of Fig 4 9 then the following call to the subroutine FUNC1 corresponding to Fig 4 9 is allowed FUNCI x loc y loc z loc g If a formal parameter is an array then the corresponding argument actual parameter can be a part of an existing array This is achieved by specifying one or more array indices in the loc construct when calling the subroutine The following convention is used if the array used as an actual parameter has N dimensions and the number of indices of the actual parameter is m then the subroutine treats that parameter as an array which has N m 1 dimensions and which starts from the element specified by the indices of the actual parameter For example the following call to the subroutine FUNC1 corresponding to Fig 4 9 is allowed FUNCI1 x loc y loc FC 2 3 loc g As mentioned FC is a three dimensional array whose first index means the sequence number of a graph and the second index is the sequence number of a free curve of that graph For example if the subroutine FUNC1 contains a line d c 4 3 then that line will be equivalent to d FC 2 6 3 i e the temporary variable d will be assigned the Y value of the third point of the free curve No 6 of the graph
15. but with additional arguments epsabs absolute accuracy requested default value 0 epsrel relative accuracy requested default value 10 leniw twice the maximum number of subintervals allowed in the partition of the given integration interval default value 400 i e the default maximum number of subintervals is 200 maxpl an upper bound on the number of Chebyshev moments that can be stored default value 100 Sum f i i1 i2 sum of terms f when the summation index i varies from il to i2 e g Sum In i i 2 10 Sum2 f t a b dt sum of terms f when the summation variable varies from a to b in increments of dt Iter f 1 i1 i2 repetition iteration of expression f when the iteration index i varies from il to i2 the returned value is the last calculated value of expression f Root f x a b root of the nonlinear equation f x 0 a and b are limits of the interval that should be searched for the root For example the expression Root sqrt x 1 x 3 x 0 1 is equal to the root of the equation Vx 1 7 0 1 e 0 60542342357183 Root2 f x a b RelErr AbsErr loc flag same as Root but with additional arguments RelErr relative accuracy requested default value 10 AbsErr absolute accuracy requested default value 10 flag the error code which is modified by the function If the equation is solved successfully then fla
16. or of a group of objects Mouse drag with the left mouse button pressed Selecting a group of text labels or free form lines Press the left mouse button at a point in the graph window where there are no objects and drag the mouse Adding a text label or a free form line to a group of selected objects or removing an object from the group Shift left click on the text label or the free form line Deleting selected text labels and free form lines Delete after selecting an object or a group of objects Selecting a point of a function graph Click the left mouse button on the curve or on its name in the legend then use the keys and lt next or previous point Home and End first and last points or T and V previous or next curve in the list of plotted curves Selecting a vertex of a free form line Click the left mouse button on the line then use the keys and lt next or previous vertex or Home and End first and last vertices Changing the coordinates of the selected point vertex Press Ctrl and use the arrow keys Copying the data and format of the selected curve or objects Ctrl C after selecting a curve an object or a group of objects Pasting a curve or graphical objects i e creating a curve or a group of text labels and free form lines from clipboard data Ctrl V when nei
17. No 2 Recursion is allowed recursion means a call of a subroutine from within the same subroutine However unlike in some other programming languages e g C all instances of one subroutine share one memory space This means that any temporary variable that has been assigned a value in one instance of a subroutine will have the same value in all other instances of that subroutine The same is true of the formal parameters too except when the formal parameters in question are pointers to arrays or functions if a subroutine is called from within the same subroutine then the values of the formal parameters of the calling instance of that subroutine and of all higher level instances of that subroutine automatically become equal to the argument values used in that call For example let us assume that subroutine FUNC I1 a has a local parameter d which indicates the call depth and which is initially equal to 0 and that the first four lines of the code of that subroutine are d d 1 b d if d lt 2 FUNCI a I d d 1 6699 Then if the formal parameter a is initially equal to 2 e g if the code of the calling formula is FUNCI 2 its value immediately after executing the third line will become 3 because such is the value passed to subroutine FUNC1 in the call to it from within FUNC1 The value of the temporary variable b which was initially assigned the value 1 will become equal to 2 Notes 1
18. causes the cells containing the corresponding x and y values to become highlighted in the table if necessary the table window is automatically scrolled to ensure that those two cells are visible And vice versa selecting a new cell in the table causes a change of the position of the red cross in the graph window In partial mode when a point has been selected or a cell highlighted the so called point dialog window is visible That window contains the x and y values of the current point as well as the point number see Fig 16 1 Notes 1 The point dialog window is always visible when a point has been selected in one of the plotted functions regardless of whether the table window is open or not 2 The result of pressing the arrow keys on the keyboard depends on which window has input focus the point dialog window or the table window If the table window has focus then pressing an arrow key causes a corresponding adjacent cell to become highlighted If the point dialog window has focus then pressing the or lt key causes the next or the previous point of the current curve to become the current point i e in this case the cell that is below or above the current cell becomes highlighted Pressing the 1 or V key when the point dialog window has focus causes the previous or the next curve in the legend to become the current one and the current point becomes the point whose abscissa is closest to the
19. longer measurements it may become evident that the model is not correct The opposite situation is also 50 possible analysis of nonlinear fitting results may lead to the conclusion that the model is unsuitable when in fact it is correct Thus when evaluating suitability of a theoretical model by the method of nonlinear fitting errors of two types are possible A type I error occurs when a true null hypothesis is rejected A type II error occurs when a false null hypothesis is accepted The probability of those errors is determined by the mentioned critical probabilities For example if the critical probabilities are 0 005 and 0 995 then probability of the type I error is 0 01 1 This means that if the standard conditions are satisfied then after doing the same set of measurements at the same conditions and fitting each experimental data set with the same theoretical function the value P x df will not belong to the interval 0 005 0 995 on the average only once in 100 times Thus the probability of the type I error can be decreased by decreasing the first critical probability and increasing the second critical probability However if the null hypothesis is false then by widening the acceptance region the probability of the type II error is increased i e there is an increased risk that a false null hypothesis will not be rejected The values of critical probabilities are therefore chosen on the basis of a trade off between the ris
20. Cancel all unsaved changes of the parameter list and of individual parameters are discarded and the parameter list dialog is closed Parameter expressions may call functions that change values of array elements or values of parameters of the other type i e expressions of global parameters may call functions that modify local parameters of the currently edited program and expressions of local parameters may call functions that modify global parameters After calculating such expressions i e after clicking Calculate or Calculate All and then clicking Cancel those changes will not be automatically cancelled If GraphiXT determines that parameter evaluations may involve a change of array element values or of values of parameters of the other type then all plotted formulas that depend on arrays or on parameters of the other type are automatically recalculated after calculating the current parameters and the corresponding curves are re plotted However in such a case the plotted formulas are recalculated using the o d values of parameters of the current type which existed before opening the parameter list dialog window or which were saved by the most recent click on the button Apply In order to re plot formulas using the updated parameter values one must click the button OK or Apply 4 3 Using arrays in programs A data array in GraphiXT is a collection of numbers elements each selected by one or
21. Independent set of values xMin 0 xMax 1 is 101 Use this set of argument values Layer No 1 node coordinates Arrays J Global parameters Local parameters Curve format and name Compile a s k gt 10000011 emi Z tse same al P E P Fig 4 1 Examples of the program editor window a The curve is displayed in a time graph window and the formula contains references to f x t model functions In this case the x variable must be assigned a value and abscissas of the curve points must coincide with the model time values In this example the formula calculates the expression 3 t In f x Os 1 3 10 f CDN o Where Jis the fx model function whose number in the function list is 6 That function is denoted fx6 b The curve is plotted in a coordinate graph window The formula contains references to a f model function and to a free curve plotted in the same window as well as comments There is an error in one line That line is highlighted and the description of the error is both in the status bar of the program editor window and in the programming tools dialog window In this case modifying the x variable is not allowed because x is the function argument whose values are computed automatically If a given inequality or equality is true then the result of the comparison statement is 1 and if it i
22. Smoothing direction Maximum interval between inflection points 5 Viincreasingx V decreasing x Number of changed points 20 Smoothed Y gt RMS error before smoothing 0 59125609013358 alter sm 0 28315353295119 Average before smoothing 3 8452158059251 after sm 3 8452158059251 Number of smoothing cycles 100 x OK Cancel Fig 15 1 An example of the smoothing dialog window 61 After smoothing the static text fields RMS error before smoothing and after sm display the values of the root mean square error before smoothing and after it This quantity is defined by the formula 1 n 2 Ja 15 1 N 4 j29 66999 1 where n is the number of points that are in the smoothing range the subscript and y are the point abscissa and ordinate and a is the average slope 1 e imi 15 2 X X is the point number x n Those quantities are computed using the points that belong to the smoothing range and one additional point on each side of the smoothing interval if some points of the curve do not belong to the smoothing interval Since smoothing decreases fluctuations of the function values y the RMS error after smoothing is usually less than the RMS error before smoothing However since the RMS error a is computed including one or two points that are outside the smoothing range the RMS error after smoothing may occasionally become greater than the RMS error before smoothing This
23. be separated from each other by tabs and their values must be separated from each other either by spaces or by tabs however the data import dialog window has input fields for entering any other set of separators or for specifying that the files do not have the header line or for specifying the number of initial lines that must be ignored In order to be able to paste free curve data from the clipboard the text data in the clipboard must conform to the same format as described above and besides there must be an additional line containing the text GraphiXT plot data without quotation marks prior to the line with column headers This clipboard data can be formed either by copying the text from any text editor or by copying a GraphiXT curve using the curve s context menu command Copy curve in the latter case the curve format is also copied to the clipboard When creating importing or pasting free curves it is possible to link them to any existing X dataset i e to make their x values equal to x values of existing curves In addition to the usual method of accessing the curve options by selecting the corresponding command in the context menu the free curves can also be accessed from the dialog window with the list of free curves which can be opened by selecting the menu command Graph options Free curves or by clicking the toolbar button E see Fig 1 1 A curve can be hidden by unchecking the ch
24. command can also be selected in the context menu that appears after right clicking the title bar of the graph window e X or Y axis can be rescaled by double clicking the axis or by right clicking it and selecting an appropriate command from the context menu After rescaling the X or Y axis its limits become equal to the least value and the greatest value of the abscissas or ordinates respectively of all curves plotted in the active graph window If no curves are plotted or only formulas with temporary sets of X values are plotted see Section 4 1 then the result of rescaling the X axis depends on the graph window type a in coordinate graphs whose window icon contains the letter x the X axis limits become equal to 0 and 1 b in time graphs whose window icon contains the letter t the time axis limits become equal to the simulation time limits The simulation time limits are set in the dialog window Time limits and amount of data see Fig 7 1 which is opened by selecting the menu command Simulation options Time limits and amount of data The simulation time limits are entered in text boxes Minimum visible time and Final time see Fig 7 1 e In order to make it easier to determine values of the plotted quantities from their graphs it is recommended to turn on auto rescale of Y axis This is done by right clicking any one of the two Y axes then selecting Y axis options in the context men
25. command of the graph context menu An existing formula can be modified by double clicking the corresponding curve or its name in the legend or by selecting the command Formula in the curve s context menu Those actions open the program editor window Examples of that window are shown in Fig 4 1 Formulas can also be created or modified using the menu command Formulas which is available in two menus Graph options and Programming or by clicking the toolbar button LE see Fig 4 1 Then the dialog window with the list of all defined formulas is opened A new formula can be created by clicking the button New in that dialog window An existing formula can be modified by selecting it in the list and then clicking the button Edit In either case the program editor window is opened Note A formula curve can be hidden in the same way as other curves i e by unchecking the checkboxes Connect points and Show points as symbols in the curve format dialog window or by clicking the button Hide in the mentioned dialog window with the list of formulas see also Section 3 If a formula curve is hidden it can only be accessed through the mentioned formula list Two types of programs can be used in GraphiXT environment formulas and subroutines The main difference between a subroutine and a formula is that a subroutine is only executed when called invoked programmatically from a formula
26. formula or subroutine returning the value of x Notes 1 Functions counter counter freg and counterle15 require presence of the high resolution performance counter it is present in all modern personal computers If that counter is absent those functions return 1 2 Although the function Return x does return the value of x that value can not be used in the calling program because any statement containing a call to Return will be unfinished it will be interrupted by that call For example the statement a exp t Return 10 sin x will be executed by first calculating the value of exp t and then terminating the current program with the return value of 10 Thus the term sin x will not be calculated and the value of the variable a will not be modified Advanced built in functions 66 99 oe a loc a memory address or sequence number of a programming object a where may be the name of a variable parameter array subroutine another built in function or a DLL function If the argument is an identifier of a variable or an array then the returned value is its memory address and if it is a function name then the returned value is the sequence number of that function in the set of all defined functions 6 a Size loc a number of elements of a one dimensional array Size2 loc a i number of elements of array a whose indices differ only by the v
27. immediately and can not be undone The two buttons Insert reference and Cancel see Fig 4 8 are only visible when a program editor window is active After clicking the button Insert reference a call to the currently selected subroutine will be inserted at the cursor position in the code of the edited program and the dialog window with the list of subroutines will be closed After clicking Cancel the subroutine list dialog will be closed without inserting the call to the subroutine The name and formal parameters of a subroutine are defined in the dialog window Subroutine name and formal parameters Its example is shown in Fig 4 9 After creating a new subroutine that Edit New Inser before Duplicate Delete Delete All No Name and formal parameters 1 FUNC a 2 ffloc b tT 3 FUNCI a loc b lacfc locff x loc lociL J FUNC2floc Array J loclArray2 J T Insert reference Fig 4 8 An example of a list of subroutines Name FUNCI Formal parameters Parameter No 1 a variable a which can not be modified in the subroutine the corresponding actual parameter is a value of a variable or an expression a Parameter No 2 a variable b which can be modified in the subroutine 4 the corresponding actual parameter is an address of a variable loc b Parameter No 3 a two dimensional array c the correspondin
28. is a built in function not x which returns 1 when x 0 and 0 when x 0 Separators Parentheses and which are used for grouping operands of binary operations as well as for indicating the start and the end of a function argument list or of an expression used as a condition in the if else construct or the loop operator while see below about the if else and while constructs curly braces and which are used to denote the start and the end of a branch of the if else construct or of the body of the while loop operator i e a sequence of statements which should be executed when a specified condition is true the curly braces are only needed when that sequence consists of two or more statements brackets and J which are used for grouping operands of binary operations as well as for indicating the start and the end of a list of array indices 66 99 the comma which is used to separate function arguments in function calls or array indices in references to arrays Basic built in functions exp x the exponential function In x the natural logarithm Ig x the decimal logarithm sqrt x the square root sin x cos x tg x the trigonometric functions arcsin x arccos x arctg x arctg2 y x the inverse trigonometric functions sinh x cosh x tanh x the hyperbolic functions jO x j1 x jn
29. kept Since the format and position of the created free form line is the same as format and position of the original curve it may be at first impossible to visually distinguish the created line from the original curve The created free form line can be separated from the original curve by mouse dragging it or by changing dimensions or axis limits of the graph window 40 12 Keyboard and mouse shortcuts The table below lists actions that can be performed not only using the mentioned menu commands but also using special key combinations and mouse Intended result Actions using keyboard and mouse shortcuts Narrowing the axis intervals Drag the mouse with the middle button pressed Rescaling the vertical axis then the axis limits become equal to the minimum and maximum values of the plotted functions Double click the left or right Y axis Rescaling the horizontal axis then the axis limits become equal to the minimum and maximum argument values of the plotted functions Double click the bottom or top X axis Rescaling both axes Double click any one of the four corners of the plotting area Editing the text of a text label Double click the text label Changing the format of a function curve Double click the curve or its name in the legend Changing the format of a free form line Double click the line Changing the position of a graphical object 1 e a text label a line or the legend
30. more indices that can be computed at run time by the user program Arrays in GraphiXT can have up to 20 dimensions indices A two dimensional array i e an array that has two indices can be visualized as a table of numbers where the row number is equal to the value of the first index and the column number is equal to the value of the second index a two dimensional array is often called a matrix Array element values can be accessed and modified in user defined programs Array elements are referenced using the format a 11 12 13 where a is the array name and il i2 i3 are the array indices positive integer numbers For example if A is a two dimensional array then the following statement sets the value of the element at row 2 and column 3 equal to the sum of two elements element at row 5 and column 10 and element at row 20 and column 30 A 2 3 A 5 10 A 20 30 Any expression can be used instead of each array index Value of each such expression is rounded to the nearest integer number For example if N is a variable that was previously assigned the value 10 6 then the following example is equivalent to the previous one A 2 N 8 A N 2 10 A 9 N 3 N 2 8 array data types are supported 64 or 32 bit floating point and 32 16 or 8 bit signed or unsigned integer However in all arithmetic operations the array element values are converted on the fly to 64 bit floating point format
31. n x Bessel functions of the first kind orders 0 1 n respectively yO x y1 x yn n x Bessel functions of the second kind orders 0 1 n respectively erf x the error function erfc x the complementary error function gamma x the gamma function Ingamma x the natural logarithm of the absolute value of the gamma function gammp a x the incomplete gamma function abs x the absolute value of a number max x y min x y the larger or smaller of two values select x y Z if x 0 then returns y otherwise returns z Idexp x y 4 2 fmod x y the floating point remainder of x y ceil x the smallest integer that is greater than or equal to x floor x the largest integer that is less than or equal to x near x the integer that is closest to x time number of seconds elapsed since midnight January 1 1970 clock number of milliseconds elapsed since the start of the program counter number of processor cycles since the computer startup or restart counter freg number of processor cycles per second counterle15 remainder of counter 109 rand a pseudorandom integer in the range 0 to 32767 srand x sets the starting point for generating a series of pseudorandom integers the next call to rand will start a new seguence of pseudorandom numbers which depends only on x the function srand x always returns zero Return x immediate termination of the current
32. nine radio buttons of the array properties dialog see Fig 4 5 Array elements can be initialized at any time using an expression entered in the text box Expression of the array properties dialog see Fig 4 5 That expression can include user defined identifiers of array indices and references to any programming object defined in the current project making possible complex initializations Identifiers of array indices are entered in the bottom text box Index identifiers see Fig 4 5 The default identifiers of the first three indices are il i2 and 13 For example if the initialization expression is as shown in Fig 4 5 then after clicking the button Initialize the element A 2 5 3 will be assigned the value 10000 2 100 5 3 20503 values of all other elements will be calculated similarly Any user defined array can be allocated or deallocated at any time using buttons Deallocate Allocate and Deallocate All Allocate All see Fig 4 4 When an array is deallocated all Array properties EX Name Array3 Dimensions 10 15 20 Import from a file Array data type 9 Floating point Integer 2 4 bytes Bbytes 3 Signed 2 bytes 4 bytes Unsigned 1 byte Use same value for all elements Value of the first element 10101 Expression 10000 i1 100 i2 13 Index identifiers i1 i2 13 Apply Initialize Cancel Fig 4 5 An example of an array properties dial
33. of hours MM is the number of minutes and SS SSS is the number of seconds and milliseconds The button 0 00 00 is used to set the current computation time to zero Note The program keeps in memory the processing time corresponding to each stored model function value After deleting a part of model data the current computation time becomes equal to the last available value 31 32 Initial simulation time Minimum visible time Time of the last computed point of fft curves Time of the currently computed fft point Final time 4e 010 Number of time values in memory Delete model data Model curve and array data amount kB Used memory 11936 kB Available physical memory gt 4194303 kB Don t delete model data corresponding to the initial time Computation time 0 00 00 000 0 00 00 Double precision f x t functions 8 bytes for each value Primary functions All functions Set precision of f x t function values L a Current simulation time 1 9872671518368e 010 Minimum visible time 0 Time of the last computed point of fft curves 1 98e 010 Time of the currently computed f t point 2e 010 Final time 4e 010 Number of time values in memory 101 Delete model data Model curve and array data amount 7951 kB Used memory 38864 kB Available physical memory gt 4194303 kB Don t delete model data corresponding to the initial
