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QCMatPack Matrix and Numerical Analysis Software - Quinn

1. aa 0 0 512 Kaiser Bessel wo eset epee 0 5 0 0 Digital Signal Processing 109 You can also apply your own custom window to the raw data Just apply your windowing function to the raw data and call the FFTCale method that does not use a windows parameter Or you can call the FFTCalc method that does specify a window parameter and in that case specify DPS DSPWIN NOWIN window since that passes through the raw data without modification Class DSP The source data for an FFT can take real only or complex form The raw data can be passed to the DSP class as simple double arrays or as DDMat and DXMat array classes DSP constructor Create a DSP object using double arrays public DSPL double re double imag he re Array of real values of the raw data imag Array of imaginary values of the raw data This parameter can be null for a FFT that starts with real data DSP constructor where the raw data is real only and passed in using a DDMat matrix object public DEPT DDMat source Parameters source Source vector of fftreal data DSP constructor where the raw data is complex and passed in using a DXMat matrix public DSP DXMat source Parameters source Source vector of fft complex data 110 Digital Signal Processing The constructors do not calculated the FFT They just store the raw data Calculate the actual FFT using one of the following FFTCale methods Method DSP fFTCalc Calculate
2. 140 Using QCMatPack to Create Windows Applications File system Import resources from the local file system LESE aO metadata OS com a06 Eclipse_QCMatPackJayaExamples eC Ib H O S RedistributableFiles E QI Expand the Quinn Curtis java directory and check the com quinncurtis matpackjava folder Using QCMatPack to Create Windows Applications 141 File system Import resources from the local file system ys Cs Quinn Curtis java a eee C E README txt 1 amp metadata 2 H com ED qunncurtis OJE chart2djava 06 chartadjava H H amp matpackjava H O spechartjava aoe docs ee al gm An IN La 5 Select Finish and the example program folders under com quinncurtis matpackjava examples will be added to the project The entire contents ofthe example program folders will be copied so com quinncurtis matpackjava examples directory structure is recreated under the Eclipse QCMatPackJavaExample folder 142 Using QCMatPack to Create Windows Applications clipse SDK E A com quinncurtis matpackjava examples ComplexMath E com quinncurtis matpackjava examples CubicSplinesTest E E com quinncurtis matpackjava examples CurveFitTest 28 com quinncurtis matpackjava examples EigenComplex E E com quinncurtis matpackjava examples EigenRealsym EB com quinncurtis matpackjava examples EigenRealUnsym EB com quinncurtis matpackjava examples FFTTest 4B com quinncurtis m
3. Initialize data curvefit new CurveFit A RHS cftype getCurveFitAlgorithm 100 Curve Fitting switch cftype case 1 curvefit polynomialSolve solveOrder Coefficients regstats break case 2 curvefit exponentialSolve Coefficients regstats break case 3 curvefit rationalSolve solveOrder Coefficients regstats break case 4 curvefit mixedSolve solveOrder Coefficients regstats break default curvefit polynomialSolve solveOrder Coefficients regstats break fstats curvefit getFStats coefdev curvefit getCoefficientDev Cubic Splines The cubic splines algorithm starts with the assumption that the curve between each contiguous pair of data points Xo Yo X1 Y1 x ls Y1 X2 y2 X y2 X3 y3 Sn l Yn 1 Xn Yn is best modeled using a unique cubic polynomial of degree 3 equation If you start with 0 n data points for a total of n 1 data points you end up with cubic equations one for each segment between the data points These cubic equations are used interpolate an estimate for f x for any x in the interval xo x Additional constraints force continuity between the first and second derivatives for each curve as you pass from one segment to the next within the interval xo Xxn The continuity of the derivatives is what gives the resulting curve a smooth look analogous to the wooden spline that was once used by draughtmen to draw a smooth line through a series of po
4. double I arbata 1 60672 9787 5 0y6 1 7 4 y MR 4 8 2 215 double ITI iData 8 7 2 2 6 1 T 4225 3A 5 4 8 1 2 0 double brData 0 0 20 10 double biData 60 60 50 DXMat A new DXMat arData aiData DXMat b new DXMat brData biData DXMat x new DXMat A lES Solve b x QR Factorization LESXOR lesqr new LESXOR A lesqr solve b x LU Decomposition LESXLU leslu new LESXLU A leslu solve b x Gauss Jordan LESXGJ lesgj new LESXGJ A lesgj solve b x double aSPDDataR 13 27 Ab 2 22 4 1 4 IS ke double aSPDDatal 0 el 2 0 4 1 4 O DXMat ASPD new DXMat aSPDDataR aSPDDatal LESXCholesky leschol new LESXCholesky ASPD b DXMat makeRHSFromCoefs ASPD leschol solve b x Gauss Jordan Class LESGJ LESXGJ Systems of Equations 63 The LESGJ and LESXGJ classes implement the Gauss Jordan with partial pivoting algorithm the most stable version of the Gauss Jordan algorithm Gaussian elimination turns the original square non singular system of equations into one that is upper triangular using a technique of subtracting multiples of one equation from another Once the system of equations is upper triangular the solution is solved for using simple back substitution The partial pivoting enhancement of the algorithm ensures that extremely 64 Systems of Equations small values a
5. etc operators make manipulating complex numbers as simple as possible Second QCMatPack implements several generalized matrix classes the most important of which are the DDMat for real number matrices and the DXMat class for complex number matrices These are used for the input and output of data to the numerical analysis routines and the manipulation of real and complex number matrices The matrix classes are designed so that there is no practical limit other than the amount of free system memory to the size of matrices The DDMat and DXMat are symmetrical in that the same matrix math and linear algebra routines are available for both real and complex numbers Third the library implements the best of the algorithms we could find in numerical analysis These include LU Decomposition QR Factorization Cholesky and Gauss Jordan for the solution of real and complex linear systems QL QR and LR routines for the solution of real and complex eigenvalue eigenvector problems FFT s for real 8 Class Architecture and complex data multiple and stepwise multiple regression curve fitting and thermocouple linearization e Fourth a pair of matrix viewer controls one for real DMatViewer and the other for complex matrices XMatViewer let you add scrollable matrices to your Windows form based application programs QCMatPack for Java Class Summary The following categories of classes realize these design considerations Complex nu
6. Return Value Returns the determinant Method DDMat equals DXMat equals Equals comparator Two matrices are considered equal if they are the same dimension and if each corresponding element at matrix index i j is equal DDMat boolean Jequals DMatBase obj DXMat boolean equals XMatBase obj Parameters obj Returns true if the source object is equal to the current object Return Value Returns true ifthe source object is equal to the current object Method DDMat getDataBuffer DXMat getDataBuffer Returns a reference to the internal DComplex data buffer that stores the matrix data DDMat double getDataBuffer 44 Matrix Classes DXMat DComplex getDataBuffer Return Value Returns a reference to the internal matrix buffer Method DDMat getDiagonal DXMat getDiagonal Returns a vector that is the diagonal of the matrix DDMat DDMat getDiagonal DXMat DXMat getDiagonal Return Value Returns a vector that is the diagonal of the matrix Method DDMat getElement DXMat getElement Returns the element of the matrix DDMat le getElement int element Set an element of the vector Element int row int column Returns the element of the matrix DXMat le getElement int element Set an element of the vector Element int row int column Returns the element of the matrix Parameters Matrix Classes 45 row The row of the