34. of the selected formula is visible only when at least one of the fitting functions listed in the window Nonlinear least squares fitting is a user defined formula with at least one local parameter defined The contents of the other two list boxes do not depend on the used fitting functions The list box Model parameters contains not only the names of model parameters but also short explanations of their meanings This list box is visible only when a simulation plug in has been loaded for example CarrierFunc dll The optimized parameters must be selected in the mentioned three list boxes After clicking OK the names and current values of those parameters will be added to corresponding columns of the grid control Varied parameters see Fig 14 1 Each new parameter is set to be varied by default i e the checkbox in the corresponding cell of the column Vary is checked The current value of a parameter and the state of the mentioned checkbox can be changed directly in the grid control Varied parameters In addition after selecting a parameter in the dialog window Nonlinear least squares fitting i e after left clicking the corresponding cell in any of the columns of the grid control Varied parameters it becomes possible to change the initial value or bounds of that parameter or to turn on off its normalization Those properties are shown in the group of controls Selected parameter on the right
35. of the list box and clicking the button OK the name of the selected dataset will be added to the list of fitted dataset names which are in the column Name of the dataset grid control in the dialog window Nonlinear least squares fitting see Fig 14 1 The cells that are to the right of that column contain the limits of the fitting interval If the datasets are of the At type model functions or free curves then there are two numeric cells tMin and tMax which contain the minimum and maximum times respectively If the datasets are f x f model functions or f x free curves then there are four numeric cells xMin xMax tMin and tMax which contain the minimum and maximum values of the coordinate as well as the minimum and maximum times respectively see Fig 14 1 Ifa given fitted data set is a free curve of the type f x then tMin tMax because such a dataset must be assigned a single time value In order to add more fitted data sets the mentioned steps must be repeated The checkbox Use indicates if a given dataset or all datasets corresponding to a given fitting function will be used during fitting After defining a new fitted dataset its definition interval is initially unlimited i e it includes all points of that dataset In this case the cells xMin xMax tMin and tMax contain minimum and maximum values of the coordinate and time in the given dataset If the
36. of the system Argument x values of all f x t const functions that belong to the same layer must be equal to each other However the sets of x values corresponding to different times or different layers may be different from each other 10 Slider bar The slider bar is at the bottom of the main window Some elements of the slider bar depend on the type of the active graph window time graph or coordinate graph see Fig 10 1 The slider bar has the following elements 1 The time slider is used to change the current time of the active graph window and of all windows that are synchronized with the active window 2 The first of the two buttons is used to open the dialog window where the time slider limits and the time multiplier can be entered the same dialog is opened by right clicking the time slider In addition if the current graph is a time graph the mentioned dialog has a checkbox for selecting an option to show the value of the current time next to the time slider or to hide it compare Fig 10 la and Fig 10 1b 3 The checkbox Auto is used to turn on or turn off the current time automatic adjustment mode After checking this checkbox the current time of the active graph window and all windows that are synchronized with it becomes equal to the time of the last point of ff model curves however afterwards the current time may be set to any other value Besides when the number of stored model time values ch
37. of time cross sections will be updated by adding the names of the selected functions to it The default name of a time cross section is formed by inserting the x value into the original name of the f x t model function see Fig 6 1 The x value of an existing time cross section can be modified by left clicking the corresponding item of the time cross section list and then clicking the button Change X see Fig 6 1 Then the new x value must be entered in the dialog New X value the same dialog window can also be opened by right clicking the item of the list and selecting the command Change X value from the context menu or by double clicking the item of the list After entering the new x value and clicking OK the name of the corresponding time cross section will be automatically modified by inserting the new value of x into it When an item is selected in the list of time cross sections three other buttons become active Insert before Duplicate and Delete see Fig 6 1 The result of clicking the button Insert before differs from the result of clicking the button New in one respect only the new time cross sections will not be appended to the end of the list but they will be inserted before the selected item instead By clicking the button Duplicate a time cross section identical to the selected one will be inserted after the selected time cross section then the x value of the new
38. one of those handles all handles are hidden so that they do not interfere with precise positioning of the vertex After releasing the left mouse button all handles are shown again In order to end modification of the line shape the left mouse button must be clicked anywhere outside of the mentioned line handles Note The line shape can also be modified by entering coordinates of the current vertex in axis units in the point dialog window that is visible in line edit mode or by pressing the Ctrl key on the keyboard and using a Free form line format and position mxm Color Width Style Black v 1 Solid x Horizontal position Vertical position Constant distance to Constant distance to Left edge of the window Top edge of the window Left T axis Top axis Vertical central line Horizontal central line Right Y axis Bottom axis Right edge of the window Bottom edge of the window Constant distance to line x const Constant distance to line Y const x bounding rectangle left edge x coordinate Y bounding rectangle top edge Y coordinate x bounding rectangle center X coordinate Y bounding rectangle center Y coordinate x bounding rectangle right edge x coordinate X bounding rectangle bottom edge Y coord x Y line will only be visible when Min lt X lt x Max line will only be visible when yMin lt Y lt yMax Fig 11 3
39. or another subroutine whereas formulas can not be referenced programmatically in this respect a formula is analogous to the main function which exists in many programming languages Up to 20 arguments can be passed to each subroutine Each argument can be either an expression or an address of a variable or a pointer to an array or a pointer to another subroutine built in function or a DLL function The next section will describe the main rules of writing simple programs formulas which do not use subroutines or data arrays After that more advanced programming techniques which involve the use of subroutines and arrays will be described 4 1 Writing simple programs and displaying calculation results User programs must be written in the GraphiXT programming language The main rules of using the GraphiXT programming language are described below Each non empty line of a program must contain an expression with the assignment operator or without it The exceptions to that rule are the lines containing only a curly brace it may be the only character in a line or only the keyword else Each expression may contain any number of arithmetic logical or comparison operations and calls to various functions The value of the last calculated expression is the final value of the computed function Below is the list of standard operators built in functions and special symbols that may be used in a program Arithmetic operators
40. process eccccccccccccsscesseenseeceeceeseesecnsecuseceseceseceseeseeeeeaeeeaeeeaeeeseecsaecsaecsaecaeceseeneenaeenas 59 15 Data Smoothin green a i a a i a a E a ee ess 61 16 Using function data tables iis sanis ius o is si Ia i a a i sa 65 Coding Credits Lessons as a a a vices i a i i Ta a I E T a a vanes 67 1 Introduction GraphiXT is a data analysis software and numerical computing environment The original purpose of GraphiXT was to facilitate study of time dependences i e kinetics of a large number of physical quantities that are related to each other Consequently graphs displayed by GraphiXT are of two types graphs of functions f t whose argument is time 7 and graphs of functions f x t whose arguments are coordinate x and time However the actual meaning of function arguments is up to the user GraphiXT exe can be used as a stand alone program or in conjunction with plug ins that solve kinetic equations describing a particular physical system Currently there is one such plug in which is designed to simulate charge carrier kinetics in multi layer systems That plug in consists of two files the function file CarrierFunc dll and the parameter editor file CarrierParms exe described in a separate user manual The program GraphiXT has been developed as an educational tool for the course Numerical simulation of charge transport processes at the Faculty of Physics of Vilnius University It can also be used as
41. reference point for calculation of a new position in the graph window after changing the window dimensions or axis limits the command Line format and position must be selected in the line s context menu the same result is achieved by double clicking the line This action opens the dialog window Free form line format and position An example of such a window is shown in Fig 11 3 The majority of the controls of this window are self explanatory The purpose of the list boxes Horizontal position and Vertical position is the same as in the text label options dialog window see Section 11 1 The only difference is that the edge coordinates of the text label are replaced by the edge coordinates of the bounding rectangle of the free form line 11 4 Converting free form lines to function graphs Although it is possible to attach a free form line as a whole to a particular vertical or horizontal line of the graph the lengths of the line segments and angles between them do not depend on the window dimensions or the axis limits of the graph For example when the graph window is increased free form lines of that graph do not become bigger This is the main difference between free form lines and function graphs curves However if the x coordinates of free form line vertices form an increasing or decreasing sequence then that line can be converted to a function curve This is achieved by selecting the command
42. same numbers that compose the column s and G is the matrix of second derivatives of the function F p at po multiplied by 1 2 Elements of that matrix are 1 OF 14 1 10 j 1 2 m 14 1 13 p J P Po Inserting the expression of F po s 14 1 11 into 14 1 9 instead of F p we obtain s G s lt F 14 1 14 The points p that satisfy this condition are inside an m dimensional ellipsoid whose center is at the point P po and whose projection to the i th parameter axis is the interval Poisi S Ppi lt poits G 1 2 m 14 1 15 where po is the optimal value of the i th parameter i e the projection of the point po to the axis p and si 16 VIG Ny Mam G 1 2 m 14 1 16 where G i 1 2 m are the diagonal elements of the inverse matrix G the matrix G is called the variance covariance matrix The quantity s i 1 2 m is the standard deviation or error of the optimal value of the i th parameter If the measurement errors of different data points are independent and distributed normally then the statistical distribution of each parameter is also normal In such a case there is a 68 3 probability that the true value the statistical average of a given parameter belongs to the interval 14 1 15 a 95 4 probability that it belongs to the interval po 25 lt p lt po 2s and a 99 7 probability that the true value belongs to the interva
43. smoothing and after sm e The text box Smoothed Y gt is used to enter the minimum smoothed y value the threshold If ordinates of two points in a row are less than this threshold then smoothing proceeds as though those two values were equal to each other i e they are not smoothed If this text box is empty then all y values are smoothed e The static text field Number of changed points shows the number of points that were modified due to three point smoothing Notes 1 If the smoothed curve is an f x model function then this text field shows the total number of modified points corresponding to all model time values not just the current time 2 If the number of smoothing cycles is greater than 1 then this text field shows the maximum number of points modified during a single smoothing cycle Exponential smoothing is defined as follows Via A 1 a j 15 4a where y is the value of the th point after smoothing i e the th smoothed value y is the 1 st smoothed value y is the value of the th point before smoothing i e the th non smoothed value The exponential smoothing factor a is always between 0 and 1 When ais close to 1 then y y When ais close to 0 then the smoothed value is close to the arithmetic average of all previous values The exponential smoothing method is represented by the following four controls e The text box Exponential smoothing
44. the secant rule The full list of built in integration summation iteration and root finding functions is given below 22 Int f x a b integral of expression f with respect to x from a to b e g Int 1 sqrt exp t x x x 0 5 Int2 f x a b epsabs epsrel limit same as Int but with additional arguments epsabs absolute accuracy requested default value 0 epsrel relative accuracy requested default value 10 limit the maximum number of subintervals in the partition of the given integration interval default value 200 Inti f x a 1 integral of expression f with respect to x from a to 00 Inti f x a 1 integral of expression f with respect to x from o to a Inti f x a 2 integral of expression f with respect to x from o to o the argument a is not used Inti2 f x a i epsabs epsrel limit where i is 1 or 2 same as Inti but with additional arguments epsabs absolute accuracy requested default value 0 epsrel relative accuracy requested default value 10 limit the maximum number of subintervals in the partition of the given integration interval default value 200 Intw f x a b w 1 integral of expression f x cos wx with respect to x from a to b Intw f x a b w 2 integral of expression f x sin wx with respect to x from a to b Intw2 f x a b w i epsabs epsrel leniw maxp1 where i is 1 or 2 same as Intw
45. time Computation time 0 00 22 074 00 00 Double precision f x t functions 8 bytes for each value Primary functions All functions Set precision of f x t function values Start simulation b Fig 7 1 Examples of the dialog window Time limits and amount of data a when there are no stored model data b when there are stored model data The group of two checkboxes and a button that is at the bottom of the dialog window Time limits and amount of data is used to modify precision of f x t model functions The amount of computer memory needed to store values of f t model functions is usually relatively small and values of f t functions are therefore stored with the maximum precision 8 bytes or 64 bits This is a precision of 15 significant digits Since the memory amount needed to store values of f x model functions is usually much larger it is sometimes necessary to find a trade off between the memory requirements and precision In most cases the default precision of 4 bytes 32 bits is sufficient This corresponds to 7 significant digits If a higher precision is needed and the available amount of physical memory is large enough the precision may be increased In order to increase the precision of all f x f model functions the checkbox All functions should be checked This action doubles the memory amount needed to store each value of all f x model functions If the checkbox Pri
46. 0 4 eri 0 6 0 8 l Fig 4 2 An example of a graph corresponding to the formula of Fig 4 1b without the erroneous line Local global and model parameters exist until they are removed by the user Those parameters can be optimized during nonlinear fitting see Section 14 Nonlinear fitting Variables that are not defined as local or global parameters exist from the first assignment of a value to them until computing the final value of the program Those variables can be used for storing intermediate values e g the a variable in Fig 4 1a They can not be optimized during nonlinear fitting If a syntax error is found during compilation of a program then the corresponding line of code is highlighted and a message with a description of the error appears in the status bar of the program editor window and in the programming tools dialog e g see Fig 4 1b Only the first detected error is pointed out after correcting it other errors will be found too If there is no error message this means that the formula syntax is correct The error message is also absent when the formula code is empty or filled with comments An example of a graph corresponding to the formula of Fig 4 1b is shown in Fig 4 2 this graph has been obtained after removing the erroneous line x 8 from the formula code In this example the horizontal line Total current has been calculated according to the formula Y FT2 This is the value of the f
47. 2 Pe A eN 13 13 2 B gt e confidence intervals of the average and of the single point at confidence levels 70 Yo 95 Yo and 99 8 The half width of the confidence interval is computed by multiplying the standard deviation 13 12 or 13 13 by the Student coefficient tan corresponding to the given number of points n and to the given confidence level a where 70 Yo 95 or 99 8 The limits of the confidence interval are computed from the assumption that the center of that interval coincides with the average value Thus the confidence interval limits of the average value are equal to e standard deviation of the average Vs Y Eta ny gt 13 14a and the confidence interval limits of a single point are equal to Ve Y ty ny 13 14b The program can also compute similar statistical parameters of the set of argument values x1 X2 Xn 45 14 Nonlinear fitting Nonlinear least squares fitting is performed by selecting the menu command Data analysis Nonlinear fitting This action opens the nonlinear fitting dialog window An example of that window is shown in Fig 14 1 The various elements of that window are used to select fitting functions formulas or model functions fitted data sets free curves or model functions varied parameters global local or model parameters and fitting options The fitting process is started by clicking the button Start fitting
48. 4 1 7 Op By inserting the expression of F 14 1 3 into 14 1 7 we obtain the system of normal equations 0 Sw v P v 2 0 i 1 2 m 14 1 8 k l Op This is a system of m equations in m unknowns corresponding to m optimal parameter values This system of equations is the mathematical formulation of the principle of least squares If the dependence of the estimated responses y p on p is nonlinear then equations 14 1 8 are nonlinear too In such a case there is usually no analytical solution of this system of equations and it must be solved numerically using an iterative procedure I e the point po in the parameter space is approached step by step The size and direction of the next step depends on the properties of the minimized function F i e its value and its partial derivatives with respect to all varied parameters at the current step One 48 such step is called iteration Different fitting methods compute the next step differently The choice of the fitting method may influence the fitting time and the total number of iterations However the final result po does not depend on the chosen fitting method except when the function F p has more than one minimum 14 1 4 Parameter confidence intervals Let us assume that there is a single set of measured data and a very large number of optimal theoretical curves obtained by repeating the same measurements at the same conditions and fitting each set of meas
49. Allt values tMin tMax t axis range of this graph trange of this fft function Apply these t and x limits to All f x datasets corresponding to the same fitting function All f x datasets plotted in the same graph All f x datasets Error factors of the selected dataset i e earn O17 TF yl el AT 1 7 sqrtff yl e e 0 01 lt ly Divide SSE by the number of points 1 z2 z FC1 Fitting data set No 2 t 0 04 Select Apply these error factors to All datasets All datasets corresponding to the same fitting function Cancel Fig 14 7 An example of the dialog window with options of fitted datasets The grid control at the top of the dialog window is used to select a fitted dataset After selecting a dataset by left clicking the corresponding row of the grid control its options are automatically shown by the controls below The controls that are in the group Fitting range of the selected dataset define the interval of time values of the graph window indicated in the selected cell of the column Graph if the selected dataset corresponds to an f t model function or a free curve of the type ft or the interval of coordinate x values if the selected dataset corresponds to an f x model function or a free curve of the type f x If any one of the two checkboxes Entire curve and Fitting range coincides with t
50. An example of a dialog window with properties of a free form line 39 the arrow keys when the Ctrl key is pressed all line handles are hidden and the current vertex is indicated by a red cross When the Ctrl key is not pressed the left or right arrow key selects the previous or next vertex respectively i e makes it the current vertex Another way to make any vertex the current vertex is by entering the vertex number in the corresponding text box Point No of the point dialog window The line handle corresponding to the current vertex has a red border The absolute position of a free form line in the graph window can be changed either by mouse dragging it or by left clicking it and entering the coordinates of the current vertex in the mentioned point dialog window or by pressing the Ctrl key on the keyboard and using the arrow keys In this mode the current vertex is indicated by a red cross the current vertex can be changed in the same way as during modification of line shape Note The point coordinates of the curves that represent various functions can be modified in the same way as vertex coordinates of free form lines see above 1 e the same point dialog window is displayed and the same keyboard shortcuts can be used However values of functions and their arguments can not be changed by dragging the plotted points with the mouse In order to change the free form line format or its
51. GraphixT A program for visualizing solutions of one dimensional kinetic equations v1 21 User s Manual by Andrius Po kus Vilnius University Faculty of Physics 2015 02 23 Copyright 2015 by Andrius Po kus E mail andrius poskus live com Web http www graphixt com Contents Ve TNETOdUCUONS isos dace obec aa od dads i hd dace whe a e a a aa ed oda ced tate wis Tact a a a as 1 DVI SEF Ibert aCe asec seco t ean asks woe eaten seniashs sob A A aaseudentany sont Seneaageunaus oe cans saesuses oe E E 2 3 Model functions and free CULVES eine snor siais ii iai a A as i ik ia K ia a a ao i 4 4 Computational programming with GraphiX T aaa aaaaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa 5 4 1 Writing simple programs and displaying calculation results ec eceeceseeeceeseesececeeeeeseceeeeaeeeeceaeeaeees 5 4 2 Lists of global and local parameters cccccccecssesseesseceececeescecsaecseceseceseceseeseeeseeeseaeceseeeseeeseeeseseaness 14 4 3 USMg arrays I pT CTAMS i035 as feet ia ais Aa ias oon senna asa i ls Lo r ia ues i e nt eect eee 15 4 4 Using subroutines in programs ccccecccesscessceeeeeeseesseeesceeseeescecsaeceaecsecseceseeeeeeseeeseeeseaeeeseeeseeesaeeaaees 18 4 5 Using DLL functions in protams sssrinin taninan ea et aaa a aa anert aa aaea 21 4 6 Built in integration summation iteration and root finding fUNCtIONS cc cecceeseeteeteeeteeeteeennees 22 4 7 Runtime error handing eie
52. SE expression 14 1 1 Parameter values A 121 8 A 0 0365 0 0040 s B 47 9 7 4 A 0 00572 0 00073 s 2 minimizing the relative deviation SSE expression 14 1 2 Parameter values A 114 32 4 0 0228 0 0067 s B 27 9 9 5 45 0 00508 0 00080 s 2 p a 14 1 5 k l Ox where o is the standard deviation of the k th measured response The SSE with weight factors equal to w 1 0 i e the quantity 14 1 5 is usually denoted by the Greek letter y chi with the superscript 2 i e 7 and called chi squared The difference between the number of fitted data points and the number of varied parameters n m is called the number of degrees of freedom As it will become clear from the following discussion the quality of fit is more conveniently characterized using the ratio of the SSE and the number of degrees of freedom This ratio will be called the reduced SSE and denoted F F n m Here F is the general SSE 14 1 3 Thus F is defined without any restrictions on the weight factors any one of the mentioned expressions of F 14 1 1 14 1 5 may be inserted into 14 1 6 F 14 1 6 14 1 3 The mathematical formulation of the principle of least squares Since at the minimum point of a function its partial derivatives with respect to all arguments are zero the condition of a minimum SSE can be formulated as follows ie G 1 2 m 1
53. Unlike formal parameters that are variables or pointers to variables e g formal parameters No 1 and 2 in Fig 4 9 formal parameters that are pointers to arrays or functions e g formal parameters No 3 and 4 in Fig 4 9 are not shared among different instances of one subroutine 2 The above example shows how to track the call depth also called nesting level However such a method of tracking the call depth would only work if there are no runtime errors when executing the subroutine FUNC1 After a runtime error execution of the current instance of FUNC1 would be terminated immediately and the local parameter d would not be decremented by 1 for more information about runtime error handling see Section 4 7 20 4 5 Using DLL functions in programs Programs may contain calls to user defined DLL functions DLL stands for Dynamic Link Library The list of all defined DLL functions is opened using the menu command Programming User DLL functions or the programming tools dialog button DLL see Fig 4 1 An example of such a list is shown in Fig 4 10 A new DLL function can be defined by clicking the button New Definition of an existing DLL function can be modified by clicking the button Edit or by selecting the command Edit function definition from the context menu or by double clicking an item of the list box In any case the dialog window DLL function definition is op
54. When that checkbox is checked the value of y in formulas 13 1 through 13 7 is replaced with In y i e the natural logarithm of the absolute value of y and the fit line is an exponential function see the dashed curve in Fig 13 1 In such a case only positive or only negative values of y are used during fitting depending on the dominant sign of y in the fitting range 13 2 Integration The integral of a function can be computed using linear or exponential interpolation This term refers to the shape of an imaginary line that connects adjacent points of the function during its integration The method of linear interpolation also called the trapezoid quadrature method is illustrated in 43 Fig 13 2 and Fig 13 3 In this case the points are connected by a straight line y 4 B x Obviously if in the entire integration range the integrated function is positive and superlinear i e if its derivative or slope increases with increasing x as in Fig 13 2 or negative and sublinear 1 e its derivative decreases with increasing x as in Fig 13 3 then a more accurate estimate of the integral could be expected when the points are connected by an exponential curve y exp A B x The latter method is most accurate when the integrated function is similar to an exponential function which approaches zero either when x gt 00 as in Fig 13 2 and Fig 13 3 or when x gt o In other words exponential interpolati
55. a scientific research tool Here are some features of GraphiXT e it is possible to create any number of synchronized graph windows where the dependence of a particular quantity or several quantities on the coordinate x at a particular moment of time is plotted e the mentioned time current time can be easily changed using a special slider it is possible to use animation mode when the program itself changes the current time with a constant rate e when GraphiXT is used with simulation plug ins the calculation results are automatically plotted during the simulation curves can be generated from user defined equations or from user defined DLL functions a built in compiler for computational programming supporting multi dimensional arrays and subroutines a simple to use array viewer editor built in mathematical functions including special functions numerical integration functions and others elementary data analysis linear fitting integration statistical analysis nonlinear least squares fitting and solution of systems of nonlinear algebraic equations curve data can be exported to text files or imported from text files plotted function data can be edited either in table format or directly in graph windows TEST ESET GS EE File Show Window Graph options Data analysis Programming Tools Simulation options Start simulation Help D agam FKEFNRIe vy X TI KA amp 2 Charge carrier concentrations Ie 1 F
56. a table has three fixed rows The first row contains column numbers the second row contains dataset identifiers IDs and the third row contains dataset names The dataset identifiers which are shown in the second row are constructed as follows X dataset IDs t model time values XLn node coordinates of the model layer No n for example XL2 Xn independent X set No n for example X10 X temporary set of abscissas Y dataset IDs FTn values of the f t model function No n e g FT10 FTXn values of the time cross section f x const t of the f x t model function No n e g FTX10 FXn values of the f x t const model function No n e g FX10 Fn values of the formula No n e g F10 FCn values of the free curve No n e g FC10 The function data can be edited by entering values directly into non empty cells of the data table or by copying and pasting Besides in partial mode the current x and y values and the current point number can be entered into the corresponding edit controls of the point dialog window New columns are automatically added when new curves are created The easiest way to create new columns is by selecting the graph or table menu command Create free curves This action opens a dialog window where both the number of new free curves and the number of points in each of them can be specified The new curves can be linked to e
57. abscissa of the point that was current before The order of datasets in the data table is in general different from the order of the function names in the legend In the table all datasets that share the same set of x values are next to each other Consequently unlinking curves or re linking formulas causes reordering of corresponding columns in the table The X dataset columns are always on the left of the y dataset columns of the curves linked to that X dataset If the X set is a set of model time or coordinate values then the column with x values is followed by the columns with values of plotted model functions that are linked to the same X dataset followed by the columns containing y values of formulas linked to that X dataset In this case the columns containing y values of formulas linked to that X dataset are sorted in the order of creation of those formulas If the X dataset is independent then the columns containing y values of free curves and formulas linked to that X dataset are sorted in the order of their linking to that X set I e the order of columns with Y values of free curves and formulas in data tables is the same as the order of the corresponding rows in the bottom grid control of the X dataset dialog which is described in Section 5 see Fig 5 1 The X datasets are sorted as follows first the model time or x values then independent X datasets in the order of their creation and finally the temporary X sets Each dat
58. additional factor Vn will appear in the denominators of the expressions of error factors 14 2 1 14 2 3 The two checkboxes at the bottom of this dialog window are used to specify that the same method should be used to calculate error factors for all other fitted datasets or for all fitted datasets corresponding to the same fitting function Those checkboxes can not be checked if the fourth error weighting option has been selected 14 2 4 General nonlinear fitting options The dialog window of general nonlinear fitting options is opened by clicking the button General options see Fig 14 1 An example of that window is shown in Fig 14 8 Below are descriptions of all controls of that window e The top two text boxes are used to specify the increment of a parameter value that should be used when computing the partial derivatives of the SSE with respect to the varied parameters Those derivatives are computed by the finite difference method OF _ F p 0 5Ap F p 9 5Ap pl Ap l The same method is used when computing the partial derivatives Oy Op they appear in the expression of a 14 1 21 Thus in order to compute the partial derivative of the sum of squared errors F with respect to a varied parameter p when that parameter is equal to po a small increment Ap of that parameter s value must be specified and two auxiliary values of SSE must be computed One of those two values of SSE corresponds to the param
59. alculated then this result is raised to the power 2 and multiplied by 3b then the value of 2a is added Thus using parentheses the same expression could be written as follows 2 a 3 b sin x 2 Comment lines may be inserted into programs Each comment line begins with see Fig 4 1b For a convenient access to programming objects as well as for defining the set of X values of an edited formula a dialog window Program editor tools is displayed when a program is being edited see Fig 4 1 It can be hidden or displayed whenever a program editor window is active In order to display the programming tools dialog the user has to select the menu command Programming Show programming tools dialog or the corresponding command of the context menu of the program editor Below are explanations of all controls of the programming tools dialog e The five buttons and the drop down list of built in functions which are at the top of the programming tools dialog window see Fig 4 1 are used to insert a function call into the formula Each of the mentioned six controls corresponds to a particular function type a f t model function a f x f model function a free curve of the current graph a user defined subroutine a user defined DLL function or a built in function By clicking any one of the mentioned five buttons a dialog window with a function list is opened for more information about the lists of subrouti
60. alue of index No i sea 23 Invert loc a inversion of a square two dimensional array matrix This function replaces the original matrix with the inverse matrix and returns the determinant of i original matrix Sa 29 Find x loc a 10 the sequence number of the largest element of a one dimensional array a whose value is less than or equal to the value of x It is assumed that elements of the array form a Ras sequence 10 is the number of the starting element 1 e the location in the array where the search starts if 10 lt 1 or if 10 exceeds the number of elements in the array then the search starts from the middle element of the array The search algorithm depends on the size of array a if the number of array elements is less than 40 then array elements are checked sequentially starting from the element No i0 otherwise the method of repeated bisection of the index range is applied If x is less than the first element of the array then the function Find returns zero Find2 x loc a i0 n m the extended version of the function Find The meaning of the first three arguments was explained in the previous paragraph n is the number of the initial elements of array a that are to be included in the search 1 e the effect is the same as though the array a was S by a smaller array which consists of the initial n elements of array a If n lt 1 or if n exceeds the a o
61. alues of array elements can be entered into the cells of the array view by typing or by copying and pasting In the case of a multidimensional array any two indices can be chosen to be variable those two indices are used to number the rows and columns of the array view The position range of values of each index the maximum number of cells that can be shown in the array view and other options of the array viewer are set in the dialog window Array view options which can be opened by selecting the menu bar command Programming Array view options when an array view window is active or by selecting the same command from the array viewer context menu For example the array view of Fig 4 6 could be produced by setting the array view options as shown in Fig 4 7 The variable indices are defined using the two rows which is in the graph No which is in the graph No c 39 g 33 5 35 Nn 29 17 Variable indices Start value End value Number of values Index No 1 1 0 Z Use maximum Index No 2 5 Use maximum All indices Transpose Maximum number of cells 1000000 v Automatically recalculate formulas on edit Column width 78 Row height 19 Apply Cancel Fig 4 7 An example of the array view options dialog corresponding to the array view shown in Fig 4 6 of input fields at the top of the array view options dialog T
62. ample the expression Root sqrt x 1 x 3 x 0 1 is equal to the root of the equation Vx 1 x 0 1 e 0 60542342357183 For more information about built in functions Int Inti Intw Sum Iter and Root see Section 4 6 Notes 1 If the value of the array element whose number is returned by the function Find x loc a i0 or Find2 x loc a i0 n m occurs more than once in the array a then the returned result may correspond to any of the occurrences depending on the values of the arguments 10 n and 66 29 m 6699 2 If elements of the array a do not form a non decreasing sequence then functions Find x loc a 10 and Find2 x loc a i0 n m usually return an incorrect result Some of the mentioned mathematical functions are special functions which are defined as integrals Below are the definitions of those functions Bessel function of the first kind order n J 2 cos n0 xsin 0 d0 n 0 1 2 4 1 TY Bessel function of the second kind order n LOS gt sin xsin 0 n0 d0 z fie Je de n 0 1 2 4 2 MG TY Error function Dy ss erf x e dt 4 3 R Complementary error function Va ae erfc x 1 erf x 1 e dt 4 4 a Gamma function r x tear 4 5 0 Incomplete gamma function _ ly a 1_ t VOD e de 4 6 0 The conditional construct if else In the si
63. an only be one legend in each graph window Each of those objects can be added to a graph window at the mouse cursor position using the graph context menu Objects of the first three mentioned types can also be inserted using the toolbar buttons T K the toolbar is at the top of the main window and the legend can be inserted by selecting the menu command Graph options Add legend in the latter case the legend will be placed at the top right corner of the graph window A more detailed description of the mentioned objects is given below 11 1 Text labels After creating a text label by one of the mentioned methods a text cursor a flashing vertical line appears at the mouse cursor position Then the text can be entered The text may consist of several lines in order to start a new line the Enter key must be pressed on the keyboard To end text editing the left mouse button must be clicked outside of the text label A text label can not be empty or consist of spaces only otherwise it will be automatically deleted The text of an existing text label can be modified by double clicking the text label or by selecting the command Edit text from the text label s context menu The position of a text label in the graph window can be changed by dragging the text label with the mouse The text format i e font properties color direction etc can only be changed after exiting the text edit mode The text format is modi
64. anges i e when a set of model function values is added to the model data during simulation or when a part of the model data is deleted or when model data are imported from text files then the current time of all graphs whose Auto mode is turned on becomes equal to the time of the last point of At model curves Y E Auo E k gt 0 0000e 000 0500000000 7 tsyne tsyncall C x syne C x sync all gt Spe S Langusge a J 0108523561 E V Auto Elk 0 0000e 000 0 500000000 Z tsync Z tsyne al Liesmo Clesmeal Pm E ira Language b g E Fauto 715 0109523561 01 Atsync Fltsyncal Elese E syne al 2 sr Language c Fig 10 1 The slider bar a when the active window is a time graph window and the value of the current time is not shown b when the active window is a time graph window and the value of the current time is shown c when the active window is a coordinate graph window 35 10 11 12 13 14 36 The checkbox lt gt is used to turn on or turn off step like change of the time axis limits of the current time graph window in response to a change of the time slider position If the entire interval of the graph time axis belongs to the slider interval then a change of the slider position causes a change of the time axis limits In other words the time interval of the time graph shifts along with the current time with the condition that the current time is c
65. be clicked Then the names of the selected function and the corresponding graph window appear in the columns Function and Graph of a grid control in the window Nonlinear least squares fitting see Fig 14 1 In order to define additional fitting functions the mentioned steps must be repeated 14 2 2 Selection of varied parameters and parameter properties The names values and uncertainties of the varied optimized parameters are shown in the grid control Varied parameters see Fig 14 1 This grid control has five columns 1 Name the names of optimized parameters 2 G L M the letter G L or M which indicates the type of the corresponding optimized parameter global local or model parameter these terms are explained in Section 4 1 in the case of a local parameter the graph number and formula ID are indicated beside it see Fig 14 1 3 Value current values of optimized parameters 4 Vary a checkbox which is only checked when the corresponding parameter is varied 5 Uncertainty the half widths of 68 3 confidence intervals of optimized parameters Optimized parameters are added by clicking the button New which is under the grid control Varied parameters This action opens the parameter selection dialog window An example of this window is presented in Fig 14 5 This dialog window contains three list boxes The list Local parameters
66. codeproject com Articles 8 MFC Grid control 2 27 The code for the message box with custom button captions used in the Lithuanian version of GraphiXT was taken from a post by Miguel Schindler on the codeguru com website http Avwww codeguru com cpp w p win32 messagebox article php c 10873 The code for the toolbar was taken in part from an article by Peter Lee on the codeguru com website http www codeguru com cpp controls toolbar article php c2537 FullFeatured 24bit Color Toolbar htm The code for the hyperlink in the About dialog window was taken from an article by Chris Maunder on the codeguru com website http Awww codeguru com Cpp controls controls hyperlinkcontrols article php c2133 The installer is based on the free script driven installation system Inno Setup created by Jordan Russell http www jrsoftware org 67
67. cross section may be modified as required By clicking Delete the selected time cross section will be deleted If at a given value of the x const value does not coincide with any stored value of the x argument of the selected function corresponding to that time then the value of the time cross section is computed by the method of linear interpolation using two nearest stored x values one of them is greater than const and the other one is less than const The only limitation on the x const value is that it must belong to the definition interval of the selected function Each time cross section belongs to a particular graph window When a graph window is closed the time cross sections plotted in it are lost Note Curves displaying the f x const t functions can also be created using formulas see Section 4 r Model f x constt functions Ls 01 Change New Insert before Duplicate No Name L l Curve format and name Delete Delete All Cancel Fig 6 1 An example of the time cross section definition dialog window 30 7 Time limits and amount of data Prior to starting simulation using the GraphiXT plug in that solves kinetic equations such as CarrierFunc dll the user has to specify the initial and final times of simulation This is done by selecting the menu command Simulation options Time limits and amount of data This actio
68. curve and all smoothing options will be saved If the selected curve is an f x model function then the smoothing subroutine will smooth its values corresponding to all stored model time values not just the values corresponding to the current time of the graph If the checkbox Close this dialog window when finished is checked then this dialog window will be closed after smoothing After clicking the button OK the curve is not smoothed the smoothing dialog window is closed and all smoothing options are saved After clicking the button Cancel the curve is not smoothed the smoothing dialog window is closed and all unsaved changes of the smoothing options are discarded The group of two checkboxes Smoothing direction are used to specify whether the curve should be smoothed left to right i e in the direction of increasing argument or in the opposite direction or in both directions In the latter case the ordinate of each point of the curve is replaced by the average of the two values obtained using both smoothing directions f Smoothing Graph 2 Graph2 7 Curve Al Smoothing range Smoothing range coincides with the t axis range V Entire curve tMin 0 2 First point No 1 20 tMax 10 Last point No 20 20 Apply the same smoothing range to other functions plotted in this graph Method Three point smoothing 7 Close this dialog window when finished
69. dataset is a free curve of the type Jx then the default value of its time is equal to the current time of the corresponding graph The limits of the fitting interval and the time values associated with f x free curves can be changed by clicking the button Dataset options This action opens the fitted dataset options dialog window An example of that window is shown in Fig 14 7 Below are explanations of all controls of that window 55 Model fft functions No and name Electric charge e cm 2 Total current density A cm 2 Total current A External voltage source current A External capacitor current A External photon flux density 1 cm 2 5 Simulation time step s Processing time of one time step of the simulation s Free curves No and name Fitting data set No 1 Fig 14 6 An example of the fitted dataset selection dialog window Fitting function Fitted dataset Free surface potential Potential of the left edge of the system V Fitting data set No 1 Free surface potential Potential of the left edge of the system V Fitting data set No 1 Electric field Electric field V cm Fitting data set No 2 t 0 04 Fitting range of the selected dataset No 3 Time of ffx free curve Entire curve Fitting range coincides with the x axis range of the graph Current time of the graph Min 0 sMar 15 t 0 04 Range of t values of f x t model function
70. datasets also see Section 8 Additional times The same temporary set of additional times is also used for computing the final model curves when the user interrupts fitting or when fitting is stopped automatically After computing the final curves the user defined set of additional times is restored if it exists If model parameters are not varied during fitting but some of fitting functions are formulas containing references to model functions then model function values are not recalculated during fitting Consequently the stored argument values of those model functions i e model values of time or coordinate x are not recalculated too If they do not coincide with abscissas of the points of fitted datasets then the model function values corresponding to those abscissas are calculated using linear interpolation In order to repeat the entire fitting process from the start the initial parameter values must be restored This is done by clicking the button gt which is under the text box Lower bound see Fig 14 1 By clicking the button lt current values of all parameters including those that are not visible in the dialog window Nonlinear least squares fitting are copied to initial values After stopping the fitting process the current fitting state including all internal parameters of the fitting subroutine is saved Consequently after clicking the button Start fitting again fitting will co
71. dditional times could have the meaning of abscissas of the measured time dependences of electric current or potential Comparison of simulation results and experimental data is easiest when simulation time values coincide with experimental time values This can be achieved using the menu command Simulation options Additional times By selecting this command the dialog window Additional times is opened An example of that window is shown in Fig 8 1 The additional times can be defined in three ways 1 By selecting the X datasets whose values should be used as additional times This is done by first selecting a time graph in the list box Graph and then using the list box Curve abscissa sets of the selected graph to select one or more X datasets By defining one or more sets of equidistant time values Each such set consists of two or more time values with all intervals between any two adjacent values equal to each other This set of values is completely defined by three numbers a the initial time b the final time c the number of time values Those three numbers are shown in the list box Sets of equidistant time values for each defined set In order to define a new set of equidistant values or to change an existing set or to delete an existing set the user has to click one of the three buttons Add Change or Delete that are under the mentioned list box 3 By specifying one or more indiv
72. diately follows the call to that subroutine If the current program is a formula i e if there is no higher level calling program then calculation of the next value in the current sequence of values begins except after manual termination as explained in the previous paragraph If such an error occurs during nonlinear fitting then the fitting is interrupted with a message about an arithmetic error Note Integration errors which are indicated by a non zero value of the error code IER assigned by built in integration functions see Section 4 6 do not cause interruption of the program execution However runtime errors that may prevent a call to a built in integration function from being initiated are possible e g insufficient memory for work arrays of the function After such an error execution of the program is terminated as described above and system variables IER AbsErr and nEval are not modified In the program editor window the first runtime error that was detected in the last computed sequence of values is shown by automatically scrolling to the corresponding line of code and highlighting it If the runtime error occurs in a subroutine then the line containing the corresponding call to that subroutine is highlighted in the code of the calling program too see Fig 4 13 26 5 X datasets It is possible for a group of free curves and formulas to share the values of their abscissas Such curves are called
73. e The first variable is the independent variable 1 e the function argument which could have the meaning of time or coordinate and all subsequent variables are the dependent variables functions of the independent variable All non empty lines that are below the header line must contain values of the mentioned variables listed in the same order as their names The argument values must be sorted in ascending order argument values that are exactly equal to each other are allowed too Values of all f model functions must be in a single file Each of the files with the f x t const model function data must correspond to a particular time value That value must be specified in the file name The file name may be a number that is equal to the time value for example 1 txt or 2E 7 txt or it may be any sequence of characters ending with the equality symbol and a number for example x t 1 txt or LayerNol_t 2E 7 txt Those files must be in one folder which does not contain any other files except for the mentioned file with f t model function data If the simulated system consists of two or more layers then the files with f x t function data corresponding to different layers of the system must be in different subfolders of that folder The names of those subfolders must be such that after sorting them in alphabetic order the sequence number of each subfolder is the same as the sequence number of the corresponding layer