7. 0 C to 400 C 0 mV to 20 872 mV 0 03 C to 0 03 C Thermocouple Linearization 123 500 C to 1372 C 20 644 mV to 54 886 mV 0 05 C to 0 06 C 600 C to 1300 C 20 613 to 47 513 mV 0 04 C to 0 02 C 1064 C to 1664 5 C 1664 5 C to 1768 1 C 11 361 to 19 739 mV 19 739 to 21 103 mV 0 001 C 0 002 C to 1064 C to 1664 5 C 1664 5 C to 1768 1 C 10 332 to 17 536 mV 17 536 to 18 693 mV 0 0002 C to 0 0002 C 0 002 C to 0 002 C There are no errors in the temperature to voltage conversion conversions Our routines use the temperature to voltage polynomials specified in NIST ITS 90 Those polynomials define the voltage to temperature curves All errors are measured with respect to those curves Cold Junction Compensation When making thermocouple measurements you must take into account the effect of the temperature of the terminal block terminating the non hot end of the thermocouple This is referred to as cold junction compensation Typically this involves making a temperature measurement of the terminal block using a non thermocouple based measuring device usually a solid state temperature sensor the Analog Devices AD590 or a precision thermistor The terminal block temperature is converted to an equivalent 124 Thermocouple Linearization thermocouple voltage use our tempToMilliVolts method and the resulting voltage added to the measured thermocouple voltage The compensated v
8. 3 0 4 0 double rl 5 0 ff 3 5 ci 2 HH 6 07 DComplex c3 DComplex add DComplex subtract cl c2 rl 6 0 DComplex multiply DComplex a DComplex b DComplex multiply DComplex a double d DComplex multiply double d DComplex a a bi c di ac bei adi bdi ac bd be ad i DComplex cl new DComplex 1 0 2 0 DComplex c2 new DComplex 3 0 4 0 double rl 5 0 ff 23 el it ri 6 07 DComplex c3 DComplex add DComplex multiply DComplex multiply cl c2 rl 6 0 0 26 Complex Numbers tic DComplex divide DComplex a DComplex b tic DComplex divide DComplex a double d tic DComplex divide double a DComplex d ac bd bc ad rer ara tare DComplex cl new DComplex 1 0 2 0 DComplex c2 new DComplex 3 0 4 0 double rl 5 0 I7 ga fel 7 22 F el 6 0 DComplex c3 DComplex add DComplex divide DComplex divide cl c2 rl 6 0 Mix all of the operators in complex math expressions DComplex cl new DComplex 1 0 2 0 DComplex c2 new DComplex 3 0 4 0 DComplex c3 new DComplex 5 0 6 0 DComplex c4 new DComplex 7 0 8 0 double rl 9 0 ff e5 0 Cel 7 It 63 11 0 64y il DComplex c5 DComplex subtract DComplex divide DComplex add DComplex multiply DComplex divide c 1 02 83 11 0 eal 233 Equality Operators Two complex numbers are equal ifthe real parts ofthe two numbers are
9. MAR ESTAS SCS vg aah ee ee es oka tae eR 12 Solutio of Linear Systems anne 13 Solution of Eigensystems a en Rio 16 Multiple Repression en ns Eis 17 FIRE Fittings peun N neuen 18 Digital Signal Processing ns ee ee de nella ee 19 Class Hierarchy of OCMatP ck 2 ee E Aa aaan aAA 20 3 Complex NUMDCIS u a ua 23 Complex Math Operators sesseseeseesseesseseesseesseserssressrssresseestesesstessesressresseserssressesee 24 Complex Math Functionssunsaen a RER RR 27 Miscellaneous Funcfions u a usausenere een nnlasnkeil 31 4 Matrix Classes DDMat and DXMat classes ccccscccesssccesscecesscecesscecssececsseeeesseees 33 Linear Algebra Terminology i370 sus rein 33 Matrix Constructors eins ehesten hoch 35 Matrix Operatorsn noeit ena anna ae 37 Matix Methode Soe ais a lan sa a E EA E 41 A Summary of all DDMat DXMat Methods 0 0 cc ceeccescceeseeeceenceeeseceeeeseeenseeesaeenes 52 Chapter 5 Solution of Linear Systems of Equations eeeennsennsennennnenen nee 59 Standard Interface for the Solution of Linear Equations eeeennnenneen 61 GAUSS OR AT insist ee ae ae E e AE 63 EU Decsmpostion az a area 64 OR Pactorl2a lat ee as ee 65 Cholesky Detemposttien unse en anal 66 Chapter 6 Eigenvalue and Eigenvector Solutions esseesseessseensnnneennnnnnnnnn 69 Standard Interface for Eigenvalue and Eigenvectors Solutions 72 QL Algorithm for Real Symmetric Matrices seesssessenssessnn
10. exhibit a discontinuity between points 2 and 3 where a step change occurs and the second derivative becomes infinite If you are close to a step change the cubic splines algorithm my produce while swings around in the evaluated function in the transition area Cubic splines can only be used to interpolate data points in the original fitted data interval defined by xo Xn If you attempt to calculate an f x value outside of the interval the algorithm will use the cubic equation of the nearest segment either the Xo xi or the x Xn segment This can result in wild swings that are not an accurate representation of the underlying function The second derivative of the function is assumed to be 0 at the endpoints xo and Xn This is known as a natural spline because it assumes the shape that a wooden spline would take if the ends were left free Class CubicSplines The CubicSplines class calculates the table of cubic spline equations used in interpolating values within the range of the original data CubicSplines constructor 102 Curve Fitting public cubicSplines DDMat x DDMat y Parameters xX A matrix of the independent variable x values y A matrix of the dependent variable y values CubicSplines interpolatePoint Method Interpolate a new y value given an x value and the previously calculated spline coefficients public double interpolatePoint double x E Parameters x The x value to interpolate
11. m 1 0 n 1 The software includes routines that will convert a base O matrix into a base 1 matrix In that case the base O matrix dimensioned to size mx n with array indexing of 0 m 1 0 n 1 is converted to a matrix size m 1 x n 1 that can be used with 1 based indexing in the range 1 m 1 n All of the data in the original matrix is shifted one column and one row so that all data fits in the new index range of 1 m 1 n While the 0 row and column indices in these converted matrices are still useable in a true base 1 algorithm they would never be accessed because they are considered out of range A base 1 matrix can be converted back to a base 0 matrix In that case the base 1 matrix dimension m 1 x n 1 is converted back to a matrix of with the dimensions m x n The data is shifted back one row and one column so that the data starts in the 0 row and column Matrix Constructors Define a matrix with the dimensions rows x columns DDMat DDMat int rows int columns DXMat DXMat int rows int columns Construct a matrix as a copy of the source matrix DDMat DDMat DMatBase source DXMat DXMat XMatBase source Construct vector as the column or row rcflag V_ROW or VCOL of the source matrix The index of the column or row to copy is rcnum DDMat 36 Matrix Classes DDMat DDMat source int rcflag int rcnum DXMat DXMat DXMat source int rcflag int rcnum Construct a ma
12. volts for example taken at discrete and possibly periodic times In such cases a table of the measured variable versus time is produced The goal of the research is to uncover the underlying function f x measured for the interval xo xn Once the underlying function is discovered the solution is usually only an approximation of the true function the function can be used to calculate the values of f x temperature in this case for x values points in time which were not part of the original measured data Polynomials are generally used as approximating functions because of their simplicity and speed of calculation on computers There are several ways to use polynomials to approximate a discrete function The first is to fit a single polynomial to a group of data points This is called polynomial curve fitting and that is what our CurveFit class does The second method starts with the assumption that the curve between each contiguous pair of data points is best modeled using a unique cubic polynomial of degree 3 equation This is called cubic splines and that is what our CubicSplines class does Polynomial Curve Fitting Polynomial curve fitting results in a single polynomial equation of order n which is the least squares approximation of the original data Eqn 1 Y C C X C X C X Ca X Predict what the value of Y will be for a given X by plugging a value for X into the equation and solving it for Y One of the most com
13. 1 2 order regstats Returns curve fit statistics The RegStats class is described under the CurveFit polynomialSolve section Return Value Returns an error code Method CurveFit mixedSolve This method fits the data to a mixed polynomial equation of the form 2 3 y apta x a x a x an x b x by x ve ba x Curve Fitting 99 c In x The mixedSolve method returns the solution coefficients ao an bj bn and c in the equation above public int mixedSolve int order DDMat regcoef RegStats regstats E Parameters order Set the maximum order of the polynomials regcoef Returns the calculated curve fit coefficients The a coefficients are returned in array element 0 order the b coefficients are returned in array elements order 1 2 order and c is returned in element 2 order 1 regstats Returns curve fit statistics The RegStats class is described under the CurveFit polynomialSolve section Return Value Returns an error code Curve fitting example Extracted from the example program CurveFitTest CurveFit curvefit null int numObs 100 int simOrder 6 int solveOrder 3 int cftype 0 double simErr 0 1 DDMat A new DDMat Independent variable matrix DDMat RHS new DDMat Dependent variable matrix DDMat Coefficients new DDMat DDMat fstats null DDMat coefdev null RegStats regstats new RegStats
14. 64 65 66 67 72 73 75 76 83 86 90 96 111 113 118 DMatViewer 8 9 10 20 103 104 117 118 119 DSP viii 6 7 9 19 21 105 107 108 109 110 111 112 113 114 115 DTriplet 19 20 21 36 54 DXMat vii 2 7 8 11 12 13 19 20 23 33 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 62 63 72 77 79 80 109 110 111 118 119 EigenBase 16 21 69 74 EigenQL 8 16 21 69 71 72 74 EigenQR 8 16 17 21 23 69 71 72 74 75 76 EigenQR2 8 12 16 17 21 23 69 71 72 74 76 77 Eigensystem solver base class Complex 8 12 16 17 21 23 69 71 72 77 78 79 Eigensystem solver base class Real 16 21 69 74 Eigenvalues 69 71 75 Eigenvectors vii 71 72 EigenXBase 16 21 69 77 78 80 EigenXLR 8 12 16 17 21 23 69 71 72 77 78 79 Error processing 20 21 47 129 130 131 Exponential curve fitting 9 18 21 93 94 96 97 98 99 Fast Fourier Transform viii 6 7 9 19 33 105 106 107 109 110 111 112 113 114 115 Gauss Jordan Complex 8 12 13 14 15 21 23 59 63 64 67 Gauss Jordan Real 8 13 14 15 21 59 62 63 64 67 Integer vector 20 21 83 86 90 IVect 20 21 83 86 90 Java Applications 6 143 Least squares linear regression 8 17 18 21 82 83 84 85 86 87 88 89 LESBase 13 14 20 59 LESCholesky 8 13 14 15 21 59 62 66 67 LESGJ 8 13 14 15 21 59 62 6
15. A EValues EVectors residuals refnum QR Algorithm for Real Unsymmetric Matrices Class EigenQR The EigenQR class will calculate all of the eigenvalues a real symmetric matrix using the QR method If you request the eigenvalues and eigenvectors from this class it in turn calls the EigenQR2 class because that is the one that calculates eigenvectors for a real unsymmetric matrix EigenQR Constructor es and Eigenvectors 75 public EigenoR DMatBase a Parameters a The source matrix for the eigensystem calculations The source matrix must be square but it does not have to be symmetric Method calcEigen Values Calculate the real eigenvalues for the source matrix A Eigenvalues that have an imaginary part are discarded Since a real unsymmetric matrix can produce real and complex eigenvalues you probably want to be calling the caleEigenValues method next that accepts a complex matrix instead public int calcEigenValues DMatBase eigval Calculate the complex eigenvalues for the source matrix A public int calcEigenValues XMatBase eigval Parameters eigval Returns the eigenvalues of the source matrix A Return Value Returns an error code Method calcEigenValuesAndVectors Calculate the real eigenvalues and eigenvectors for the source matrix A Eigenvalues that have an imaginary part are discarded Since a real unsymmetric matrix can produce real and complex eigenvalues you probably want to be calling
16. A The numbers of the right hand side 1 2 0 in this case are usually referred to as the rhs or the matrix variable b The x s or unknowns are referred to using the matrix variable x When written in matrix format the system of equations above is written as Ax b where 1 2 A and b 0 The solution x to this particular set of equations is x 2 2 60 Systems of Equations The matrix representation is generalized for systems of equations of any size whether 2 x 2 or 10000 x 10000 The real number example above can be further generalized to the domain of complex numbers Solving large systems of equations is a critical element in almost every engineering discipline Typical applications include circuit analysis finite element analysis digital signal processing forecasting simulation econometrics linear programming robotics computer animation genetic engineering nuclear engineering etc The list is endless The first formal method for solving systems of simultaneous equations is usually attributed to Frederick Gauss the 19 century German mathematician His general algorithm was further improved over the years and appears here in the numerically most stable form Gauss Jordan with partial pivoting Other algorithms included in the software include LU Decomposition QR Factorization or Decomposition and Cholesky Decomposition Each algorithm has specific benefits and weaknesses you need to consider when
17. C DXMat C Returns the specified column or a sub range of the column as a vector DDMat C int col Returns the specified column as a vector C int col int start int num Returns a sub range of the specified column as a vector DXMat C int col Returns the specified column as a vector C int col int start int num Returns a sub range of the specified column as a vector Parameters col specified column start specified row to start copying from start number to copy Return Value Returns the specified column or a sub range of the column as a vector Method DDMat clone DXMat clone Returns an object that is a clone of this array object 42 Matrix Classes DDMat java lang Object clone DXMat java lang Object clone Return Value Returns a clone of this matrix object Method DDMat copy DXMat copy Copy a matrix DDMat void copy DMatBase src DXMat void copy XMatBase src Parameters STC source array Method DDMat determinant DXMat determinant DDMat D DXMat D Returns the determinant of the current matrix DDMat double D Returns the determinant of the current matrix double determinant Returns the determinant of the current matrix static double determinant DDMat A Returns the inversion and determinant of the specified matrix Matrix Classes 43 DXMat DComplex D DComplex determinant static DComplex determinant DXMat A
18. Get Set Returns the Standard error of see estimate Get Set significance of regression flag sigfla Get Set sum of residuals squared sumressq Method CurveFit exponentialSolve This method fits the data to an exponential equation of the form y ac The exponentialSolve method returns the solution coefficients a and b in the equation above All values of y must be positive public int exponentialSolve DDMat regcoef RegStats regstats Parameters regcoef Returns the calculated curve fit coefficients The a coefficient is returned in the array element 0 and the b coefficient is returned in the array element 1 98 Curve Fitting regstats Returns curve fit statistics The RegStats class is described under the CurveFit polynomialSolve section Return Value Returns an error code Method CurveFit rationalSolve This method fits the data to a rational polynomial equation of the form 1 2 3 ag tay x a x a x t aq x 1 b x b x b x bir The rationalSolve method returns the solution coefficients ao an and b bn in the equation above public int rationalSolve int order DDMat regcoef RegStats regstats E Parameters order Set the maximum order of the polynomial regcoef Returns the calculated curve fit coefficients The a coefficients are returned in array element 0 order and the b coefficients are returned in array elements order