74. e analysis dialog window specifying the analysis x range and other analysis options and clicking the button Calculate after clicking OK analysis is not performed but its options are saved and the dialog window is closed The same dialog window can be opened by selecting the command Curve data analysis from the curve s context menu After doing the calculations the text editor window Info with the analysis results is automatically opened for example see Fig 13 1 In addition after linear fitting a free curve representing the linear or exponential fit function is automatically created for example see the dashed curve in Fig 13 1 The Info window can be opened at any time by selecting the menu command Window Open information window keyboard shortcut Ctrl T Contents of the Info window are saved in the project file gxt 13 1 Linear fitting During linear fitting the program computes the least squares estimates of the coefficients A and B of the linear equation y A B x 13 1 and plots the resulting straight line The essence of the method of least squares is the following Let us assume that a data set consists of the argument values x1 X2 Xn 19 Xn and corresponding values of the function y x A typical example is a set of measurement data In such a case n is the number of measurements The measured function values will be denoted y yo Yn The theor
75. e assignment operator Besides the variable x may only be assigned values in time graphs see Fig 4 1 Identifiers of variables arrays and functions are case insensitive Identifiers can not be longer than 50 characters otherwise the trailing part of the identifier starting with character No 51 will be ignored The operator priority when it is not indicated by parentheses is determined according to the usual rules The assignment operator has the lowest priority i e the assignment operation is done last The priority of logical AND is higher than the priority of logical OR for example the expression a amp b c amp d is equivalent to a amp b c amp d The comparison operation priority is higher than the logical operation priority but lower than the arithmetic operation priority excluding the assignment operation Multiplication and division priority is higher than addition and subtraction priority and the priority of raising to a power is higher than multiplication and division priority Addition has the same priority as subtraction This means that a sequence of addition and subtraction operations will be evaluated left to right Similarly multiplication has the same priority as division The priority of function calls is higher than the arithmetic operation priority For example the expression 2 a 3 b sin x 2 would be evaluated in this order first the built in function sin x is c
76. e curve or formula can be accessed The built in arrays are the following e gt g y XC g c p X value of the data point No p of free curve No FC g c p Y value of the data point No a of free curve No NFC g the number of free curves in the cen No g XF g c p X value of the data point No p of n No c which is in the graph No FA g c p Y value of the data point No ap of formula No c which is in the graph No NF g the number of as an the a No g TA t model time value No XL n t x coordinate of node N x of layer No at the moment of time No t FT f t value of the model At function No f corresponding to moment of time No t FX f t x value of the model f x function No f at node No x and at moment of time No t LFT f the sequence number of the layer corresponding to the baal JA function No f IFT f the sequence number of the model f t function No f inside the corresponding layer LFX f the sequence number of the layer corresponding to the model f x function No P IFX f the sequence number of the model f x function No f inside the corresponding layer I2FT n f the final overall sequence number of the At function No f of the layer No n I2FX n f the final overall sequence number of the f x 4 function No f of the layer No
77. e formula s G i 1 2 m 14 1 17 The confidence intervals may only be calculated according to 14 1 16 or 14 1 17 when 1 the theoretical model is correct 2 errors of the measured responses are independent and distributed normally Then the expression 14 1 16 is applicable for the case of any weight factors wz with the condition that the observed scatter of the measured responses around the theoretical ones correctly reflects the statistical properties of the measured responses and the expression 14 1 17 is applicable when w 1 0 where o is the standard deviation of the k th measured response this corresponds to the SSE expression 14 1 5 The mentioned two conditions will be called the standard conditions If the SSE is defined according to 14 1 5 then a third standard condition must be added 3 the values of o which are used to calculate wz are correct However it is often not known beforehand if the standard conditions are true For example the theoretical model may reflect the true relation between the variables x and y only approximately or it may be totally wrong Then the obtained optimal parameter values and their confidence intervals are probably worthless Consequently it is desirable to be able to test the hypothesis that the theoretical model is correct This is the so called null hypothesis The standard procedure of testing the null hypothesis is based on estimation of the probabilit
78. e function DLL1 of Fig 4 11 is used as an argument of a subroutine then the corresponding formal parameter must be defined as follows loc f or equivalently loc f x y z 4 6 Built in integration summation iteration and root finding functions GraphiXT has eleven built in integration summation iteration and root finding functions Seven of them were listed in Section 4 1 The remaining four functions Int2 Inti2 Intw2 and Root2 are extended versions of the previously mentioned functions Int Inti Intw and Root with additional arguments allowing more control over the computation process the simpler versions of those functions use default values of the additional arguments The integration functions were taken from the QUADPACK library for numerical integration of one dimensional functions that library is in public domain web page http www netlib org quadpack and the nonlinear equation solver is based on one of subroutines of the HOMPACK suite for solving nonlinear systems of equations which is also available from the Netlib repository web page http www netlib org hompack Those subroutines were translated from Fortran into C using the Fortran to C converter f2c exe also in public domain web page http www netlib org f2c mswin Below are short descriptions of those functions The function Int is a translated double precision v
79. ecalculate all X values of this dataset is checked then the X values will be recalculated immediately after clicking the button OK Otherwise clicking the button OK will only cause the new parameter values to be copied into the X set grid control If the checkbox Abscissas of skipped points must be the same for all linked curves is checked then the visible points of all free curves and formulas linked to the selected X dataset will always share the same set of abscissas Explanation In general some points may be skipped during plotting so that the X dataset properties Ea x dataset ID 4 x dataset name X Minimum value 0 Maximum value 5 Number of values 101 5 Recalculate all Xx values of this dataset V Abscissas of skipped points must be the same for all linked curves Largest number of points shown 501 Fig 5 2 An example of the dialog window X dataset properties total number of displayed points does not exceed the specified maximum number for each plotted curve that number can be entered in the curve format dialog window which is opened from the curve context menu In such a case when this checkbox is unchecked the visible points of linked curves may have different abscissas even when the mentioned maximum number is the same for all those curves When there are no skipped points the state of this checkbox has not effect on the plotted data If the checkbox Abscissas of sk
80. eckboxes Connect points and Show points as symbols in the curve format dialog window or by clicking the button Hide in the mentioned dialog window with the list of free curves Hidden curves are not plotted in the graph window their names are not shown in the legend and their data are not taken into account when rescaling the axes However it is still possible to analyze the data of hidden curves e g to use them for nonlinear fitting smoothing etc or to use them in formulas If a curve is hidden then its options can only be accessed by opening the corresponding list of curves menu command Graph options Free curves or Graph options Select The main differences between model functions and free curves are the following e Model function data do not belong to any graph window Consequently if a curve representing a model function is deleted or if the graph window containing that curve is closed the model function data is not lost and can be plotted again at any time In contrast each free curve belongs to a particular graph window Ifa free curve is deleted or if its graph window is closed the free curve data are lost e The shape of At model curves is affected by the number of model time values in memory and the shape of f x t const model curves depends on the current time of coordinate graphs whereas the shape of free curves is not affected by the mentioned factors e The argument f valu
81. ed responses hence the errors are functions of the parameters y y p The goal of fitting is finding the optimal parameter vector po which minimizes the difference between the measured responses and estimated ones The corresponding errors y po vz are called residual errors Prior to discussing methods of nonlinear fitting it is necessary to define the concept of the difference between measured responses and estimated responses precisely This can be done using a geometric analogy If the set of the measured responses is treated as a point v in an n dimensional Euclidean space whose Cartesian coordinates are v A 1 2 n and the set of the estimated responses 46 is treated as a point y in the same space with coordinates y k 1 2 n then the distance between those two points is VF where F gt y p y 14 1 1 k l The quantity F is called the sum of squared errors SSE and used as a criterion of the quality of the fit In order to improve the fit one has to decrease the distance between the two points Thus the less is F the better is the fit The optimum fit corresponds to the minimum value Fo F po This definition of the optimum fit is the basis of the method of least squares 14 1 2 Choice of weight factors The quality of the fit can also be characterized using the sum of squared relative errors I e SSE could be defined as ray aim 14 1 2 k l Vk The mos
82. ened see Fig 4 11 The DLL function definition consists of the following four components e the DLL file name e the function name in the DLL file i e the original name of the function e the identifier used when calling that function in programs i e the alias of that function e the number of function arguments 0 to 20 It is possible to choose one of two calling conventions passing copies of the argument values to the DLL function or passing the argument addresses to the DLL function In order to pass argument addresses the number or arguments must be entered with the minus sign All mentioned components of the DLL function definition are shown in the list of DLL functions see Fig 4 10 The number of arguments is shown in parentheses after the original name of the function see Fig 4 10 In the case of a failure to load a DLL function three question marks would appear after its name in the list box Note In programs calls to unloadable DLL functions are treated as syntax errors The first argument of a DLL function is a 32 bit address of a 32 bit integer variable that argument is the only one that is omitted when inserting the call to that function in a program All other arguments must be 64 bit floating point numbers type double of the C programming language or their 32 bit addresses C type double The return value of the function must be a 64 bit floating point number r
83. er than 1 and less than 1 and its normalized bounds are equal to 1 and 1 14 2 3 Definition of fitted data sets and their properties The names and limits of the fitted data sets are shown in the dialog window Nonlinear least squares fitting in the grid control Fitted datasets corresponding to the selected fitting function see Fig 14 1 In order to add a new fitted dataset the fitting function that will be used when fitting that dataset must first be selected in the grid control Used formulas and model functions by left clicking the corresponding row of that grid control and then the button Add dataset must be clicked This action opens the fitted dataset selection dialog window An example of that window is shown in Fig 14 6 That window contains two list boxes The top list box contains the names of model functions and the bottom list box contains the names of free curves The latter list only contains the names of the free curves that belong to the same graph window as the selected fitting function The list of model functions contains the names of all f t model functions or all f x 1 model functions depending on the graph type the model functions that are used as fitted datasets are not required to be plotted in graph windows unlike the model functions that are used as fitting functions After selecting the necessary function in the top or bottom list box i e after left clicking the corresponding item
84. ere are two list boxes for each of the two coordinates of the text label see Fig 11 1 The top list box is used when the horizontal or vertical position of the text label must not depend on the coordinate axis limits of the graph However if for example the text label is used to present information about a particular curve or a particular point of a curve then it may be desirable to make the position of the text label dependent on the position of that curve or that point That position expressed as the number of pixels from the left and top edges of the graph window may depend on the graph axis limits The curve or the point in question the reference point can even become invisible if the axis limits are changed In such a case the bottom list box should be used When an item is selected in that list box the text field X or Y displays the x or y coordinate of the reference point That coordinate is expressed in the coordinate axis units i e in the same units in which the axis tick labels are expressed see Fig 11 1b When this option is selected the visible distance from the reference point to the text label expressed in pixels will be constant but the absolute position of the text label in the graph window measured in pixels from the edge of the window will be variable depending on the position of the reference point If the reference point is not visible i e if its x or y coordinate does not belong to the corre
85. ersion of the Fortran subroutine QAGS the double precision version is named DQAGS According to the QUADPACK readme file QAGS is an integrator based on globally adaptive interval subdivision in connection with extrapolation de Doncker 1978 by the Epsilon algorithm Wynn 1956 The function Inti is a translated double precision version of the Fortran subroutine QAGI the double precision version is named DQAGI According to the QUADPACK readme file QAGI handles integration over infinite intervals The infinite range is mapped onto a finite interval and then the same strategy as in QAGS is applied The function Intw is a translated double precision version of the Fortran subroutine QAWO the double precision version is named DQAWO According to the QUADPACK readme file QAWO is a routine for the integration of COS OMEGA X F X or SIN OMEGA X F X over a finite interval A B OMEGA is specified by the user The rule evaluation component is based on the modified Clenshaw Curtis technique An adaptive subdivision scheme is used connected with an extrapolation procedure which is a modification of that in QAGS and provides the possibility to deal even with singularities in F The function Root is a translated Fortran subroutine ROOT According to the subroutine s description which is included in the comments section of the original source code the method used for the solution is a combination of bisection and
86. es of all f t model functions are equal to each other Similarly argument x values of all f x t const model functions corresponding to the same layer and the same time are equal to each other This means that if an argument value of a particular model function is changed the corresponding argument value of all other model functions changes as well Conversely free curve argument values do not depend on argument values of any other functions or free curves e A free curve can be added to a graph window of any type either a time graph or a coordinate graph For example a free curve obtained by copying a curve representing an f x t const model function plotted in a coordinate graph can be pasted into a time graph or vice versa Conversely model functions can only be plotted in a graph of the appropriate type 4 Computational programming with GraphiXT In previous sections two types of curves that GraphiXT can display were described model functions f t and f x t const and free curves The third type is the curves computed according to user defined formulas Here the term formula means a set of computer instructions code written in the GraphiXT programming language which is described in Section 4 1 and defining a curve i e a one argument function Each formula belongs to a particular graph window A new formula can be created using the menu command Graph options Create a formula or the same
87. essed The visible position of the line in the graph window measured in pixels from the window edge depends on the window dimensions and on the 38 Line position and format X 24697159565581 Color Red 7 Width 1 Style Dashed Fig 11 2 An example of a dialog window with properties of a vertical line axis limits If the coordinate of the straight line does not belong to the interval of the corresponding axis then that line is not shown 11 3 Free form lines A free form line is a polygonal chain also called a piecewise linear curve i e a connected series of line segments After creating a new free form line by one of the mentioned methods the first point of that line is placed at the mouse cursor position Then a required number of vertices may be added A vertex is added by clicking the left mouse button In order to end the line the right mouse button must be clicked The number of vertices can only be changed during creation of a line Afterwards it is possible to change vertex positions but not their number The lengths of the line segments and angles between them can be changed using the command Change line shape of the line s context menu When that command is selected a small square the line handle appears on each vertex Then positions of all vertices can be changed by dragging the corresponding line handles with the mouse After pressing the left mouse button on any
88. eter value that exceeds po by 0 5Ap and the other auxiliary SSE corresponds to the parameter value that is less than po by 0 5Ap The increment Ap is computed based on the numbers entered in the top two text boxes of the nonlinear fitting options window see Fig 14 8 The first text box is used to enter the relative parameter change The absolute change Ap is obtained by multiplying the relative change by the current parameter value If the absolute change of the normalized parameter value obtained in this way is less than the number entered in the second text box then Ap is set equal to the latter value Therefore if Ap must be constant during entire fitting a very small number e g 1e 100 must be entered in the first text box a zero value is not allowed Note the numbers entered in those two text boxes are also used when constructing the initial simplex 14 2 4 P Po e A group of five text boxes Ending criteria define the conditions that have to be satisfied to stop the 58 fitting procedure Fitting is stopped when the optimized parameters stop varying or when the SSE stops decreasing The minimum change of SSE between iterations must be specified for each of the General nonlinear fitting options x Relative parameter change for finite difference estimation of partial derivatives 0 001 Minimum absolute change of normalized parameters for estimation of partial derivatives 1e 006 Ending criteria Mini
89. eters has been modified by the user otherwise the last column is empty The parameters Min Max and N as well as the X dataset name can be modified by entering their values directly into corresponding cells The ID of an X set is constructed as follows t model time values XLn node coordinates of the layer No n for example XL2 Xn independent X set No n for example X10 e After selecting a cell or a group of cells in the X set grid the bottom grid control is filled with IDs and names of all free curves and formulas linked to the selected X dataset If two or more rows have been selected in the X set list then the curve list contains the IDs and names of curves linked to the X set whose name is in the X set list row containing the focus rectangle and if there is no focus rectangle then the listed curves correspond to the first selected X set Note Formulas can be linked to any of the three mentioned types of X sets whereas free curves can only be linked to independent X sets fe A X datasets m Graph No Graph name Properties of the selected X set 1 Graphi f Recalculate the selected X sets 2 Apply changes Delete the selected X sets x datasets of the selected graph No 2 Number of linked free curves 0 1 F2 Number of linked formulas ee 2 F3 F3 Unlink the selected curves Delete the selected curves
90. etical value of y at a given argument value x is a function of the unknown coefficients A and B see 13 1 hence we can write V x y x A B k 1 2 n The problem of estimating the coefficients A and B is formulated as follows The most likely values of A and B are the values that minimize the expression EE GraphiXT Electrophotographic layer discharge gxt 2 Charge carrier concentrations EEE Data analysis EE ptions Start simulation Help Graph 2 Charge carrier concentrations z I 2 Charge carrier concentrations Curve Electron concentration 1 cm 3 Analysis range 3x1014 Entire curve V Analysis range coincides with the x axis range 2 5x 1014 xMin 4 850252 First point No gg 7301 2x 1014 xMax 12 881 Last point No 258 301 15x 1014 E Apply the same analysis range to other functions plotted in this graph Type of analysis Linear fitting V Close this dialog window when finished 4 Fit the natural logarithm Extend the fit line to X axis limits Calculate Cancel EJ Potential of the left edge of the system V 1x104 5x108 l t 1 Free surface potential gt B Graph Charge carrier concentrations Curve Electron concentration 1 cm 3 t 0 13 g Range from X 99 4 9 to X 258 12 85 Number of points 160 Type
91. f elements of array a then all elements of array a are included in the search The argument m determines the a deona if m 1 then a ak sequential search is performed and if m 2 then the method of repeated bisection of the index range is applied Sea bb Hist loc a loc b loc c calculates frequencies of values of elements of array in intervals bins defined by array b the frequency in this context means the number of times a value belonged to a given bin The calculated frequencies are assigned to elements of array c The values of array b must be sorted in ascending order equal values are not allowed A aie is placed into a bin if it is less than or equal to the upper edge of that bin and greater than its lower edge All values that are less than or equal to the first element of array b are placed into the first bin and the values that are greater than the last element of array b are not counted The function Hist returns the number of elements of array a whose values do not exceed the value of the last element of array b Hist2 loc a loc b lce o n i the extended version of the function Hist The meaning of the first three arguments was explained in the previous paragraph The argument n is the maximum number of 66599 1 initial elements of array a that must be processed and the argument indicates if elements of array c shou
92. f the fitted curve that belongs to the current fitted dataset If that curve is not an f x t model function then iEnd iStart nPoints 1 If that curve is an f x t const model function then iStart and iEnd are equal to sequence numbers of the first and last points of the subset of the fitted curve corresponding to the current time and belonging to the current fitted dataset if that dataset spans two or more model time values then iEnd lt iStart nPoints 1 because nPoints is calculated including the points that correspond to all model time values that are used for fitting If the program is executed not during nonlinear fitting then iStart 1 iEnd nPoints iXpoint when fitting an f x t const model function 1Xpoint is equal to the sequence number of the current fitted data point in the subset of the fitted dataset corresponding to the current time counting only the points that are used for fitting If the program is executed not during nonlinear fitting or if the fitted curve is not an f x t const model function then 1Xpoint 0 11 iBase the sequence number of the first data point of the current fitted dataset in the union of all datasets corresponding to the current fitting function excluding unused datasets and the points that are not used for fitting If the program is executed not during nonlinear fitting then iBase 0 Those identifiers excluding x may not be used on the left hand side of th
93. f there is only one curve linked to that X set then that curve can not be unlinked The button Delete the selected curves causes deletion of the selected curves from the graph window Note If the selected X set is independent i e if its ID is of the type Xn then it will be automatically deleted after deleting the last curve linked to it The button Apply changes becomes active when at least one number in the columns Min Max or N has been modified i e when there is the asterisk in at least one cell of the column Mod After clicking that button the current values of each modified X set will be replaced with equidistant values calculated on the basis of the numbers in columns Min Max and N If one or more X sets have been selected in the middle grid control three other buttons in the top part of this dialog window become active The button Recalculate the selected X sets is used to recalculate all selected X sets regardless of whether their parameters have been modified or not Note The X values can only be recalculated when the corresponding dataset is independent i e with ID of the type Xn and when no free curves are linked to it i e when only formulas are linked to it If the X dataset is a set of model time or coordinate values then clicking the button Recalculate the selected X sets only causes the three mentioned numbers to be updated if simulat