19. Multiple Regression MulReg StepwiseMulReg SplitMulReg Regression analysis is a statistical tool for the evaluation of the relationship of one or more independent variables X1 X gt Xx to a single dependent variable Y for a fixed number of observations The primary goal of regression analysis is to obtain predictions of Y using the known values of X These predictions are calculated using a regression equation They are subject to error and are only estimates of the true values of Y There can be other goals of regression analysis for example to determine which of several independent variables are important and which are not for describing and predicting a dependent variable The MulReg class solves the generalized least squares multiple regression problem The StepwiseMulReg class includes forward backward and stepwise multiple regression algorithms that only includes independent variables the x values in the final solution that pass a F statistic significance test The SplitMulReg splits the original data into two develops a model using one dataset and uses that to predict the values in the second dataset 18 Class Architecture MulReg The least squares multiple regression algorithm calculates the coefficients for a set of specific independent variable x values The method also calculates a large number of parameters describing the quality of the regression the R value R squared value standard error of the estimate SEE th
20. N point DFT A 1024 point DFT calculated using the equation above would require 4 million floating point multiplications and 4 million floating point additions In the early 1960 s researchers notably Cooley and Tukey noticed patterns in the DFT calculation that when exploited properly could be used to reduce the number of complex multiplications to N Log N The number of floating point multiplications in a 1024 point DFT is reduced by 99 to 40 000 The entire class of Fast DFT algorithms is known as FFT Fast Fourier Transform As a historical note while Cooley and Tukey are usually given credit for the invention of FFT based on the publication of their algorithm in 1965 it is now known that the German mathematician Carl Frederick Gauss invented a near identical version ofthe FFT in the early 1800 s He used it as an aid in calculating the trajectories of asteroids and planets Between 1800 and 1965 the FFT algorithm has been re invented a number of times FFT Harmonics The underlying assumption of an FFT is that an arbitrary sampled signal can be recreated as a summation of other periodic waveforms The FFT functions will calculate the periodic waveforms that sum up to make the original signal The FFT functions give you enough information to calculate the frequency magnitude and phase shift of each harmonic component The frequency of the harmonic components in the final answer depends on the sample rate used when sampling
21. NOWIN RECTANGULAR GAUSS HAMMING HANN BARTLETT BARTLETT HANN NUTTALL BLACKMAN HARRIS BLACKMAN NUTTALL FLATTOP PARZEN WELCH BLACKMAN KAISER _BESSEL TRIANGULAR and EXPONENTIAL before the FFT and the FFT results can be interpreted using magnitude phase and frequency routines The PowerSpectrum method returns the power spectrum of the original signal Miscellaneous Chart Classes BMTimer DTriplet XTriplet 20 Class Architecture MatError MatSupport RegStats IVect Various classes exist to support the main matrix and algorithm classes BMTimer A simple timer class used to benchmark algorithms in the software It is based on the system clock and while it has microsecond resolution its accuracy is only 20 milliseconds DTriplet The DTriplet class is used to transfer data to from real matrices It specifies a row column and value XTriplet This XTriplet class is the DComplex version of DTriplet MatError Processes errors in the matrix and algorithm classes MatSupport Common functions used by multiple classes are moved into the MatSupport class avoiding duplication RegStats The RegStats class is used to return summary statistics for the multiple regression and curve fitting classes IVect IVect is a simple integer vector class used to pass integer data to and from some classes Class Hierarchy of QCMatPack DComplex MatBase DMatBase DDMat XMatBase DXMat MatViewerBase DMatViewer XMatViewer LESBase LE
22. StepRegTest example program C StepwiseMulReg mulreg new StepwiseMulReg DDMat A new DDMat Independent variable matrix DDMat RHS new DDMat Ddependent variable matrix Initialize A and RHS matrices mulreg new StepwiseMulReg A RHS for int i 0 i lt mulreg getIndependentVariables numColumns 1 i mulreg setConsider i StepwiseMulReg REG VARCONSIDER RegStats regstats new RegStats DDMat coefficients new DDMat regmode GetRegMode switch regmode case 0 mulreg solve coefficients regstats break case 1 mulreg forwardSolve coefficients regstats break case 2 mulreg backwardSolve coefficients regstats break case 3 mulreg stepwiseSolve coefficients regstats break default mulreg solve coefficients regstats break 90 Multiple Regression Class SplitMulReg What is the goal of your regression If you want to find the best model that gives the smallest error for the existing data you should be using one of the stepwise multiple regression algorithms If you want to create a model that is the best predictor for new sample data then you should try the SplitMulReg algorithm This means that you are looking for a model that gives the best prediction of Y given X X2 Xx for some new observations The SplitMulReg algorithm is a modification of a stepwise regression that randomly assigns the raw sample data into two groups a training group and t
23. a new y value for Return Value Returns the interpolated y value Cubic splines example Extracted from the example program CubicSplinesTest void testCubicSplines dataY new DDMat sampleN dataX new DDMat sampleN for int i 0 i lt sampleN itt double x double i 2 0 dataX setElement i x dataY setElement i 3 5 Math cos x Concatenate dataY and dataX to display in a single matrix viewer Curve Fitting 103 DDMat dataXY new DDMat DDMat colConcat dataX dataY dataXY dMatViewerl new DMatViewer dMatViewerl setViewMatrix dataXY 10 2 dMatViewerl setMatName Data dMatViewerl setPosition 10 10 200 300 dMatViewerl setDecimals 3 dMatViewerl getColumnHeads setElement 0 X dMatViewerl getColumnHeads setElement 1 Y this add dMatViewerl dMatViewerl updateDraw csplines new CubicSplines dataX dataY void interpolatePoints interpolateY new DDMat interpolatedN interpolateX new DDMat interpolatedN actualY new DDMat interpolatedN residualY new DDMat interpolatedN This interpolates on the same iterval as the original data for int i 0 i lt interpolatedN i double x double i 20 0 interpolateX setElement i x interpolateY setElement i csplines InterpolatePoint x actualY setElement i 3 5 Math cos x residualY setElement i interpolateY getElement i
24. applet versions If you want to run one of the examples as an applet run the Applet java file in the example programs folder For Eclipse right click on the Applet1 java file in the Package Explorer and choose Run As Java Applet In a Java applet the JFrame window used in the Application file example is not since that would force the graph to be created in a window separate from the browser Instead the example program JPanel is created in the content pane of the Applet window The Applet java file below is extracted from the LESTest example program package com quinncurtis matpackjava examples LESTest import java awt import javax swing JPanel 144 Using QCMatPack to Create Windows Applications public class Appletl extends javax swing JApplet static final long serialVersionUID 1802260290307532227L public void initi initDemo publie void starei public void initDemo This skips the frame window and places the MainPanel directly in the content pane of the Applet Container contentPane this getContentPane JPanel panell new LESTest panell setPreferredSize new Dimension 900 640 this setSize new Dimension 900 640 contentPane add panell BorderLayout CENTER The program above can be run as is in an Applet viewer What you really want to do is to run the program from a web browser To do that you need to create a jar file ofthe applet and add the proper initialization co