94. factor is used to enter the parameter a of the formula 15 4a e The group of two radio buttons and a text box Smoothing margin is used to define the number of initial points that are not smoothed Those points are used to compute the initial smoothed value i e the initial value of y which is used on the right hand side of the equality 15 4a when smoothing the first point The mentioned two radio buttons are used to select one of two possible methods of defining the smoothing margin explicit or automatic In the case of explicit method the value of the smoothing margin must be entered in the text box that is to the right of the radio button In the case of automatic method the size of the smoothing margin will be computed automatically using the formula N 2 0 1 this value is rounded to the nearest integer Note If two smoothing directions are used then the points that belong to the smoothing margin when traversed in one direction may not belong to the smoothing margin when traversed in the opposite direction In such a case the ordinates of those points are replaced by corresponding smoothed values e Ifthe checkbox Use current value is checked then the previous non smoothed value y on the right hand side of the equality 15 4a is replaced by the current non smoothed value y i e the smoothed values are computed according to this formula Yra AV 1 Y 15 4b Although exponential smoothing elimi
95. fied by selecting the command Text label options from the text label s context menu This action opens the dialog window Text label options examples of that window are in Fig 11 1 Most of the controls of that window are self explanatory Below are explanations of the list boxes Horizontal position and Vertical position The selected highlighted items in the list boxes Horizontal position and Vertical position define the change of the text label position in the graph window after changing the window dimensions or changing axis limits or margins of the graph The default position of a text label is such that the left edge of the text label is at a fixed distance from the left edge of the graph window and the top edge of the label is at a fixed distance from the top edge of the graph window here the distance is measured in pixels Such position of the text label is defined by selecting the top item in the mentioned list boxes see Fig 11 1a However such position of the text label may be unacceptable if the label is associated with particular lines or points shown in the graph For example if the text label contains the title of a graph axis then it should be at fixed distance from that axis If the axis title must be at the center of the axis then the item Vertical central line or Horizontal central line must be selected depending on whether that axis is X axis or Y axis respectively Th
96. formula curve will change too Note If the edited program is a subroutine then those three option buttons are absent see Fig 4 13 12 e After clicking the button Compile the current formula and all other modified programs are checked for syntax errors and compiled but the curves that depend on those programs are not recalculated e After clicking the button Apply the current formula and all other modified programs are checked for syntax errors compiled and re calculated and the curves that depend on those programs are re plotted clicking this button is equivalent to clicking the toolbar button e By clicking the button Arrays a dialog window with the list of all user defined arrays is opened In the mentioned dialog window new arrays can be defined or existing arrays can be modified for more information about the list of arrays see Section 4 3 e By clicking the button Global parameters a dialog window with the list of global formula parameters is opened Global parameters are parameters that are visible in all programs When the value of a global parameter is changed all formulas depending on that parameter are automatically recalculated except when the global parameter is changed by a program and some of the formulas depending on that parameter belong to other graph windows in which case the latter formulas must be recalculated manually By optimizing global parameters it is poss
97. g is assigned the value zero otherwise it is set to an integer number from 1 to 4 depending on the type of the difficulty that was encountered Argument No 1 of all built in integration summation iteration and root finding functions defines the expression to be processed and argument No 2 defines the variable of integration equation or index of summation or index of iteration The expression defined by argument No 1 stays in scope of the calling program so that all previously defined variables can be used in it The variable of integration equation or index of summation takes priority over all other variables including system variables and it is only visible in the processed expression so that any identifier can be used in place of that variable In example 5 below the value of the system variable t the time value is not affected by the summation Multiple integrals can be computed by defining the integrand 1 e the integrated expression as a call to an integration function see Example 2 below The original QUADPACK integration functions have an integer argument that serves as an error code If a difficulty is encountered during integration then the error code is assigned a non zero value ranging from to 6 and indicating the type of the error In such a case GraphiXT outputs an error message and highlights the line of the program code where the error occurred for more information about runtime error handlin
98. g see Section 4 7 Every call to any built in integration function causes an update of the following three system variables IER the mentioned error code AbsErr estimate of the modulus of the absolute error nEval number of integrand evaluations 23 Those three variables are set to zero before starting computation of each sequence of formula values If a call to a built in integration function can not be initiated e g due to insufficient memory for work arrays of the function then the system variables IER AbsErr and nEval are not modified Examples 1 The integral of the function exp x exp t x with respect to x from I to 10 result Int exp x x exp t x 2 x 1 10 2 The double integral of a user defined function FUNC I a b result Int Int FUNC1 a b a al a2 b b1 b2 3 Example of integration from 5 to co result Inti exp x x exp t x 2 x 5 1 when integrating from o to a finite bound the last argument should be 1 4 Example of integration from co to o the corresponding graph is shown in Fig 4 12 result Inti exp x x exp t x 2 x 0 2 in this case argument No 4 must be 2 and argument No 3 can be any value 5 The sum of products of corresponding elements in row 2 of matrix A and in column 3 of matrix B result Sum A 2 t B t 3 t 1 Size loc A 1 1 Fig 4 12 contains an example of a graph of convolution of two func
99. g actual parameter is an address of an array loc c Parameter No 4 a subroutine f whose first argument is a value of a variable or an expression the second argument is the address of a one dimensional array and the third argument is the address of a two dimensional array i loc f x loc J loc J 4 Fig 4 9 An example of the dialog window Subroutine name and formal parameters 19 dialog window is opened automatically The name and formal parameters of a subroutine that is being edited can be changed by selecting the menu command Programming Name and formal parameters or by selecting the same command from the program editor context menu or by clicking the button Name and formal parameters of the programming tools dialog window see Fig 4 13 In the list of formal parameters comment lines are allowed Each comment line starts with a double slash see Fig 4 9 If a given argument actual parameter of the subroutine is the address of a variable array or function then the built in function loc must be used when calling that subroutine This is reflected in the list of formal parameters see Fig 4 9 The rules of using the construct loc in the list of formal parameters are explained in the comments that are visible in Fig 4 9 If the formal parameter is a function then the corresponding actual parameter must be loc lt function name gt For example if x
100. ging the current time marker with the mouse or by selecting the command Current time value and line format from the current time marker s context menu 3 Model functions and free curves In addition to model functions GraphiXT can display the so called free curves which are not associated with any model A free curve can be created in four ways a by menu command Graph options Create free curves b by drawing a free form line and then converting it to a free curve this is done using the line s context menu c by importing curve data from text files this is done using the menu command File Import free curve data from text files d by pasting the curve data from the clipboard In order to be able to import free curve data from a text file its format must conform to the following requirements The first non empty line must contain column headers Each column header must be the name of a corresponding variable The first variable is the independent variable function argument and all subsequent variables are the dependent variables functions of the independent variable All non empty lines that are below the header line must contain values of the mentioned variables listed in the same order as their names The argument values must be sorted in ascending or descending order argument values that are exactly equal to each other are allowed too By default the argument and function names must
101. graph window i e it is equal to the mentioned constant const The checkbox t sync for turning on or turning off the time synchronization mode This is the operation mode when current times of two or more graph windows are equal to each other By checking this checkbox the active graph window is included into the group of time synchronized graph windows and the current time of that window becomes equal to the current time of that group The checkbox t sync all is used for turning on or turning off time synchronization of all graph windows of the active project The checkbox x sync for turning on or turning off the coordinate synchronization mode This is the operation mode when respective X axis limits of two or more coordinate graph windows are equal to each other This checkbox is only enabled when the current graph is a coordinate graph not time graph By checking this checkbox the active graph window is included into the group of coordinate synchronized graph windows and the X axis limits of that window become equal to respective limits of that group The checkbox x sync all is used for turning on or turning off coordinate synchronization of all coordinate graph windows of the active project The button J for turning on slider animation In animation mode the current time of the active graph window and all windows that are time synchronized with it increases with a cons
102. h iteration because it is needed for solving the system of linear equations 14 1 23 besides the diagonal elements of the inverse matrix are needed for computing half widths of parameter confidence intervals 14 1 16 or 14 1 17 The normalization method depends on the parameter bounds 1 If the parameter is unlimited in both directions then it is normalized by dividing it by the absolute value of its initial value If the initial value is zero then the parameter is not normalized even when the checkbox normalize is checked 2 If the parameter is bounded in one direction only then it is normalized by subtracting the value of the bound from the current value of the parameter and then dividing this difference by the absolute value of the initial value of the same difference In this case division by zero is not possible because the initial value of a varied parameter is not allowed to be equal to one of the bounds 3 If the parameter is bounded in both directions then it is normalized by subtracting the arithmetic average of the two bounds from the current value of the parameter and then dividing the obtained difference by 0 5 x upper bound lower bound i e by the half width of the parameter definition interval Thus if the parameter is unbounded or bounded in one direction only then its normalized initial value is 1 If the parameter is bounded in both directions then its normalized initial value is always great
103. hanged either by moving the time slider or by entering its value in the text box at the bottom of the main window This change of the time limits can be either smooth or step like If the checkbox lt gt is unchecked then the change of the time limits is smooth so that the difference between the current time and any one of the two time limits stays constant If that checkbox is checked then the limits of the time axis don t change while the current time is between them Those limits only change when the current time becomes greater than the upper limit or less than the lower limit In this case the magnitude of the mentioned change is always equal to a multiple of the difference between the upper and lower limits of the time axis Two text boxes for entering the time limits of the active time graph window and all windows that are synchronized with it see Fig 10 1a and Fig 10 1b or one text box for entering the current time of the active coordinate window and all windows that are synchronized with it see Fig 10 1c Static text field which is only visible when the active window is a coordinate graph window see Fig 10 1c This field contains the stored model time value that is closest to the current time of the active graph window This field is not empty only when at least one model time value is stored in computer memory The indicated time value corresponds to f x t const model functions plotted in the active coordinate
104. he bottom of the main window see Fig 1 1 By checking t sync the active graph window is included into the group of time synchronized graph windows and the current time of that window becomes equal to the current time of that group By checking t sync all all graph windows of the project are synchronized The synchronized time graph windows share not only their current time but also their time axis limits Respective X axis limits of several coordinate graph windows can be forced to be equal to each other using checkboxes x sync and x sync all see Fig 1 1 The checkbox Auto which is to the right of the time slider see Fig 1 1 is used to turn on or turn off automatic change of current time By checking this checkbox the current time of the active graph and the graphs that are synchronized with it becomes equal to the time of the last point of ft model curves however afterwards the current time of the active graph may be set to any other value When the number of model time values that are kept in memory changes for example after adding new points during simulation or after deleting part of the data or after importing model data from text files the current time of all graph windows with the Auto mode turned on becomes equal to the time of the last available point of f t model curves If the entire interval of the graph time axis belongs to the slider interval then a change of the slider posi
105. he values of constant indices are entered into the corresponding cells of the grid control All indices 4 4 Using subroutines in programs According to Wikipedia a subroutine is a sequence of program instructions that perform a specific task packaged as a unit This unit can then be used in programs wherever that particular task should be performed When discussing subroutines it is important to distinguish between formal and actual parameters of a subroutine Another definition from Wikipedia a formal parameter is the variable as found in the function definition while argument sometimes called actual parameter refers to the actual value passed For example let us suppose that subroutine FUNC defines the following function of one variable f x In 1 x Then the variable x which is used in that definition is the formal parameter of the subroutine Now if there is a variable a defined in the calling program then in order to calculate the value of In 1 a the subroutine FUNC must be called as follows FUNC a In this call the value of a is the actual parameter or argument of the subroutine FUNC In GraphiXT each subroutine must return a value and calls to subroutines may be used in arithmetic expressions In this respect subroutines are similar to built in functions see Section 4 1 In fact subroutines are user defined functions As in the case of formulas the value returned by a subroutine i
106. he x axis range of the graph is checked then the mentioned interval is determined automatically If neither of those checkboxes is checked then the interval limits must be entered in the two text boxes that are below The controls that are in the group Time of f x free curve can only be activated when the selected dataset corresponds to a free curve of the type f x These controls define the time value that must be associated with the selected f x dataset during nonlinear fitting If the checkbox Current time of the graph is checked then the mentioned time is determined automatically If that checkbox is not checked then the time value must be entered in the text box t The controls that are in the group Range of t values of f x t model function can only be activated when the selected dataset corresponds to an f x model function These controls are used to define the time limits the coordinate limits are defined using the previously mentioned controls If any one of the three checkboxes All t values t axis range of this graph and t range of this f t function is checked then the time limits are determined automatically in the latter two cases it is also necessary to select a graph or an ft dataset in the corresponding drop down list If neither of those checkboxes is checked then the time limits must be entered in the two text boxes that are to the right of the checkbox All t values
107. his problem can be solved analytically i e 4 and B can be expressed using elementary functions but when y x is nonlinear this problem can only be solved numerically applying an iterative procedure If y x is the linear function 13 1 then the SSE expression 13 2 can be written as follows F A B gt A Bu y n4 BY xp Dy 2ABY x 2AS y 2BY X Y 13 3 k l k l k l k l k l k l It is known that partial derivatives of a function with respect to all arguments at a minimum point are zero After equating to zero the partial derivatives of the expression 13 3 with respect to A and B we obtain a system of two linear algebraic equations with unknowns A and B Its solution is B 13 4 2 2 k l k 1 1X Be A y x 13 5 Na M I The B coefficient is called the slope of the straight line and the A coefficient is called the intercept The standard deviations or errors of those two coefficients are calculated according to formulas u Fae 3 6 n n 2 D AB nin __ 13 7 n n 2 D where Fmin iS the minimum value of the SSE 13 3 i e the value of SSE when A and B are equal to their optimal values 13 4 and 13 5 x is the average argument value 1 n T Us 13 8 A k l and D is the variance of the argument values 2 1 2 D x 7 5x 13 9 nra n The dialog window Data analysis contains the checkbox Fit the natural logarithm see Fig 13 1
108. ible to fit several data sets by different formulas simultaneously and solve systems of nonlinear algebraic equations see Section 14 In the mentioned dialog window new global parameters can be defined or existing global parameters can be modified for more information about the list of parameters see Section 4 2 e By clicking the button Local parameters a dialog window with the list of local parameters is opened Local parameters are parameters that are visible in one program only Consequently local parameters of different programs may have identical names In the mentioned dialog window new local parameters can be defined or existing local parameters can be modified for more information about parameter lists see Section 4 2 Note If a local parameter identifier is the same as a global parameter identifier then the local parameter value is used In addition to global and local parameters parameters of yet another kind are used Those are the model parameters which depend on the currently loaded GraphiXT plug in such as the charge transport simulator CarrierFunc dll Unlike global or local parameters model parameters can not be referenced by their identifiers in programs However programs can include references to model functions which depend on model parameters 200 Displacement current A cm 2 Conduction current A cm 2 t 1e 011 Total current A cm 2 150 100 50 50 0 0 2
109. idual time values Those values are listed in the list box Individual times In order to specify a new value or to change an existing value or to delete it the user has to click one of the three buttons Add Change or Delete that are under the mentioned list box 2 The final set of additional times which is passed by GraphiXT to the simulation plug in is the union of all sets of time values defined in the window Additional times a Additional times Ex Select the sets of curve abscissas that should be included in the set of additional times Graph Curve abscissa sets of the selected graph 1 Free surface potential Fitting data set No 1 4 Graph4 T Sets of equidistant time values Individual times No 1 0 01 lt t lt 0 03 N 5 t1 0 04 No 2 0 08 lt t lt 0 13 N 11 t2 0 06 Add Change Delete Add Change Delete Cancel Fig 8 1 An example of the dialog window Additional times 34 9 Importing model data from text files Model data are imported from text files using the menu command File Import model data from text files In order to be able to import model data from a text file its format must conform to the following requirements The first non empty line of the file must contain column headers Each column header must be the name of a corresponding variabl
110. in Section 14 1 4 The latter text field is only visible if the bottom checkbox of the General nonlinear fitting options dialog window has been checked see Fig 14 8 In this case the positions and names of the mentioned text fields are as shown in Fig 14 1 Otherwise the last seventh text field is not visible and the names of text fields No 5 and No 6 are Sum of squared errors taking into account weight factors and SSE df respectively After stopping the fitting process all plotted curves that depend on the varied parameters are automatically updated If some of varied parameters are model parameters then the curves are updated at each iteration Note Since the parameter confidence intervals can only be computed when the SSE derivatives with respect to unknown parameters are known see Section 14 1 4 and since the simplex method does not use derivatives in the case of the simplex method the parameter confidence intervals are computed after stopping the fitting process at this stage the button Stop fitting turns into the button Stop computing errors In the case of the LM method the confidence intervals are periodically updated together with parameter values during fitting If the varied parameters include at least one model parameter then the model functions are recalculated at each iteration In this case a temporary set of additional times is used which consists of the time values of all fitted
111. ion is in progress The button Delete the selected X sets is used to delete all selected X sets excluding the sets of model time values and model layer coordinate values This causes deletion of all formulas and free curves linked to the selected X sets Notes 1 All free curves and formulas of the current graph excluding the formulas that are not linked to any X set can be deleted by selecting all X datasets and then clicking the button Delete the selected X sets all X sets can be selected by clicking the top left cell of the X set grid control 2 Formulas whose abscissas correspond to the X axis range of the graph as in the example of Fig 4 1b can not be deleted by this method because they are not linked to any X dataset the set of their argument values is temporary because it depends on the X axis range of the graph Those formulas can only be deleted using the formula context menu or using the dialog window with the list of formulas which is opened by selecting the menu command Graph options Formulas or Programming Formulas The button Properties of the selected X set is used to open a dialog window where all properties of the selected X set are displayed and can be modified An example of that dialog window is shown in Fig 5 2 The four editable text boxes in the top half of this dialog window provide an alternative way to modify the parameters of the selected X dataset If the checkbox R
112. ions were true then the probability to obtain such a large value of y would be less than 1 0 995 0 005 It such cases one may guess that at least one of the standard conditions is probably false If the value of y is too large and if it is known that the second and the third standard conditions are satisfied then one may conclude that the theoretical model is incorrect If the value of y is too small this usually means that the values of o used to compute the SSE are greater than the true standard deviations i e the third standard condition is not satisfied In such a case no conclusion can be made about correctness of the model because the value of P x df which would be obtained if the correct weight factors were used is not known If that probability is between 0 005 and 0 995 i e neither too small nor too large and if it is known that the second and the third standard condition are satisfied then one may guess that the first standard condition is satisfied too 1 e that the theoretical model is correct The mentioned guesses may be incorrect For example the conclusion that the model is correct is only based on statistical properties of the data it is possible that the model is not suitable but the number of fitted points is too small or random measurement errors are much larger than discrepancies caused by imperfections of the model so that those discrepancies are not visible After performing more accurate or