25. as a vector R Initialize a row or a column to the specified value rCInitialize Matrix Classes 57 Returns the vector norm for a row or a column in the matrix rCNorm Returns the sum of squares for a row or a column in the current rCSSQ matrix Initialize a sub range of a row or a column to the specified value rCSublnitialize w the matrix to empty 0 rows and 0 columns reset Overloaded Resize the vector This is a non destructive resize resize Original data to the extent that it fits into the new vector is preserved Overloaded These methods will create a text file and output the save dataset to that file in a CSV Comma Separated Value format A CSV file can be read by popular spreadsheet and word processing programs This version of Save assumes CSV values are organized in a ROW_MAJOR format Scale a column in the current matrix scaleCol Scale all of the elements of the current matrix scaleMat Scale a row in the current matrix scaleRow Set the current matrix as using a base one matrix setBaseOneMatrix Overloaded Set a column in the current matrix setCol Set an element of the matrix setElement Overloaded Initializes the vector using the specified 1D array setMatrix Source array Overloaded High speed routine for copying a matrix Uses setMatrix System Array Copy Overloaded Set a row or a column setRC Overloaded Set a row in the current matrix Overloaded Set the size of the matrix setSiz
26. b matrix scalar operator overload multiply DXMat a double b matrix scalar operator overload multiply DXMat a DXMat b matrix matrix operator overload Two or more matrices of proper dimenson can be multiplied The number of columns of the left matrix must be the same as the number of rows of the right matrix Given a matrix A mx n and a matrix B n x p the size of the matrix C that is the product of A B C is mxp double double DDMat A DDMat B DDMat C ITI aData 1 2 3 4 4 5 6 7 7 8 9 10 3 x 4 bData 5 6 7 8 9 9 7 8 6 4 3 2 1 0 9 3 7 0 4 7 4 x 5 new DDMat aData new DDMat bData DDMat multiply aA Bi 77 3 x 5 Matrix Classes 39 double 11 abata 1411 2 3 21 1445 6 7 17 8 9 10113 FF 8 x 4 double bData 115 8 1 toe tee Optio EEE TI lita U 4 5 DDMat A new DDMat aData DDMat B new DDMat bData DDMat C DDMat multiply A B 3 x 5 double UDO abata 11 2332 1 3 Tete 200 FR a DDMat A new DDMat aData double c 3 1415 DDMat B DDMat multiply A c double abate 1 2 3 4 144 576 U 118 9 10113 7 8 4 double bbate 115 8 7 8 oy 128 Orde 130 271 Oy oe Ted A a DDMat A new DDMat aData DDMat B new DDMat bData DDMat C DDMat multiply A B 3 x 5 Matrix Division static DDMat divide DDMat b DDMat a solves the linear system of eq
27. b to b The mixed fit curve fit algorithm solves for a linear combination of the polynomial exponential and rational polynomial curve fits represented using the equation y f x gt y ao a xi a x tritt b x by x bn x c In x CurveFit constructor public CurveFit DMatBase iv DMatBase dv iv A vector of the independent variable x values dv A vector of the dependent variable y values The four curve fit algorithms all use the constructor above Call the solve method appropriate to the algorithm you want to use Method CurveFit polynomialSolve This method fits the data to a polynomial equation of the form n 1 2 3 y a ta x a x trat an xX where n is considered the order of the polynomial The polynomialSolve method returns the solution coefficients of the polynomial ao an summary statistics for the curve fit public int polynomialSolve int order DDMat regcoef RegStats regstats E Curve Fitting 97 Parameters order Set the maximum order of the polynomial regcoef Returns the calculated curve fit coefficients The coefficients are returned in an array elements 0 to order regstats Returns curve fit statistics Return Value Returns an error code RegStats class Public Instance Properties Get Set overall F statistic ovfst Get Set multiple correlation coefficient R r value Get Set coefficient of determination R rsq Squared
28. basics Many graduate school textbooks are devoted entirely to the uses and interpretation of the results of regression The mathematics behind the actual calculations involved in least squares multiple regression can be found in textbooks devoted to linear algebra and matrix math We provide several algorithms multiple regression the standard least squares approach which regresses only the independent variables you specify and three other methods forward selection backward elimination and stepwise selection which automatically choose the final regression coefficients starting with your best guess and eliminating variables with marginal F Test statistics They have different properties and each one can be the best for a particular regression problem and specific data set As a rule in case of not very highly correlated independent variables the model obtained using general standard least squares regression will produce the smallest error if checked on the same data set on which it was built The other methods produce results slightly less accurate but simpler and more reliable models Least Squares Multiple Regression Class MulReg MulReg The MulReg class uses the least squares method of fitting data to an equation It chooses regression equation coefficients that minimize the sum of the squares of the distances between the observed responses and those predicted by the fitted model This sum of squares is called the residual sum of s
29. complex number so that it has a magnitude of 1 The normalized value z of the complex number a bi is a Magnitude a bi b Magnitude a bi 1 DComplex normalize static DComplex normalize DComplex a DComplex cl new DComplex 3 0 4 0 DComplex c2 cl normalize DComplex c3 DComplex normalize cl The normalized value of cl is 3 0 5 4 0 51 0 6 0 81 DComplex getPolarAngle Method The polar angle r in radians of a complex number a bi is r Math Atan2 b a Calculate the polar angle of a complex number getPolarAngle getPolarAngle DComplex a static doubl DComplex cl new DComplex 3 0 4 0 double rl cl getPolarAngle double r2 DComplex getPolarAngle cl The polar angle of cl is Math Atan2 4 0 3 0 9273 radians DComplex sqrt Method Calculate the square root of a complex number static DComplex sqrt DComplex c static DComplex sqrt double xr double xi 30 Complex Numbers DComplex cl new DComplex 3 0 4 0 DComplex c2 cl sqrt DComplex c3 DComplex sqrt cl The square root of cl is 1 0 2 01 DComplex pow Method Raise a complex number to real power static DComplex pow DComplex c double exponent DComplex pow double exponent double rl 2 0 DComplex cl new DComplex 3 0 4 0 DComplex c2 cl pow rl DComplex c3 DComplex pow cl rl The value of cl is 7 0 24 03 DComplex
30. curve fitting CubicSplinesTest Discrete function approximation using cubic splines FFTTest Calculating FFTs and related functions TCLinTest A thermocouple linearization program MatErrorTest Demonstration of QCMatPack error handling Introduction 5 There are two versions of the QCMatPack class library the 30 day trial versions and the developer version Each version has different characteristics that are summarized below 30 Day Trial Version The trial version of QCMatPack for Java is downloaded as a zip file named Trial QCMatPackR1x7x zip Once you download the zip file you un zip it to a local hard drive and run the Setup exe program from the resulting QCMatPackInstall folder The 30 day trial version stops working 30 days after you run the Setup exe program for the first time Developer Version The developer version of QCMatPack for Java is downloaded as a zip file name something similar to JAVMATDEV1R1x0x353x1 zip Once you download the zip file you un zip it to a local hard drive and run the Setup exe program from the resulting QCMatPackInstall folder You can use your original download links to download free updates for a period of 2 years Chapter Summary The remaining chapters of this book discuss the QCMatPack for Java numerical analysis software designed to run on any hardware that has Java 1 4 or higher interpreter available for it Chapter 2 presents the overall class architecture of the QCMatPa
31. equal and the imaginary parts of the two numbers are equal static boolean isEqual DComplex a DComplex b Complex Numbers 27 Determine whether two complex numbers are equal static boolean isEqual DComplex a DComplex b double tolerance Determine whether two complex numbers are almost i e within tolerance DComplex cl new DComplex 1 0 2 0 DComplex c2 new DComplex 1 0 2 0 if DComplex isEqual c1 c2 not true based on the values above label7 setText Comparison gt cl toString is equal to a2 testring J static boolean notEquals DComplex a DComplex b DComplex c1 new DComplex 1 0 2 0 DComplex c2 new DComplex 3 0 4 0 if DComplex notEquals cl c2 true based on the values above label8 setText Comparison gt Cl rostrang is not equal to e2 testrang The other equality operators lt lt gt and gt are not applicable to complex numbers That is because you cannot compare two complex numbers and decide if one is greater than the other For example which is the larger number 100 51 5 1001 5 1001 100 5i At best you can only compare the magnitudes of complex numbers In that case the complex numbers in the previous example are all equal but fail the equality test If you find you need to do some sort of greater than or less than test on two complex numbers it is up to you to decide wh
32. exp Method Raise the value e to a complex power static DComplex exp DComplex c double rl 2 0 DComplex cl new DComplex 3 0 4 0 DComplex c2 cl pow rl DComplex c3 DComplex pow cl rl The value of e is 13 128783081462162 15 200784463067956 i Complex Numbers 31 Miscellaneous Functions The DComplex class also some support functions which make the input and output of complex numbers easier toString and parse DComplex toString Method This method converts the current value of the member variables into a string of the form a bi The complex number 1 1 2 21 is represented in string notation as 1 1 2 21 The default numeric format of the two double values in the string are controlled by the G numeric format option of the double toString string format method A second toString method is supplied that can take the floating point format string as an argument Get the string representation of a complex number java lang String toString java lang String toString java lang String s DComplex Parse Method This method converts a string in the format a bi into a DComplex object Get the string representation of a complex number static DComplex parse java lang String s Parse a complex representation in this fashion f f 4 Matrix Classes DDMat and DXMat classes MatBase DMatBase DDMat XMatBase DXMat There are two mat
33. i lt mz i for int je J lt ny I A setElement i j i n j A setElement 2 12 22 generates a row index out of range error catch java lang ArrayIndexOutOfBoundsException exp exp printStackTrace 13 Using QCMatPack for Java to Create Applications and Applets Eclipse These directions were created using Eclipse Platform Version 3 0 2 Our standard development environment is Eclipse and our example programs are organized along the lines of the Java example programs that ship with Eclipse From the Eclipse Java Perspective IDE select File Switch Workspace and browse to and select the C Quinn Curtis java directory Select Workspace Directory 2 x Select the workspace directory to use DO QCRTGraphcFinstall OD QCRTGraphinstall DB QCRTGraphNetDemoProgram D acw32 DI ocw32New OD Quinn Curtis a H metadata r D docs lib DB Quinn Curtis CE Development Quinn Curtis CF x E DEE E F Make New Folder Cancel This will establish the current workspace directory as Quinn Curtis java From the Eclipse main menu select File New Project Java Project Give the project the name Eclipse QCMatPackJavaExamples any name will work 134 Using QCMatPack to Create Windows Applications New Java Project A xj Create a Java project Create a Java project in the workspace or in an external location fi Project name Eclipse_QCMatPackJavaExample
34. is possible to transpose a complex matrix using the simple description above it is not mathematically correct The proper mathematical transpose of a complex matrix is more properly called the conjugate transpose In a conjugate transpose the complex conjugate of the source row values become the new column values of the destination matrix For example if element 5 3 of the source matrix has the complex value 7 21 the element 3 5 of the transposed matrix will have the value 7 21 the complex conjugate of the original value Symmetric Matrix A real matrix is considered to be symmetrical if it is square and the elements above the diagonal are the mirror image of the elements below the diagonal i e A i j Alj i For example in a symmetric matrix the following element pairs one above the diagonal and the other below the diagonal must be equal 5 3 3 5 MAOIs 1 H17 TEN While a complex matrix can be symmetric A i j A j 1 it is not a useful condition Like the complex transpose a mathematically symmetric complex matrix uses the complex conjugate A truly symmetric complex matrix known as a Hermitian matrix requires that A i j Conjugate A j i Using this definition the diagonal values of a Hermitian matrix must be real no imaginary component since that is the only way a diagonal element can equal its own complex conjugate Positive Definite Matrix a positive definite matrix is a Symm
35. matrix inverse The checkSolution method will check the accuracy of a solution by calculating right hand side values based on the original coefficient matrix and solution and comparing the results to the actual right hand side values The problem presented in the first page of the chapter is solved below using each of the four algorithms double jl aData 1 2 2 2 I I Fol double bData 1 2 0 62 Systems of Equations DDMat A new DDMat aData DDMat b new DDMat bData DDMat x new DDMat A IES Solve b LU Decomposition LESLU leslu new LESLU A leslu solve b x Gauss Jordan LESGJ lesgj new LESGJ A lesgj solve b x QR Factorization LESOR lesqr new LESOR A lesqr solve b x While the Cholesky algorithm is used in exactly the same way it will fail for the example above because the coefficient matrix A is not symmetric positive definite A symmetric positive definite matrix is used in the example below c Cholesky Decomposition Cholesky requires a symmetric positive definite matrix double aSPDData 13 oy lb Ea 22 4 1 4 157 DDMat ASPD new DDMat aSPDData LESCholesky leschol new LESCholesky ASPD leschol solve b x The complex versions of the same algorithms follow exactly the same logic except that DComplex objects are used instead of doubles and complex DXMat matrices are used instead of real DDMat matrices
36. methods and properties of the two classes all have the same names to make converting from one to the other as simple as possible The major difference between the two classes is that the DDMat generally uses double arguments and the DXMat generally uses complex class DComplex arguments This rule is not absolute since some complex matrix functions do use double arguments and return double values Because of the symmetry between the two matrix classes constructors and methods will be documented in parallel to save space Linear Algebra Terminology Matrices and linear algebra use mathematical terms not in common use with the average programmer In order to qualify the special characteristics of different types of matrices we use many of these terms 34 Matrix Classes Matrix order The order of a matrix is the dimensions of the matrix i e the number of rows and columns it has Two matrices are of the same order if they have the same dimensions The number of rows in a matrix is usually referred to using the letter m the number of columns the letter n Transpose The transpose of a real matrix is formed by taking an m x n matrix and converting it to an n x m matrix making the rows of the original matrix the columns of the transposed matrix For example the row elements of the original matrix 0 0 0 1 0 2 O n 1 become the column elements of the transposed matrix 0 0 1 0 2 0 n 1 0 While it