113. ipped points must be the same for all linked curves is checked then the maximum number of X values that will be visible in the graph window is shown in the bottom text box If the total number of X values in the range of the graph X axis exceeds this number then the program will plot every second point or every third point etc so that the number of visible points for each linked curve will never exceed the number entered in this text box 29 6 Time cross sections f x const t of f x t model functions In addition to fft model functions time graphs can display the so called time cross sections fix const t of f x t model functions Each time cross section displays the time dependence of a particular model quantity at a given point of the simulated system In order to add a time cross section or to modify the x value of displayed time cross sections the command Graph options Select f x const t functions must be selected from the main menu or the graph context menu This action opens the dialog window with the list of all time cross sections plotted in the active graph An example of such a list is shown in Fig 6 1 New time cross sections are created by clicking the button New entering the required x value in the dialog window New X value and clicking OK Then the list of all f x model functions is opened After selecting the required functions in that list and clicking OK the list
114. is because smoothing may cause discontinuities at the edges of the smoothing interval After smoothing the static text fields Average before smoothing and after sm display the average function value in the smoothing range before smoothing and after it This quantity is defined by the formula L f ydr 15 3 xX X n 1 x y This integral is computed using linear interpolation in other words the trapezoid quadrature method If only a part of the curve is smoothed then the integration interval is slightly expanded by adding one or two points that are outside of the smoothing interval one additional point on each side of the smoothing interval Note If the smoothed curve is an fx 4 model function then the values shown in the mentioned four static text fields correspond to the current time of the graph The drop down list box Number of smoothing cycles is used to specify the number of smoothing cycles that should be done one after another For example if the number 10 is selected then the final result after clicking the button Smooth will be the same as clicking the button Smooth 10 times with the number 1 selected in that drop down list The other controls that are visible in the dialog window Smoothing depend on the selected smoothing method Below are descriptions of the smoothing methods and corresponding controls 62 The three point smoothing method is represented b
115. ks of the type I error and the type II error The standard values of the critical probabilities are 0 005 and 0 995 probability of the type I error is 1 or 0 025 and 0 975 probability of the type I error is 5 14 1 5 The Levenberg Marquardt method The program GraphiXT applies two methods of nonlinear fitting the Levenberg Marquardt method LM method and the simplex method The LM method is one of the most efficient and popular methods of nonlinear fitting In its description that follows the following notations will be used _1OF Suo 2 Op Ta Op Op where i and j are the numbers of the varied parameters i j 1 2 m The general expressions of the aj coefficients can be obtained by differentiating the expression of the SSE 14 1 3 By retaining only the first derivatives in the obtained expression the following approximate expression of a is obtained a gt 2 14 1 21 k l p Op j Further on the a coefficients will be defined by the equality 14 1 21 rather than by 14 1 20 Using the LM method the increment of each parameter during each iteration is computed as follows First the coefficients 4 and a are computed then elements of the matrix a are modified in the following way Qi q amp l 2 14 1 22 14 1 20 a a i j where 4 is the parameter that controls the fitting procedure The increments of varied parameters 6p i 1 m in the current iteration are the soluti
116. l po 3s lt p lt poi 35 Thus s is the half width of the 68 3 confidence interval of the i th parameter If the standard deviation o of each measured response yis known then each error factor wz in the expression 14 1 3 should be set equal to 1 o Then if the errors of all measured responses are independent and normally distributed the statistical average 1 e the average over a large number of equivalent fits or the ensemble average of each term in the sum 14 1 3 is equal to n m n the statistical average of the value Fo of that sum is n m and the statistical average of the value of Fy 14 1 10 is 1 In this case when n the value of Fo corresponding to the given experimental data set approaches n and the corresponding value of Fj approaches 1 However the number of fitted points n is frequently too small to make those statistical properties evident in the available data Consequently the computed value of Fj may deviate from 1 significantly When n is small and the theoretical model is 49 correct then Fj is usually less than 1 and the corresponding half widths of parameter confidence intervals 14 1 16 are less than the true values In this case more reliable estimates of s are obtained by substituting 1 instead of Fj in the expression 14 1 16 Thus when the error factors wz are equal to 1 Gj the 68 3 confidence intervals of parameter estimates should be calculated using th
117. ld be set to zero prior to processing If i is equal to zero then elements of array c are set to zero before calculating the frequencies otherwise they are not modified initially and they are incremented during processing If i is zero and n is less than 1 or greater than or equal to the size of array a then behavior of Hist2 is identical to Hist Int f x a b integral of expression f with respect to x from a to b e g Int 1 sqrt exp t x x x 0 5 Inti f x a 1 integral of expression f with respect to x from a to 00 Inti f x a 1 integral of expression f with respect to x from o to a Inti f x a 2 integral of expression f with respect to x from o to o the argument a is not used Intw f x a b w 1 integral of expression f x cos wx with respect to x from a to b Intw f x a b w 2 integral of expression f x sin wx with respect to x from a to b Sum f i i1 i2 sum of terms f when the summation index i varies from il to i2 e g Sum In i i 2 10 Sum2 f t a b dt sum of terms f when the summation variable varies from a to b in increments of dt Iter f 1 i1 i2 repetition iteration of expression f when the iteration index i varies from il to 12 Root f x a b root of the nonlinear equation f x 0 a and b are limits of the interval that should be searched for the root For ex
118. linked curves and each set of abscissa values is called an X dataset or an X set When an abscissa of a point is changed the corresponding abscissas of all other curves linked to the same X set are changed too When two or more free curves are pasted or imported from a text file they are initially linked to the same X set Linked curves can be unlinked using the checkboxes Unlink this curve or Unlink all curves of the curve format dialog window that dialog window can be opened using the curve s context menu command Curve format and name In addition it is possible to specify that abscissas of skipped points must be the same for all curves linked to a particular X set this is done using a corresponding checkbox of the curve format dialog window Properties of X sets can also be changed using a menu command Graph options X datasets After selecting that command a dialog window similar to the one shown in Fig 5 1 is opened Below are descriptions of all controls of that window e The top grid control contains the list of all graphs of the current project e After selecting a graph in the top grid control the middle grid control is filled with information about all X datasets belonging to that graph X dataset identifier ID X dataset name minimum X value Min maximum X value Max number of values N and the asterisk in the last column if any of the previous three param
119. luates the assignment statement repeatedly while the expression in parentheses is non zero that expression is the so called condition of the loop operator If the assignment statement must be evaluated a predetermined number of times then a temporary variable must be used which is incremented by one after each evaluation In such a case the body of the while operator consists of more than one statement and curly braces must be used they are used exactly as in the if else construct In the following example the factorial of the number 10 is calculated fact 1 i 1 while i lt 10 fact fact i i i l Note In the above example the last calculated expression is the comparison operation i lt 10 Its last value is zero Therefore if those lines of code were the last lines of the program it would return zero In order to ensure that the program returns the value of the factorial an additional line is needed for example it could be a fact or just fact 10 Identifiers of system variables t time a in time graphs t is equal to the current value of the formula argument b in coordinate graphs if the formula argument values are equal to the stored model coordinate values or if the formula contains references to linearly interpolated values of fx t model functions i e FXn then t is equal to the stored model time value that is closest to the current time of the gra
120. lue of the f t model function No n e g FT10 FXn the linearly interpolated value of the f x model function No n e g FX10 FCn the linearly interpolated value of the free curve No n e g FC10 TER AbsErr and nEval are modified in each call to any built in integration function Their meanings are the error code estimate of the absolute error and number of integrand evaluations respectively for more information about built in integration functions see Section 4 6 Those three variables are set to zero before starting computation of each sequence of formula values NLSF if the formula is computed during nonlinear fitting and if the fitted dataset belongs to a free curve then NLSF is equal to the sequence number of that curve in the corresponding graph window and if the fitted dataset belongs to a model function then NLSF is opposite to the sequence number of that model function in the set of all model functions of the same type f a or f x t const If the program is executed not during nonlinear fitting then NLSF 0 iDat and nDat are equal to the current sequence number and the total number of the fitted datasets that correspond to the current fitting formula respectively only counting the datasets that are used for fitting If the program is executed not during nonlinear fitting then iDat and nDat are equal to zero iStart and iEnd are equal to sequence numbers of the first and last points of the subset o
121. m GraphiXT can display time graphs of functions or coordinate graphs of f x functions corresponding to the current time i e f x t const The latter time can be smoothly varied Then GraphiXT displays the change of the coordinate dependence of the plotted quantity with time Argument values of all model time functions ff are equal to each other Similarly model coordinate x values corresponding to the same moment of time and the same layer of the model are equal to each other Here the term layer is used in an abstract sense it means a set of model parameters and functions corresponding to a defined set of x values However during simulation of charge carrier kinetics in a multi layer system the mentioned abstract layer corresponds to a real layer of the system In this case the x values have the meaning of node coordinates with the first and last values corresponding to opposite surfaces of that layer In the current version of GraphiXT v1 21 the maximum allowed number of layers is 10 The sets of x values corresponding to different moments of time or different layers may be different 2 User interface The default language of the GraphiXT user interface is English It is possible to select another language after installing GraphiXT This selection is done either in the dialog window that pops up when GraphiXT is started for the first time or later on by clicking the language button which is in the bottom right co
122. m axis Right edge of the window Bottom edge of the window Constant distance to line Xx const Constant distance to line Y const x label left edge coordinate Y label top edge Y coordinate X label center x coordinate Y label center Y coordinate x label right edge coordinate Y label bottom edge Y coordinate X Val An E X 0 13318314338723 Y 03 text will only be visible when xMin lt X lt Max text will only be visible when yMin lt Y lt yMax u Fig 11 1 Examples of the dialog window Text label options a The default position of the text label b The text label position is specified in units of coordinate axes of the graph 11 2 Vertical and horizontal straight lines A vertical or horizontal straight line must intersect the plotting area of the graph i e it must not be in the graph margin After creating a straight line by one of the mentioned methods its position and format can be changed by double clicking the line or by right clicking it and selecting the command Line position and format from the context menu This action opens the dialog window Line position and format whose example is shown in Fig 11 2 The top text box is used to enter the x coordinate of a vertical line or y coordinate of a horizontal line This coordinate is expressed in the coordinate axis units i e in the same units in which the axis tick labels are expr
123. mary functions is only checked then only the precision of the primary functions and corresponding x values is increased The primary functions are the functions whose values are used to compute all other functions of the model and to determine the evolution dynamics of the state of the simulated system For example during simulation of charge carrier kinetics in a semiconductor the primary fix model functions are concentrations of free charge carriers and traps of all types and charge states The button Set precision of f x t function values is used to change precision of individual f x model functions and of node coordinates of individual layers Note If the precision of a particular f x A model function is increased then the precision of associated x values should be increased too The current state of the simulated system i e the state that corresponds to the current simulation time is always stored with the maximum precision regardless of precision of stored function values Consequently if simulation is carried out without deleting the data defining the current state of the simulated system then precision of function values would not have any effect on the simulation process However if some of the last stored time values are deleted and simulation is resumed not from the end but e g from the middle then the initial state of the system is computed from the model function values stored in memory
124. mplest case the format of the if else statement is the following if expression_1 assignment_statement_1 else if expression 2 assignment_statement_2 else assignment_statement_n For example if t lt 3 y 0 5 else if t lt 6 y 2 else y 3 The program evaluates each of the expressions in parentheses those expressions are also called conditions until an expression with a non zero value is encountered Then the statement or several statements corresponding to that condition are processed and control returns to the point after the if else statement i e the subsequent conditions are not evaluated The else if and else branches are optional only the initial if branch is required If neither condition evaluates to a non zero value then the else branch is executed if it is present The keywords if and else and assignment statements may be written on different lines e g if t lt 5 y 0 5 else y 3 if else if or else branch may consist of several statements Then the start and end of the statements belonging to that branch must be indicated by curly braces e g A if t lt a b exp 0 5 t a 1 y 2 b 2 else b t atl y 2 In b The loop operator while In the simplest case the format of the while loop operator is the following while expression assignment_statement For example while t lt 3 t t 1 1 The program eva
125. mum relative parameter change between iterations 1e 009 Minimum relative change of SSE between iterations LM method 1e 007 Simplex method 1e 011 Minimum absolute change of the ratio SSE lt number of points gt between iterations LM method D Simplex method D SSE error factors are equal to inverse standard deviations of fitted Y values Cancel Fig 14 8 An example of the general nonlinear fitting options dialog window two fitting algorithms Levenberg Marquardt LM method and simplex method The value of the absolute change refers not to the change of SSE but to the change of the ratio F n where F is the value of the SSE and n is the total number of fitted points When n is large this ratio is approximately equal to the reduced SSE see 14 1 6 e The checkbox at the bottom of the nonlinear fitting options dialog window is used to select the method of computing the confidence intervals of varied parameters If it is checked and if 7 lt 1 then the half widths of parameter confidence intervals are computed according to the formula 14 1 17 If this checkbox is not checked or if 7 gt 1 then the half widths of parameter confidence intervals are computed according to the formula 14 1 16 In addition if this checkbox is checked the probability P x df is computed too see Section 14 1 4 Since the parameter confidence intervals are not used for fitting this option has no effect on the fitting pr
126. n opens the dialog window Time limits and amount of data Its examples are shown in Fig 7 1 Below are explanations of all controls of that dialog window The text box Initial simulation time is used to enter the time value corresponding to the initial state of the simulated system see Fig 7 1a This time can only be modified when there are no stored model data Otherwise that field contains the current simulation time see Fig 7 1b which can not be changed Notes 1 The current simulation time should not be confused with the previously mentioned current graph time Those two times may be different 2 The current simulation time is stored with 128 bit precision two 64 bit floating point variables This corresponds to approximately 30 significant digits However all displayed time values are rounded to 14 significant digits The text box Minimum visible time is used to enter the minimum time value that should be used as an argument of model functions corresponding to the first point visible in time graphs i e the minimum stored value of model time If model data is already present in computer memory then a change of that time will only affect the stored model data when it is greater than the first stored value of model time Then by clicking OK or Apply all previous model time values will be deleted along with corresponding function values The text box Time of the last computed point of f t curves c
127. nates random variations of the smoothed quantity y it also causes a decrease of non random systematic changes Adaptive exponential smoothing is a modification of the exponential smoothing method that addresses this shortcoming In the case of adaptive exponential smoothing the smoothed values are also computed according to the formula 15 4a but the smoothing factor a is modified after each smoothed value so as to ensure that y reflects the systematic change of y as closely as possible I e if variation of the function y when its argument is increased or decreased becomes less random i e more directional then a is increased and if variation of y becomes more random then a is decreased The new value of a is computed as follows After computing y the residual forecast error is computed eri Via Via 15 5 Those errors and their absolute values are exponentially smoothed E41 Be 1 Be 15 6a len 1 Fle d A le 15 6b The new value of ais defined as According to Trigg D W Leach A G Exponential smoothing with an adaptive response rate Operational Research Quarterly vol 18 no 1 1967 p 53 59 63 m lel 15 7 The typical value of fis 0 1 The method of adaptive exponential smoothing is represented by the following two text boxes 64 The text box Adaptive smoothing parameter is used to enter the adaptive smoothing parameter i e the parameter of the fo
128. ndled by displaying a dialog window with an option to stop the computation when a sequence of formula values is computed longer than 2 seconds The user can either continue waiting that dialog will be closed automatically after finishing the computation of the current sequence of formula values or stop the computation by clicking the button Stop calculation After clicking the button Stop calculation the current value of the computed formula and all subsequent values in the current sequence of values are set E 2 es D a wade BFE vY x TI KA Insert reference to a function ft Built in function Runtime error when referencing array A index is out of bounds Nano kami psn Vj Auto F lt gt 0 0000e 000 10 0000 VJ tsyne V t sync all Fig 4 13 An example of a runtime error 25 equal to the indeterminate number 1 4 IND except during initialization of an array in this case values of the current and subsequent elements of the array are not modified When a runtime error occurs execution of the current program is terminated immediately I e all operations that should be normally done after the point where the error occurred are skipped In such a case the value returned by that program is set equal to the indeterminate number 1 IND If the current program is a subroutine then execution of the calling program continues from the instruction that imme
129. nes and DLL functions see Section 4 4 and Section 4 5 respectively After selecting a function its name will be inserted at the current position of the text cursor e Clicking the button Curve format and name opens the dialog window where the curve format and name are specified the same dialog window can be opened from the curve context menu Note If the edited program is a subroutine then this button is replaced by the button Name and formal parameters see Fig 4 13 e The group of three option buttons see Fig 4 1 define the rule that must be used to calculate abscissas of the curve points If the top option button is selected then the abscissa of the first point is always equal to the lower limit of the graph X axis and the abscissa of the last point is always equal to the upper limit of the graph X axis the abscissas will be re calculated each time when the X axis range changes Such X datasets will be called temporary X datasets If the middle option button is selected then the abscissas do not depend on any other data similarly to abscissas of free curves If the bottom option button is selected then the abscissas of the curve representing the current formula coincide with the indicated set of values which could be for example abscissas of a particular free curve or formula or of a group of linked free curves and formulas If any of the latter values is changed the abscissa of the corresponding point of the
130. ntinue as if it was not interrupted The current fitting state is also written to the project file gxt by selecting the menu command File Save project Consequently fitting can be resumed after closing the current GraphiXT project and then opening it again The current values of varied parameters and their standard deviations as well as additional information about the fitting process number of iterations value of SSE etc can be copied to the text editor window Info at any time by clicking the button Copy to the Info window see Fig 14 1 60 15 Data smoothing The smoothing dialog window is opened by selecting the menu command Data analysis Smoothing An example of that window is shown in Fig 15 1 Below are descriptions of all controls of that window The drop down list boxes Graph and Curve are used to select a graph window and a model function or a free curve to be smoothed The controls that belong to the group Smoothing range are used to define the part of the curve that should be smoothed The drop down list box Method is used to select the smoothing method In the current version of the program v1 21 the following three choices are available 1 three point smoothing 2 exponential smoothing 3 adaptive exponential smoothing those methods are described below After clicking the button Smooth the selected curve will be replaced by the smoothed
131. ocess Note This checkbox can only be checked when there are no fitted datasets whose SSE is divided by the number of points this is one of the options that can be specified in the previously mentioned dialog window Fitted dataset options 14 2 5 The fitting process In a general case fitting is done in two stages At first a certain number of iterations are done by the LM method and then a certain number of iterations are done by the simplex method Each one of those two stages is finished when a specified maximum number of iterations of that type is reached or when the program determines that one of the mentioned ending criteria is satisfied see Section 14 2 4 General nonlinear fitting options The maximum number of iterations of each type is selected in two drop down lists that are at the bottom of the dialog window Nonlinear least squares fitting see Fig 14 1 It is also possible to use a single fitting algorithm during the entire fitting process this is achieved by selecting 0 in one of the mentioned drop down lists Note If the SSE has only one minimum in the m dimensional space of varied parameters then the optimum values of varied parameters do not depend on the choice of the fitting method That choice only affects the shape of the path to that minimum that is followed in the course of fitting as well as the number of iterations and duration of fitting If the SSE has more than one minimum then the fitting s
132. of analysis Linear fit ln Y A B X Analysis results Parameter Value Error A 34 6240 0 010512124 B 0 256880150 0 001146293 1 B 3 8929 0 017371392 4 w m Speed J 2 VJ Auto E lt gt 0 129643084 0 13 Mitsyne Wl tsyncal xsyne E x sync all p m tt Language Fig 13 1 An example of the analysis dialog window and the text editor window Info with the analysis results 42 F 4 B Y pv x4 4 8 1 13 2 k l Expression 13 2 is the sum of squared deviations of theoretical values from the measured ones hence the term least squares That sum is also called the sum of squared errors SSE This expression always has a minimum at certain values of A and B However even if the form of the theoretical function y x correctly reflects the true relationship between the measured quantities y and x those optimal values of A and B which correspond to the minimum SSE do not necessarily coincide with the true values of A and B for example because of measurement errors The method of least squares only allows estimation of the most likely values of A and B Everything that was stated above about the method of least squares also applies to the case when the theoretical function is nonlinear Regardless of the form of that function and of the number of unknown coefficients a SSE expression of the type 13 2 must be minimized However when y x is the linear function 13 1 t