37. of multiple right hand sides using back substitution Can be used for solving over determined systems of equations where the number of rows is greater than the number of columns Produces a solution in finite time on the order of 2n 3 operations positive definite matrices Cholesky The fastest of the four algorithms Produces an intermediate matrix factorization that can be used for Requires that the coefficient matrix is asymmetric positive definite matrix high speed solution of multiple right hand sides using back substitution Produces a solution in finite time on the order of n 3 operations or about twice as fast as LU decomposition Standard Interface for the Solution of Linear Equations The linear equations solving classes all use the same interface You define the problem by passing the coefficient matrix A in the constructor of the class you are using In the case of LU QR and Cholesky the coefficient matrix is immediately factored into intermediate matrices The programmer then calls the solve method passing in a matrix of one or more right hand sides b The solution is calculated using the intermediate matrices and back substitution and returned in the same solve method call The Gauss Jordan algorithm does not factor the coefficient matrix into intermediate matrices and the 100 of the algorithm takes place when the solve method is called The inverse method returns the coefficient
38. one variable x and another one y The relationship is considered linear because it is assumed that the underlying equation relating the two variables is of the form y mx b Simple linear regression would take a list of x y pairs 2 would be the minimum number of pairs and calculate the slope coefficient m and the offset coefficient b Regression does not always return the exact values for the slope and offset because the data is usually empirical The calculated slope and offset represent the best trend line which can be fitted through the data The estimate is calculated using an algorithm which minimizes the sum of the squares of the vertical deviations separating the trend line from the actual data points This algorithm is known as the method of least squares and it is by far the most common technique used in regression analysis Simple linear regression only handles equations for two variables Multiple linear regression deals with equations of at least two variables It is quite possible to provide estimates of equations of 100 or more variables using multiple regression Large economic forecasting models use multiple regression to analyze data involving hundreds of variables The general form of a regression equation or model for k independent variables is given by the equation Y bot b X bo Xo Dk Xk where bo b b2 bk are the regression coefficients that need to be estimated The variable Y is the dependent
39. the original raw waveform The FFT 106 Digital Signal Processing can only return frequencies that are simple integer fractions of the original sampling frequency sample frequency SF The range of frequencies will range from 0 DC level or average value of the signal to SF 2 or the Nyquist frequency for the sample rate of SF For example assume that 16 samples of a periodic signal are taken at a rate of 480 Hz The FFT algorithm will calculate the degree to which the following frequencies contribute to the signal makeup Harmonic Frequency 0 16 480 0 Hz DC level 1 16 480 30 Hz 2 16 480 60 Hz 3 16 480 90 Hz 4 16 480 120 Hz 5 16 480 150 Hz 6 16 480 180 Hz 7 16 480 210 Hz 8 16 480 240 Hz NS U PUNTO If you were interested in the degree to which 60 Hz AC line frequency noise was affecting your signal then you could concentrate on harmonic 2 which in the example above is at 60 Hz The general equation for calculating the frequency for a given harmonic index is F H N SF where F frequency of harmonic H H harmonic index range 0 to N 2 N number of sampled data points SF sample frequency Assume that you are sampling waveform at 50KHz and you want to resolve individual frequency components down as low as 8 Hz How many data points do you need to sample The formula becomes 8 Hz 1 N 50KHz or N 1 8 50KHz 6250 The FFT algori
40. updateDraw setViewMatrix EVectors 10 3 11 Thermocouple Linearization TCLin Thermocouples are the most widely used transducers for measuring temperature They are economical rugged and have stable characteristics over years Further available types can measure temperatures ranging from the cryogenic to jet exhaust Thermocouple devices are two pieces of metal in isothermal contact physical contact at the same temperature Because the number of free electrons in a metal depends on its temperature and composition thermocouples exhibit a voltage potential that s a repeatable function of temperature Over the past hundred years several metallic pairs have become standard thermocouple types because of their accuracy and usefulness across common temperature ranges THERMOCOUPLES Type Composition Temperature range B Pt 30 Rh versus Pt 6 Rh 0 to 1820 E Ni Cr alloy versus a Cu Ni alloy 270 to 1000 J Fe versus a Cu Ni alloy 210 to 1200 K Ni Cr alloy versus Ni Al alloy 270 to 1372 N Ni Cr Si alloy versus Ni Si Mg alloy 270 to 1300 R Pt 13 Rh versus Pt 50 to 1768 S Pt 10 Rh versus Pt 50 to 1768 T Cu versus a Cu Ni alloy 270 to 400 NIST ITS 90 The relationship between a thermocouple s output voltage and its temperature is a nonlinear function best approximated with a high order polynomial Once you measure a thermocouple s output voltage you can calculate a temperature by plugging th
41. values It includes overrides of the most common math operators making it very simple and logical to write mathematical expressions using complex numbers For example the following line in bold is a valid expression involving DComplex complex number arguments DComplex cl new DComplex 1 0 2 0 DComplex c2 new DComplex 3 0 4 0 double rl 5 0 fF e3 gl o 1 6 0 DComplex c3 DComplex add DComplex multiply DComplex multiply cl c2 rl 6 0 The DComplex class also includes a large number of mathematical functions used in conjunction with complex numbers Conjugate Magnitude Abs Sqrt Exp Pow and Arg The DXMat class is the main class used to represent vectors and matrices of DComplex complex numbers The linear equations classes LESXLU LESXGJ LESXCholesky and LESXQR are specific to the DComplex complex number type The eigensystem classes EigenQR2 EigenXLR also uses the DComplex number type Matrix Classes MatBase DMatBase DDMat XMatBase DXMat The MatBase abstract base class is the base class for all matrix types The DMatBase class is the abstract representation of a real matrix The DDMat class is the concrete representation of a real matrix using doubles for numeric storage The XMatBase class is the abstract representation of a complex number matrix The DXMat class is the concrete representation of a complex number matrix using DComplex objects for numeric storage MatBas
42. variable in the equation above The variables X1 X2 Xx are the independent variables The objective of regression is to calculate the coefficients bo bi b2 bk in order to produce the best estimates of the dependent variable Y given the values of the independent variables X1 Xo Xk The independent and dependent variables are all measured values To calculate the coefficients you must first acquire enough data so that you have at least one more set of observations than the number of variables you are trying to fit One of the most important measures of the quality of the regression equation fit is known as the correlation coefficient The correlation coefficient is usually abbreviated as the r value R of the 82 Multiple Regression regression The closer this value is to 1 the better the fit Another measure of the fit is called the coefficient of determination usually referred to as the r R value This value is equal to the square of the correlation coefficient Again the closer this value is to 1 the better the fit The standard error of the estimate usually abbreviated SEE is yet another measure of the quality of the regression fit It is a measure of the scatter of the actual data about the regression line The smaller the SEE value the closer the actual data is to the theoretical regression line If you would like to learn more about the uses of regression any good college textbook on statistics should cover the