133. og window 16 its elements are lost but the array object is not deleted When an array is allocated it is automatically initialized using the expression that was entered in the text box Expression of the array properties dialog window see Fig 4 5 New arrays are not allocated by default If a user defined program contains a reference to an array that has not been allocated then execution of that program would cause a runtime error This is one of runtime errors handled by GraphiXT other runtime errors which are handled by GraphiXT are listed in Section 4 7 Similarly to global and local parameters arrays can be copied from one project to another using the five buttons Copy Copy All Cut Paste and Paste before see Fig 4 4 In addition the array list has the Up and Down arrow buttons which provide a more convenient way to rearrange the list the keyboard shortcuts are Ctrl 1 and Ctrl respectively Another difference between the list of parameters and the list of arrays is that changes done to the list of parameters can be cancelled whereas any change of the list of arrays is applied immediately and can not be undone Accordingly the array list dialog window does not have the buttons OK Apply and Cancel There are several built in arrays which allow access to any model data value in user defined programs Similarly any data point of any fre
134. ogram editor is opened immediately after such an error then a message containing the value assigned to the error code by the DLL function appears in the status bar of the program editor window Thus the error code can be used to inform the user about the number of arguments that must be passed to the function and about the calling convention that must be used In addition the error code can be used to provide information about various other error conditions that may occur when executing a DLL function Unlike the calls to subroutines and built in functions the calls to DLL functions do not require usage of the construct loc lt argument name gt when passing argument addresses to the function although such usage of that construct would not be treated as a syntax error E g the call to the function DLL3 of Fig 4 10 could look like this DLL3 a b c although the syntax DLL3 loc a loc b loc c is valid too However if a DLL function that requires argument addresses is used as an argument of a subroutine then the corresponding formal parameter of that subroutine must be defined using the construct loc in place of each argument of the DLL function For example if the function DLL3 of Fig 4 10 is used as an argument of a subroutine then the corresponding formal parameter of that subroutine must be defined as follows loc f loc loc loc or equivalently loc f loc x loc y loc z If th
135. on file the menu command Simulation options Model function file must be selected this action also causes the reference to the plug in user s manual to be added to the help system The model parameter editor is loaded using the menu command Simulation options Parameter editor file Prior to starting simulation with the currently loaded plug in the user has to set the initial and final times of simulation the menu command Simulation options Time limits and amount of data Each graph window has the so called current time associated with it If the active window is a coordinate graph window i e if functions f x t const are plotted in it then its current time is shown at the bottom of the main window see Fig 1 1 If the active window is a time graph window i e if functions f t are plotted in it then the limits of its time axis are shown at the bottom of the main window A vertical line indicating the current time the current time marker can also be shown in time graphs see Fig 1 1 Regardless of the graph type its current time can be changed by dragging the time slider which is at the bottom left of the main window see Fig 1 1 The slider limits can be changed by right clicking on it It is possible to synchronize two or more graph windows by forcing their current times to be equal to each other This is achieved using checkboxes t sync and t sync all which are at t
136. on is most accurate when the logarithm of the absolute value of the function is similar to a linear function If the integrand function itself is similar to a linear function or if in some parts of the integration range it is positive and sublinear or negative and superlinear then the preferred method is linear interpolation For example as it is evident from Fig 13 4 if the function is positive and sublinear then exponential interpolation dashed line in Fig 13 4 is less accurate than linear interpolation Ll Fig 13 2 Linear interpolation points are connected with a straight line In this example the integral value obtained by the method of linear interpolation is greater than the true value Since the integrand function is positive and superlinear i e its first derivative increases with increasing x a more accurate estimate could be expected when points are connected with an exponential curve e in this example B lt 0 0 X Fig 13 3 Linear interpolation points are connected with a straight line Since in this example the integrand function is negative the integral value is also negative and the absolute integral value obtained by the method of linear interpolation is greater than the true absolute value Since the integrand function is negative and sublinear 1 e its first derivative decreases with increasing x a more accurate estimate could be expected when points are connected with a negative exponential curve e
137. on of this system of linear equation dV asp f i 1 2 m 14 1 23 j l Below is the sequence of steps when applying the LM method 1 Calculate the SSE F p corresponding to the initial parameter values 2 Set 4 equal to 0 001 3 Compute the coefficients J and aj 4 Compute the coefficients Ci and solve the system of linear equations 14 1 23 5 Compute the SSE F p gp corresponding to the new parameter vector p dp 6 If F p op F p then increase by a factor of 10 and go back to Step 4 7 If F p Gp lt F p then decrease A by a factor of 10 update the parameter vector and the SSE i e p lt p amp and F p lt F p amp and go back to Step 3 Fitting is stopped when the SSE stops decreasing or when increments of all parameters become less than some predefined small value Since application of the LM method involves computing the partial derivatives of the SSE with respect to the parameters the fitting functions must be smooth 14 1 6 The simplex method A simplex is the geometrical figure consisting in m dimensions of m 1 points or vertices and all their interconnecting line segments and polygonal faces A one dimensional simplex is a straight line segment In two dimensions a simplex is a triangle In three dimensions it is a tetrahedron not necessarily the regular tetrahedron The first step of the simplex method of nonlinear fitting also called the Nelder 51 Simplex a
138. one or more model functions defined by that plug in This action opens a dialog window with the list of all graphs of the active project and the list of all formulas or all model functions plotted in the selected graph Examples of that dialog window are shown in Fig 14 4a and Fig 14 4b After selecting a graph in the top list box the contents of the bottom list box are updated The bottom list box contains the names of all formulas or model functions plotted in the graph that has been selected in the top list box although as mentioned in Section 3 model functions do not belong to any graph window the model functions that are used as fitting Selection of a model function Ss Selection of a fitting formula a z Graphs Erai No Title No and title 1 Free surface potential 2 Charge carrier concentrations 2 Charge carrier concentrations 3 Graph1 3 Graphi e 4 Graph4 Formulas Model functions plotted in the selected graph No and name No Name 1 al New Edit Format Copy Hide Show Delete Delete All Now Delete rom the mast Delete Al Cancel Cancel a b Fig 14 4 Examples of the fitting function selection dialog window a when the fitting function is a user defined formula b when the fitting function is a model function 53 functions must be plotted After selecting the fitting function in the bottom list box the button OK must
139. ontains the maximum stored model time value that time can not be edited By clicking the button with the up arrow which is above the text box Time of the last computed point of f t curves the number displayed in that box is copied to the box Minimum visible time The text box Time of the currently computed f t point is used to enter the time value corresponding to the next set of model function values that should be computed If that time is less than the time of the last computed point of At curves then by clicking OK or Apply all later model time values will be deleted along with corresponding function values Note The simulation plug in can modify the time of the currently computed f t point during simulation The actual time value of the next plotted point of AA curves may therefore be different from the value entered in that text box The text box Final time is used to enter the maximum model time value When the simulated process time exceeds that value simulation is automatically stopped Then the time of the last computed point of f t curves is equal to the value entered in the text box Final time The static text field Number of time values in memory displays the current number of stored model time values i e the current maximum number of points in the computed AA curves The purpose of the static text fields Model curve and array data amount Used memory and
140. ph otherwise t is equal to the current time of the graph x coordinate iPoint the number of the current point in the current sequence of formula argument values nPoints the total number of points in the current sequence of formula argument values iTime the number of the time value a in time graphs if the formula argument values are equal to the stored model time values then iTime iPoint otherwise iTime 0 b in coordinate graphs if the formula argument values are equal to the stored model coordinate values or if the formula contains references to linearly interpolated values of fx t model functions i e FXn then iTime is equal to the number of the stored model time value that is closest to the current time of the graph otherwise iTime 0 nTimes the number of stored model time values curGraph the sequence number of the graph to which the currently computed formula belongs The graph sequence number is also shown in the title bar of the graph window see Fig 1 1 If the current formula is an initialization expression of a parameter or an array then this variable is equal to the sequence number of the active graph window curForm the sequence number of the currently computed formula among all formulas that belong to the same graph window If the current formula is an initialization expression of a parameter or an array then this variable is equal to zero FTn the linearly interpolated va
141. problem Further on the case of two variables will only be discussed The independent variable will be denoted x and the dependent variable will be denoted y The independent variable is typically a quantity whose value can be controlled precisely i e its errors are negligible and the dependent variable y is the quantity that is determined empirically measured for each value of x Unlike with the values of x there is frequently a large uncertainty associated with the measured values of y for example due to imperfection of the measuring procedure For brevity the values of the dependent variable will be called responses The model is a function y x that makes it possible to predict estimate each response y y x k 1 2 n given the values of all parameters p i 1 2 m where n is the number of responses measurements and m is the number of parameters The measured data consist of the set of values x of the independent variable and corresponding measured responses vz The difference y vz of the k th estimated response and the corresponding measured response will be called the k th error When discussing nonlinear fitting it is convenient to treat the set of parameters as a one column and m row matrix consisting of the parameter values p i 1 2 m This column of numbers is called the parameter vector and denoted p further on vectors will be denoted in lowercase using bold italic font The estimat
142. re eerie ee TA devel i E ss REE ai 25 Sus G ISTA EESE PEE E E E a aaa 27 6 Time cross sections f x const t of f x model FUNCTIONS eee ceeceseceteceteceeeceeeeeeeeeeseeeseeeeeestees 30 7 Time limits and amount Of data dada kaikas i aaa aaa aaa aaa aaa aa aaa aaa E 31 Additional MES a is iki cosets sss ai ooo id ia i cape a S a a ai a a ia 34 9 Importing model data from text files 0 0 ceccecccesecesseeseeeseeeseecaecaecaecesecesecsseeseeeeeseeeseeeseecseecnaeesseenseenaeees 35 LOSS lider bars saaa aaa EE anes a a a o sear e A cates ideh cates stanton a A 35 IEEE Graphicakobjects a a I 37 TIE EXC ADE Sector ects ca Sos Were 205 seg aa i i i a S a i ta aa aa 37 11 2 Vertical and horizontal straight lines cc eeccesseesceesseceseeesceescecseecaecesecesecnaeeneeeseeeeeeeseaeeeseesaeseaaees 38 VES Freeform ES iorn cots ss a ihtideavine dies nda sate a a E oo A von shoes N OA EE E vas danse eens 39 11 4 Converting free form lines to function graphs ccccccccesseesecsteceseceseceseceeceeeeseeeeeaeeeseeeseeesaeeaeees 40 12 Keyboard and mouse Shortcuts ensar pa si ai a i a died ase i a i si 41 13 Blementary data analysis vis sasi ias ss i cay Va a eek sa a i a a a a a eR 42 1371 Linear Atte seksis asas as k a ai i ES T a a i k a Reeds 42 IEA EO EKET ATOI Ma EE O Hota sane daevane N tesa dou caeanthulbdgeuate Hines vaste EAE 43 13 3 Statistical analysis i enen eeeavodseegunteaderets dos ciuh e dina deeagouasGec
143. ree surface potential fo le R h o o o o o o o Ri Electron concentration 1 cm 3 _ Potential of the left edge of the system V Hole concentration 1 cm 3 Electron concentration 1 cm 3 t 0 03 800 4 Hole concentration 1 cm 3 t 0 03 0 005 01 O15 02 025 03 035 04 045 05 ts V 4 Electric field S m 32 t 3 Charge carrier concentrations at specific points La e x Electric field V cm Electron concentration 1 cm 3 x 8 Electron concentration 1 cm 3 x 10 Hole concentration 1 cm 3 x 8 Hole concentration 1 cm 3 x 10 T 03 035 04 05 05 Speed Y 2 7 Auto Uk 0 109523561 0 11 W tsyne V t syne all wsync x syne all Coe ars Language Fig 1 1 An example of the GraphiXT main window Operating system Windows XP SP2 or a newer Windows version The setup procedure is straightforward a standard installer is used Further on the f x f or At function values computed by the GraphiXT plug in such as CarrierFunc dll will be referred to as model data or model functions and the input variables used to compute those functions will be referred to as model parameters here the term model means a set of rules and relations between input parameters coordinate x and time that describe the simulated physical syste
144. rmulas 15 6a b The text box Smoothing margin is used to enter the number of initial points that are not smoothed a more detailed explanation of the smoothing margin is presented above 16 Using function data tables GraphiXT can display function data in table format Each data table is associated with a particular graph window and there is only one data table for each graph window Each table shows the data plotted in the associated graph window The table window of the active graph window can be opened by clicking the toolbar table button HJ see Fig 1 1 An example of a GraphiXT window with two open data tables is presented in Fig 16 1 When a table window is active the toolbar table button is replaced by the graph button I see Fig 16 1 It is used to activate the graph window corresponding to the current table When a table window is active the menu Graph options is replaced by the menu Table and graph options see Fig 16 1 Switching between the two windows is also possible using the corresponding menu commands Each table can be in one of two modes which determine the contents of the table 1 in partial mode the table only shows the data of the points that are visible in the graph window this is the default mode 2 in complete mode all the data associated with the graph window are shown in the table In partial mode the data values corresponding to the points that have been clipped or
145. rner of the GraphiXT main window see Fig 1 1 Note The program defaults and the information about the currently selected language are stored in the files GraphiXT defaults and GraphiXT lang which are in the GraphiXT installation folder usually C Program Files GraphiXT Since writing to files in C Program Files requires administrator privileges GraphiXT always runs with the highest permission level that it can Since the GraphiXT user interface is relatively simple it is possible to start using this program without any additional information If a user selects an improper option a message usually appears informing the user about the error Options of all lines and objects shown in the graph window i e curves coordinate axes additional lines text labels and the legend can be changed using the context menu which appears after right clicking the object The main options of the graph can be changed using the menu Graph options The majority of the commands of that menu are also included in the graph context menu which appears after clicking the right mouse button anywhere in the graph window If the model function file and the model parameter editor are loaded then the parameter editor is invoked using the menu command Simulation options Model parameters and simulation can be started or stopped using the menu command Start simulation or Stop simulation In order to load the model functi
146. rresponding to small values of the independent variable The points with much smaller absolute errors in this example those are the points that correspond to large values of the independent variable will have only a weak influence on the magnitude of F Even a large relative change of those points which exceeds their measurement error by an order of magnitude or more would not have a large influence on the optimum parameter values I e those points are essentially ignored In order to avoid such a situation in this example it would be better to use w 1 vp because then values of all terms in the SSE would be approximately equal As illustrated by this example the optimum choice of weight factors depends on the relationship between the magnitudes and errors of the measured responses If the absolute error of a measured response is constant then the SSE should be computed according to 14 1 1 and if the measurement error is proportional to the magnitude of the measured quantity then the SSE should be computed according to 14 1 2 If the measured quantity is distributed according to the Poisson distribution e g the number of particles emitted during radioactive decay of atomic nuclei then the optimum value of the error factor wx is 1 v because in the case of the Poisson distribution the standard deviation of each point from the corresponding statistical average is equal to the square root of that average In this case SSE is equal
147. rse values of a dataset that is linked to the same X dataset as the fitted dataset The four radio buttons that are on the left side of this group of controls are used to select one of the mentioned four error factor computation methods If one of the first three error weighting options is selected then the values of the constants e and f must be entered in the text boxes e and f The constant e can be defined in one of two ways by entering its value in the top one of the two text 57 boxes e see Fig 14 7 or by specifying that it should be equal to a predefined fraction of the average absolute value of the measured responses corresponding to the selected fitted dataset in the dataset options dialog that average value is denoted lt y gt In the latter case the mentioned fraction must be entered in the bottom one of the two text boxes e see Fig 14 7 In the example of Fig 14 7 the constant e is computed using the second method If the fourth error weighting option is selected then the error weighting dataset must be selected from the list that is opened by clicking the button Select see Fig 14 7 That list contains the names of all free curves formulas and model functions that are linked to the same X dataset as the fitted dataset After selecting an error weighting dataset from that list each term in the sum of squared errors will be divided by a squared value of the element of the error
148. s false then the result is 0 Although the comparison operators are most frequently used in the conditional construct if else and in the loop operator while they may also be used in arithmetic expressions For example if a 3 and b 2 then the expression a gt b 1 is equivalent to the expression 1 1 because the inequality a gt b is correct The expression a gt b 1 is equivalent to the expression 3 gt 3 because when parentheses are not used the arithmetic operations are computed before comparison operations see below about operator priority Since the inequality 3 gt 3 is incorrect the result of the latter statement is 0 Logical operators amp logical operation AND also called conjunction logical operation OR also called disjunction If both operands are non zero e g a gt b amp b gt 1 or a 2 amp b when a 3 and b 2 then the result of the logical operation AND is 1 If at least one of the operands is zero then the result of the logical operation AND is 0 The result of the logical operation OR is 1 when at least one operand is non zero For example if a 3 and b 2 the value of the expression b 2 gt a 2 lt a is 1 because the second inequality is correct If both operands are zero then the result of the logical operation OR is 0 Note There is no negation operator Instead there
149. s the value of the last computed expression However execution of a subroutine or a formula can be terminated at any time by calling the built in function Return x which was described in Section 4 1 A new subroutine can be only created in the list of subroutines That list is opened using the menu bar command Programming Subroutines or the context menu of the program editor or the array 18 editor or by clicking the button subroutine of the programming tools dialog see Fig 4 1 The list of subroutines can also be opened by clicking the toolbar button see Fig 4 1 An example of a dialog window with such a list is shown in Fig 4 8 A new subroutine can be created by clicking the button New or Insert before then a program editor window is opened An existing subroutine can be edited by clicking the button Edit or by selecting the command Edit subroutine from the subroutine context menu or by double clicking an item of the subroutine list box Similarly to arrays subroutines can be copied from one project to another using the five buttons Copy Copy All Cut Paste and Paste before see Fig 4 8 The subroutine list can be rearranged using the Up and Down arrow buttons the keyboard shortcuts are Ctrl and Ctrl V respectively As in the case of the list of arrays any change to the list of subroutines is applied
150. sacadacdusetuacesttade a Ea 45 L4Nonlinear fttit taiso hina i a ne een eae blaine a a sa 46 14 1 Elements of the theory of nonlinear fitting ccecsecseeseessecnseceseceseceseeseeesseeeeeeeeseeeaeecsaeeatees 46 14 1 1 The used terminology and formulation of the Problem ccccccccccccesscecseessetseeseeseeneeneenseeneeees 46 14 1 2 Choice 0f WeIGhtJACIOP S S sassin asas sa as iai ves a Va i Coan a ii T a 47 14 1 3 The mathematical formulation of the principle of least squares JLL aa aaa aaa aaa 48 14 1 4 Parameter confidence intervals cccccccccscesesssscesscetscessceeceeeceeseeeseeeseceseesseeessecsaecssecaecnseeneeeaeeaas 49 14 1 5 The Levenberg Marquardt method ccccccccccccscccesccesccesscensceesceeeeeeseseseesaeesseesseecsaesssecaesuseeeeenaeeaas 51 14 16 The Simple methil siste aina ania ia ii a a EE vi i Ad 51 14 2 Nonlinear fitting with GraphiXT LLL aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa 53 14 2 1 Definition of fitting functiONS LLLL aaa aaaaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa 53 14 2 2 Selection of varied parameters and parameter properties ccccccccsccccsecssecssecsteestecnseceseseseeneeess 54 14 2 3 Definition of fitted data sets and their PrOPerties cccccccccccccecseenseesseesseessecssecsseensecneseeeneeas 55 14 2 4 General nonlinear fitting options ccccccccccececccecscesscessceesceeeceeeceeseseaeeeseeeseeessecssesssecaesesesuseenaeenas 58 14 2 5 The fitting
151. side of the dialog window Nonlinear r A Selection of varied parameters j x Rares Graph4 F1 Global formula parameters Local parameters of the selected formula ct b Deselect All Select All Model parameters T Temperature a x0 System left edge coordinate I0 Initial photon flux density til Time of the first pulse Fmax Number of photons in one pulse TI Time interval between pulses wil Weight factor abs1 Absorption coefficient w Layer thickness epshf High frequency dielectric permittivity 1 Mobility parameter a mu a exp b sgrt E m Mobility parameter b mu a exp b sgrt E D1 Diffusion coefficient Select All Cancel Fig 14 5 An example of the parameter selection dialog window 54 least squares fitting see Fig 14 1 If the text box Upper bound or Lower bound is empty then the selected parameter is unlimited in the positive or negative direction respectively Parameter normalization is a parameter transform that decreases differences between values of different varied parameters When those differences are very large i e several orders of magnitude then elements of the matrix of second derivatives of the SSE 14 1 13 also differ from each other by several orders of magnitude This difference may cause a loss of precision when computing elements of the inverse matrix when the LM method is applied the inverse matrix is computed at eac
152. skipped or which belong to hidden curves are absent in the table I e in partial mode there is an exact correspondence between the data points visible in the graph and the data values shown in the table In complete mode all data values of each function plotted in the graph window including the hidden curves are present in the table regardless of whether those points are visible in the graph window The table mode can be changed by selecting the table window menu command Show all data or Show only the data visible in the graph or by clicking the corresponding toolbar button which appears to the right of the button I or H when the table window is open see Fig 16 1 The toolbar button W is used to change the table mode into complete After clicking that button it is replaced by the button Lae which is used to change the table mode into partial In partial mode a focus cell of the data table always corresponds to the current point of the graph That point is indicated in the graph window by a red cross see Fig 16 1 Selecting a point in the graph A GraphiXT Electrophotographic layer discharge gxt 2 Charge carrier concentrations Ls X File Show Edit Window Table and graph options Data analysis Pr ee concentration 1 em43 n a gaw he E Oav x O Point No 165 301 8 2 Y 1 333359e 014 kK 2 Charge carrier concentrations 3 XL FX6 FX12
153. so that precision of those values may have a noticeable effect on simulation results In this case the optimal option is Primary functions because this would cause all function values to be computed with the maximum precision afterwards the values of the secondary functions will be rounded but this rounding will not have any effect on simulation results because the secondary functions are not used to compute any other quantities Note If the computer memory already contains stored model data then an increase of precision of f x t model functions would cause an increase of the memory used to store all previous values of those functions not just the values that will be computed in the future However the actual previously computed values will not change thus they will not become more precise in the true meaning of this word In order to ensure that all stored values have the same precision the entire simulation must be repeated from the start 33 8 Additional times The previously described dialog window Time limits and amount of data makes it possible to specify only the initial and final times of simulation All intermediate times are computed by the currently loaded GraphiXT plug in that solves kinetic equations such as the charge transport simulator CarrierFunc dll However the user may define additional time values that should be inserted into the final set of stored model times For example those a
154. sponding axis interval then the text label that is associated with that point is also not visible If the bottom item is selected in the bottom list box then it is possible to enter the reference x or y coordinate in the text box X or Y 37 Font Times New Roman z Italic Underline Font size Direction Horizontal Color Black Border Horizontal position Vertical position Constant distance to Constant distance to Left edge of the window Top edge of the window Left T axis Top X axis Vertical central line Horizontal central line Right V axis Bottom X axis Right edge of the window Bottom edge of the window Constant distance to line const Constant distance to line Y const x label left edge coordinate Y label top edge Y coordinate x label center X coordinate Y label center T coordinate x label right edge coordinate Y label bottom edge Y coordinate Y x Y text will only be visible when xMin lt X lt Mas text will only be visible when yMin lt Y lt yMax a Font Italic F Bold Underline Font size Direction Bottom to top Color Black Border Horizontal position Vertical position Constant distance to Constant distance to Left edge of the window Top edge of the window Left Y axis Top axis Vertical central line Horizontal central line Right axis Botto
155. t or by selecting the command Edit parameter from the parameter context menu An example of the parameter properties dialog window is also shown in Fig 4 3 Global parameters SR Edit New Insert before Duplicate Delete Delete All 0 Name and value PAR1 2 PAR2 3e 015 a 2e 015 b 3 Parameter name and value Name 2e 015 Expression parl par2 b Calculate Cancel Copy Copy All Cut Paste Paste before Calculate Calculate All OK Apply Cancel Fig 4 3 An example of a list of global parameters and a dialog window with parameter properties 14 The purpose of buttons of the parameter list dialog window is self explanatory see Fig 4 3 A group of five buttons Copy Copy All Cut Paste and Paste before makes it possible to transfer parameters between different projects or to rearrange the current list A single parameter can also be copied pasted or cut using the keyboard shortcuts Ctrl C Ctrl V and Ctrl X The difference between Paste and Paste before is that Paste places the new parameter at the end of the list while Paste before places the new parameter before the currently selected one If a parameter is selected then the keyboard shortcut Ctrl V performs the action Paste before and if there is no selected parameter then
156. t model function No 2 at the current moment of time thus by using formulas it is possible to display current values of time functions ft in coordinate graphs The red curve Conduction current is a free curve obtained by copying and pasting a certain f x t const model function in the formula text of Fig 4 1b this free curve is referred to as FC1 The blue curve Displacement current represents the difference between the mentioned two functions FT2 FC1 4 2 Lists of global and local parameters The list of global or local parameters is opened by selecting the menu bar command Programming Global parameters or Programming Local parameters or by selecting the same command from the context menu of the program editor or the array editor or by clicking the button Global parameters or Local parameters of the programming tools dialog see Fig 4 1 The list of global parameters can also be opened by clicking the toolbar button 2 1 see Fig 4 1 The list of local parameters can only be opened when a program editor window is active An example of a dialog window with a list of global parameters is shown in Fig 4 3 the list of local parameters has the same format Parameter properties i e name value and expression can be modified in the parameter properties dialog window which is opened by double clicking the corresponding item of the parameter list or by clicking the button Edi
157. t general expression of the sum of squared errors F is F gt wily p 14 1 3 k l 14 1 1 corresponds to w 1 and 14 1 2 corresponds to w 1 v x The factor w k 1 2 n before the k th squared error is called the weight factor of that squared error The factor w i e the square root of the weight factor will be called the error factor i e omitting the words weight and sguared In order to utilize the measurement data as fully as possible the weight factors must be chosen so as to ensure that contributions of all available points to the value of Fo are approximately equal If the used theoretical model is correct i e if the only reason of non perfect fit is the errors in the measured responses then the standard deviation of each measured response from the optimum fit line is approximately equal to the measurement error of that point Thus the error factors should be inversely proportional to the corresponding measurement errors For example let us assume that as the independent variable increases from the minimum to the maximum value the measured responses decrease 100 times but their relative error stays constant i e the absolute error of each measured response is proportional to the measured value If all error factors are equal to 1 as in 14 1 1 then the main contribution to the value of F will come from the points with the maximum absolute error In the discussed example those are the points co
158. t the beginning ofa step max F min Reflection a b Reflection and expansion Contraction c LA Proportional contraction d T F min Fig 14 3 Possible outcomes for a step in the simplex method when there are three varied parameters The simplex at the beginning of the step is shown at the top The simplex at the end of the step can be any one of a a reflection away from the high point Fmax b a reflection and expansion away from the high point c a contraction along one dimension from the high point or d a contraction along all dimensions towards the low point Finin An appropriate sequence of such steps will always converge to a minimum of the function F This diagram is based on Fig 10 4 1 of the book Press W H Teukolsky S A Vetterling W T Flannery B P Numerical Recipes in C 1997 Mead method is to define an initial simplex in the parameter space In each iteration that simplex is modified so as to decrease the value of the SSE corresponding to a particular vertex of the simplex The simplex method does not use derivatives with respect to varied parameters Hence there are fewer requirements for the fitting functions than in the LM method e g those functions may be non smooth However when the fitting functions are smooth the simplex method is much slower than the LM method All possible transformations of a three dimensional simplex during fitting are shown in Fig 14 3
159. tant rate Then this button turns into the Pause button MU By clicking the latter button the slider animation is turned off The second of the two buttons is used to open the dialog window for entering the time slider animation options such as the average rate of slider position change the time interval between slider position updates i e time slider steps and the change of the current time corresponding to one such step The same dialog is opened by right clicking the animation speed slider which is described below The animation speed slider is used to modify the rate of change of the current time in the time slider animation mode The leftmost position of the speed slider corresponds to the minimum speed and the rightmost position corresponds to the maximum speed The default minimum speed is such that the maximum duration of animation is 30 s and the default maximum speed is such that the maximum duration of animation is 1 s those times can be changed in the mentioned dialog window of slider animation options The button Language for selecting the user interface language 11 Graphical objects In addition to curves that represent various functions each graph window can also contain objects of the following types 1 text labels 2 vertical or horizontal straight lines 3 free form lines 4 the legend The number of objects of the first three mentioned types is unlimited There c
160. the keyboard shortcut Ctrl V performs the action Paste A parameter can be selected or unselected by left clicking an item of the parameter list Most of the buttons perform actions on the selected parameter only There are three buttons which perform actions on all parameters of the current list Delete All Copy All and Calculate All After clicking the button Calculate All the parameter expressions will be evaluated in the same order in which the parameters are listed As evident from the example of Fig 4 3 each parameter may depend on other parameters of the same list Consequently the values assigned to parameters after clicking the button Calculate All may depend on parameters order in the list Action Calculate of the parameter list is especially suited for some one time or infrequent evaluations For example in order to invert a user defined square matrix A a parameter must be defined with expression Invert loc A Here Invert is the name of the built in function used for matrix inversion all built in functions are listed in Section 4 1 Then after calculating that parameter it will be assigned the value of the determinant of matrix A and that matrix will be replaced by its inverse matrix After clicking the button OK or Apply the current list of parameters is saved and all formulas depending on those parameters are recalculated After clicking the button
161. ther a function curve nor a free form line is selected i e when the point dialog window is not visible Cutting 1 e copying and deleting the selected curve or objects Ctrl X after selecting a curve an object or a group of objects Copying the contents of the active graph window to the clipboard in bitmap and Windows metafile formats Ctrl C when neither a curve nor an object is selected and the point dialog window is not visible Opening the text editor window Info Ctrl T when neither a function curve nor a free form line is selected i e when the point dialog window is not visible Saving the project file the default file name extension is gxt Ctrl S when neither a function curve nor a free form line is selected i e when the point dialog window is not visible 41 13 Elementary data analysis GraphiXT can be used for elementary data analysis and for nonlinear fitting In this section the elementary analysis will be described and the next section will deal with nonlinear fitting In the current version of the program v1 21 the following types of elementary analysis can be performed 1 linear fitting 2 integration 3 statistical analysis The elementary analysis of plotted function data is done by selecting the menu command Data analysis Elementary analysis then selecting the curve in th
162. tion causes a change of the time axis limits In other words the time interval of the time graph shifts along with the current time if the current time is changed either by moving the slider or by entering its value in the text box at the bottom of the main window This change of the time limits can be either smooth or step like depending on the state of the checkbox lt gt see Fig 1 1 If it is unchecked then the change of the time limits is smooth so that the difference between the current time and any one of the two time limits stays constant If that checkbox is checked then the limits of the time axis don t change while the current time is between them Those limits only change when the current time becomes greater than the upper limit or less than the lower limit In this case the magnitude of the mentioned change is always equal to a multiple of the difference between the upper and lower limits of the time axis General suggestions e In order to tile the windows so that they do not overlap and fill the entire area of the main window the menu command Window Tile must be selected The window order is determined by the order in which they were activated prior to selecting that command the active window will be the uppermost one in the first column the previously activated window will be below it etc e The title of the active graph can be changed by selecting the menu command Window Rename The same
163. tions computed using the built in function Inti Example No 4 In this example the convolution of two identical Gaussian functions is computed I exp x exp t x dx The result is a Gaussian function whose variance is twice the variance of the Gaussian function exp Fe S es Inti exp x x exp t x 2 x 0 2 FL LS amp x exp t t 0 8 0 6 0 4 0 2 10 5 0 5 10 t Fig 4 12 An example of using the built in function Inti to compute a convolution of two functions 24 4 7 Runtime error handling A runtime error is an error that occurs during execution of a program and which can not be anticipated at compile time In order to minimize the risk that a runtime error in a user program will crash the GraphiXT executable causing loss of all unsaved GraphiXT data GraphiXT checks for the following types of errors at runtime 1 out of bounds array indices e g negative indices or indices greater than the number returned by the built in function Size2 loc a i 2 references to un allocated arrays 3 abnormally long computation e g due to an infinite loop 4 infinite recursion 5 runtime errors related to limitations of certain built in functions 6 runtime errors in user defined DLL functions see also Section 4 5 An abnormally long computation of a formula value such as caused by an infinite loop is ha
164. to 2 pay i 14 1 4 k l Vk When some of the measured responses v are zero F can not be computed according to 14 1 2 or 14 1 4 because then some of the terms in the SSE would have zero denominators In such a case a small term may be added to 4 or the SSE may be computed according to 14 1 1 Fig 14 2 illustrates influence of a choice of the weight factors on fitting results in the presence of large statistical scatter in the measured responses Evidently a correct choice of the weight factors can improve accuracy of parameter estimates The parameter estimates are most accurate when the error factors are equal to inverse standard deviations of the measured responses The corresponding SSE is equal to 47 Model AN A exp A10 B exp 45f Nonlinear fitting results A 124 11 A 0 0318 0 0048 s 7 B 40 0 6 9 2 0 00553 0 00056 s True parameter values A 100 4 0 0284 s B 40 A 0 00481 s 0 100 200 300 400 500 600 t s Fig 14 2 The decay curve of a mixture of radioactive isotopes of silver Ag and Ag The solid line is the optimal theoretical curve the model function with the optimal parameter values and their standard deviations are given at the top of the graph This curve was obtained by minimizing the statistical SSE defined by 14 1 4 The dashed lines are optimal curves corresponding to other definitions of the SSE 1 minimizing the absolute deviation S
165. u then checking the checkboxes rescale after a change of curves and rescale after a change of current time or of X axis limits in the dialog window Y axis options The same dialog window can also be opened by selecting the command Graph options Margins and axis options from the menu bar or from the graph context menu e The current time can be changed in three ways a by dragging the time slider which is at the bottom left of the main window b by activating any one of coordinate graphs and entering the time value into the current time edit box it is seen in Fig 1 1 c by entering the needed value in the dialog window that is opened by selecting the menu command Graph options Axis and slider limits the same command can also be selected from the graph context menu Modification of the current time can be made more convenient when the current time marker is shown in time graphs In order to display the current time marker the time graph must be activated and the menu command Graph options Show current time marker must be selected the same command can also be selected from the graph context menu After selecting that command a vertical line indicating the current time will appear in the active time graph it will only be visible when the current time belongs to the time axis range of that graph When the current time marker is visible the current time can be changed in two additional ways by drag
166. ubroutine may find any one of them depending on initial parameter values and other fitting options In this case the minimum that was found may depend on the choice of the fitting method After entering all fitting options that were described in Sections 14 2 1 through 14 2 4 and selecting the number of iterations the fitting process can be started This is done by clicking the button Start fitting see Fig 14 1 During fitting that button turns into the button Stop fitting which must be clicked in order to interrupt fitting During fitting the current values of varied parameters which are shown in the column Value of the dialog window Nonlinear least squares fitting are periodically updated Additional information about the fitting process is shown in six or seven static text fields which are at the bottom of the dialog window Nonlinear least squares fitting Those text fields show 59 1 the total number of fitted points n 2 the number of varied parameters m 3 the number of degrees of freedom df n m 4 the current iteration number 5 the current value of the SSE 14 1 3 taking into account the mentioned error factors w 6 the current value of the reduced SSE 14 1 6 7 the probability to obtain the value of SSE that does not exceed its current value if the standard conditions are satisfied i e if the null hypothesis is true the standard conditions were formulated
167. urement data by a given model however as mentioned only one of those experimental data sets is available We will also assume that the model is correct i e there are no systematic errors Although all measurements are done at the same conditions the optimal curves corresponding to different data sets will be different due to random measurement errors If the SSE is computed for each theoretical curve using the given experimental data set then the majority of obtained values will belong to the interval F lt F p lt F 14 1 9 where Fy and Fy the optimal values of the SSE and the reduced SSE corresponding to the given experimental data set From the definition of F 14 1 6 it follows that Fo n m As it will be shown below the inequality 14 1 9 defines an m dimensional ellipsoid whose center is at p po that point corresponds to the given data set fitting results If the measurement errors are distributed normally then the projection of that ellipsoid to each parameter axis defines the 68 3 confidence interval of that parameter If the value of F is known then the width of that interval can be computed as follows The function F p is expanded in Taylor series in the vicinity of the point po retaining only the zero and second order terms Using the rules of matrix multiplication and addition this expansion is F p s Fyt s G s 14 1 11 s P Po 14 1 12 where s is the transposed vector s i e the row of the
168. weighting dataset that corresponds to the same X value The name of the currently selected error weighting dataset is shown in the text box z see Fig 14 7 If there are several fitted datasets then the final optimal values of varied parameters will be most affected by the dataset whose contribution to the overall SSE value F is greatest If equal values of weight factors are applied to all fitted datasets and if y values corresponding to different datasets differ by several orders of magnitude then the datasets with the least y will have negligible influence on the optimal parameter values When the error factors defined by 14 2 2 or 14 2 3 are used values of the SSE terms corresponding to different fitted datasets are made more similar to each other Consequently the use of those error factors makes it possible to decrease the differences between contributions of different datasets to the overall SSE However those differences may still be unacceptably large if different datasets contain a very different number of observations If values of all terms in the SSE are roughly equal then the datasets with the largest number of points will make up the major part of the overall SSE In order to eliminate this effect the checkbox Divide SSE by the number of points should be checked Then the part of the SSE corresponding to the selected dataset will be additionally divided by the number of points of that dataset n I e an
169. xisting X datasets menu command Create free curves and link them to an existing X set 66 Coding credits The code of the program editor is based on the code included in the article Crystal Edit syntax coloring text editor by Andrei Stcherbatchenko http Avww codeproject com Articles 272 Crystal Edit syntax coloring text editor The code for built in integration functions was taken from the QUADPACK library for numerical integration of one dimensional functions available from Netlib http www netlib org guadpack The code for built in function Root and Root2 was taken from the HOMPACK library for solution of systems of nonlinear equations available from Netlib http www netlib org hompack The code for built in functions erf x error function and erfc x complementary error function was taken from the free math library FDLIBM available from Netlib http www netlib org fdlibm The code for built in functions gamma x gamma function Ingamma x natural logarithm of the absolute value of the gamma function and gammp x incomplete gamma function is based on the code provided in the book Press W H Teukolsky S A Vetterling W T Flannery B P Numerical Recipes in C 1997 The code for the function data tables array viewer and for the grid controls in several dialog windows is based on the code included in the article MFC Grid control 2 27 by Chris Maunder http Awww
170. y the following three controls The text box Maximum interval between inflection points is used to specify for how long the program should remember the position of the last inflection point This parameter is expressed as a number of points N If inside a sequence of N 1 points beginning with the last inflection point another inflection point is found then the part of the curve consisting of those two inflection points and all points between them is smoothed and the second mentioned inflection point becomes the starting point of the next sequence of N 1 points to be tested for inflection Conversely if after scanning those M points no other inflection point is found then that part of the curve is not smoothed and the program begins the search for an inflection point to be used as a starting point of the next N 1 point sequence This smoothing method is based on dividing all the points of the smoothed sequence into triplets the third point of each triplet coincides with the first point of the next triplet Each of those triplets is first replaced by three points obtained by linear fitting then the first and third points are modified so as to ensure that the curve remains continuous and that its integral over each triplet is the same as before smoothing Thus this smoothing method does not change the integral of the smoothed function this can be ascertained by comparing the numbers shown in the text fields Average before
171. y to obtain the value of Fo less than the one obtained during nonlinear fitting when the standard conditions are satisfied 1 e when all differences between the optimal theoretical curve and the experimental data are only due to random measurement errors which are independent and distributed normally This probability is easiest to compute when the weight factors are equal to w 1 0 i e when F and F x Then assuming that the standard conditions are true the statistical distribution of the quantity F y is well defined and depends only on the number of degrees of freedom n m That distribution is called the 7 distribution The number of degrees of freedom will be denoted df df n1 m 14 1 18 and the mentioned probability will be denoted P g df An approximate expression of that probability is df 1 1 Ho 2 x 7 P 7 df 20 p De dr 14 1 19 xo df 3 2 rera e where yis the incomplete gamma function which is defined by the expression 4 6 After calculating this probability it is compared with certain critical values One of them is close to zero for example 0 005 and another one is close to 1 for example 0 995 If P x df lt 0 005 this means that the value of x is too small if the standard conditions were true then the probability to obtain such a small value of y would be less than 0 005 If P x df gt 0 995 this means that the value of y is too large if the standard condit

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