43. 0 57i 4 0 23 0 26i 0 23 0 26i 0 60 0 23i 5 0 49 1 00 09 062 0 81 5 0 80 0 57i 5 0 50 0 20i 0 50 0 20i 0 06 0 24i 6 0 14 0 89 0 89 0 36 0 31 6 0 30 1 10i 6 0 26 0 24i 0 26 0 24i 1 00 0 00i 7 0 85 0 38 0 58 0 26 0 43 7 0 30 1 10i 7 0 72 0 13i 0 72 0 13i 0 07 0 02i 8 076 076 0 38 08 0 74 8 0 41 0 79i 8 0 71 0 63i 0 71 0 63i 0 85 0 13i 9 0 75 067 009 0 07 0 40 9 0 41 0 79i 9 0 09 0 08i 0 09 0 08i 0 39 0 14i x A r a po 3 Real Matrix Complex Matrix Complex Matrix Typical use of the matrix viewers do not have you using the constructors You would normally add a matrix viewer to a form by dropping it from the Toolbox which automatically calls the default constructor DMatViewer constructors 118 Matrix Viewers public DMatViewer DMatBase mat int rows int cots XMatViewer constructors public XMatViewer XMatBase mat int rows int cols mat The source matrix DMatViewer uses a DDMat matrix while XMatViewer uses a DXMat matrix row The number of display rows in the matrix viewer cols The number of display columns in the matrix viewer If you add a matrix viewer to a form from the Toolbox initialize the viewer matrix rows and columns using the SetViewMatrix method XMatViewer setViewMatrix Method public void setViewMatrix DMatBase mat int rows int cols mat The source matrix DMatViewer uses aDDMat matrix while XMatViewer uses a DXMat matri
44. 1997 Wilkinson J H and Reinsch C Linear Algebra Springer Verlag 1971 INDEX Applets 1 6 133 143 Benchmark timer 19 20 21 BMTimer 19 20 21 C viii 1 35 40 62 89 113 Cholesky factorization Complex 8 12 13 14 15 21 23 59 63 66 67 Cholesky factorization Real 8 13 14 15 21 59 62 66 67 Complex matrix base class 12 13 20 33 35 42 43 51 52 64 65 66 67 73 75 76 77 78 79 111 112 113 114 118 Complex matrix class vii 2 7 8 11 12 13 19 20 23 33 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 62 63 72 77 79 80 109 110 111 118 119 Complex numbers 2 4 7 8 11 12 13 20 23 24 25 26 27 28 29 30 31 33 36 37 38 40 43 44 46 50 62 Cubic splines 9 18 19 21 93 101 102 103 CubicSpines 9 18 19 21 93 101 102 103 Curve fitting 9 18 21 93 94 96 97 98 99 CurveFit 9 18 21 93 94 96 97 98 99 DComplex 2 4 7 8 11 12 13 20 23 24 25 26 27 28 29 30 31 33 36 37 38 40 43 44 46 50 62 DDMat vii 2 7 8 10 12 13 19 20 33 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 55 62 72 74 77 84 85 87 88 89 91 96 97 98 99 102 103 109 111 112 113 114 115 118 119 131 Digital signal processing viii 6 7 9 19 21 105 107 108 109 110 111 112 113 114 115 DMatBase 12 13 20 33 35 42 43 51 52
45. 3 64 67 LESLU 8 13 14 20 59 62 64 65 LESQR 8 13 14 15 16 21 59 62 65 66 LESXBase 13 14 21 59 LESXCholesky 8 12 13 14 15 21 23 59 63 66 67 LESXGJ 8 12 13 14 15 21 23 59 63 64 67 LESXLU 8 12 13 14 21 23 59 63 64 65 LESXQR 8 12 13 14 16 21 23 59 63 65 66 Linear equation solver base class complex 13 14 21 59 Linear equation solver base class Real 13 14 20 59 LU decomposition Complex 8 12 13 14 21 23 59 63 64 65 LU decomposition Real 8 13 14 20 59 62 64 65 MatBase 12 20 33 MatError 20 21 47 129 130 131 Matrix base class 12 20 33 Matrix viewer complex 8 9 11 20 117 118 Matrix viewer real 8 9 10 20 103 104 117 118 119 MatSupport 20 21 MulReg 8 17 18 21 82 83 84 85 86 87 88 89 Multiple regression 8 17 18 21 82 83 84 85 86 87 88 89 NIST ITS 90 viii 6 9 121 122 123 Polynomial curve fitting 9 18 21 93 94 96 97 98 99 QL eigenvalue eigenvector algorithm 8 16 21 69 71 72 74 QR eigenvalue eigenvector algorithm 8 16 17 21 23 69 71 72 74 75 76 77 78 80 QR factorization Complex 8 12 13 14 16 21 23 59 63 65 66 QR factorization Real 8 13 14 15 16 21 59 62 65 66 OR2 eigenvalue eigenvector algorithm 8 12 16 17 21 23 69 71 72 74 76 77 Rational fraction curve fitting 9 18 21 93 94 96 97 98 99 R
46. 8 095 mV to 0 mV 0 05 C to 0 03 C B 250 C to 700 C 0 291 to 2 431mV 0 02 C to 0 03 C 2 3 4 0 C to 760 C 760 C to 1200 C 0 mV to 42 919 mV 42 919 mV to 69 553 mV 0 04 C to 0 04 C 0 04 C to 0 03 C 700 C to 1820 C 2 431 mV to 13 820 mV 0 01 C to 0 02 C Thermocouple Type Voltage Range Error Range Thermocouple Type Temperature Range Voltage Range Error Range Thermocouple Type Temperature Range Voltage Range Error Range Thermocouple Type Temperature Range Voltage Range Error Range Thermocouple Type Temperature Range Voltage Range Error Range Thermocouple Type Temperature Range Voltage Range Error Range E 200 C to 0 C 8 825 mV to 0 mV 0 01 C to 0 03 C k 200 C to 0 C 5 891 mV to 0 mV 0 02 C to 0 04 C N 200 C to 0 C 3 990 to 0 mV 0 02 C to 0 03 C R 50 C to 250 C 0 226 to 1 923 mV 0 02 C to 0 02 C S 50 C to 250 C 0 235 to 1 874 mV 0 02 C to 0 02 C T 200 C to 0 C 5 603 mV to 0 mV 0 02 C to 0 04 C 0 C to 1000 C 0 mV to 76 373 mV 0 02 C to 0 02 C 0 C to 500 C 0 mV to 20 644 mV 0 05 C to 0 04 C 0 C to 600 C 0 to 20 613 mV 0 02 C to 0 03 C 250 C to 1200 C 1 923 to 13 228 mV 0 005 C to 0 005 C 0 0005 C to 0 001 C 250 C to 1200 C 1 874 to 11 950 mV 0 01 C to 0 01 C
47. ARE in its unmodified form via electronic means Internet BBS s Shareware distribution libraries CD ROMs etc You may not charge any fee for the copy or use of the evaluation SOFTWARE itself You must not represent in any way that you are selling the SOFTWARE itself You must distribute a copy of this EULA with any copy of the SOFTWARE and anyone to whom you distribute the SOFTWARE is subject to this EULA C Redistributable License The standard Developer License permits the programmer to deploy and or distribute applications that use the Quinn Curtis SOFTWARE royalty free We cannot allow developers to use this SOFTWARE to create a graphics toolkit a library or any type of graphics component that will be used in combination with a program development environment for resale to other developers If you utilize the SOFTWARE in an application program or in a web site deployment should we ask you must supply Quinn Curtis Inc with the name of the application program and or the URL where the SOFTWARE is installed and being used 3 RESTRICTIONS You may not reverse engineer de compile or disassemble the SOFTWARE except and only to the extent that such activity is expressly permitted by applicable law notwithstanding this il limitation You may not rent lease or lend the SOFTWARE You may not use the SOFTWARE to perform any illegal purpose 4 SUPPORT SERVICES Quinn Curtis Inc may provide you with support services related to th
48. D9C840 044E 11D1 B3E9 00805F499D93 WIDTH 800 HEIGHT 600 codebase http java sun com jpi jinstall 14 win32 cab Version 1 4 0 mn gt lt PARAM NAME code VALUE com quinncurtis matpackjava examples LESTest Appletl class gt lt PARAM NAME codebase VALUE http quinn curtis com classes gt lt PARAM NAME archive VALUE LESTest jar qcmatpackjava jar qcmatviewjava jar gt lt PARAM NAME type VALUE application x java applet version 1 4 gt lt PARAM NAME scriptable VALUE false gt lt COMMENT gt lt EMBED type application x java applet version 1 4 Using QCMatPack to Create Windows Applications 147 CODE com quinncurtis matpackjava examples LESTest Appletl class CODEBASE http quinn curtis com classes ARCHIVE LESTest jar qcmatpackjava jar qcmatviewjava jar WIDTH 800 HEIGHT 600 scriptable false pluginspage http java sun com jpi plugin install html gt lt NOEMBED gt No Java 2 SDK Standard Edition v 1 4 support for APPLET lt NOEMBED gt lt EMBED gt lt COMMENT gt lt OBJECT gt This HTML page LESTestApplet2 htm can be loaded and the applet run at our web site using the following link http quinn curtis com LESTestApplet2 htm Note that while similar some of the attributes and tags have a different use The best place to read about the differences is in the Sun article referenced at http docsun cites uiuc edu sun_docs C solaris_9 SUNWaadm J
49. LESXCholesky LU Decomposition LESLU LESXLU QR Factorization LESQR LESXQR and Gauss Jordan LESGJ LESXGJ Each class provides for the solution of one right hand side multiple right hand sides and the matrix inverse LESLU The LESLU class uses the LU decomposition algorithm to solve a system of real number linear equations It is a very stable efficient method of solving systems of equations It is particularly useful for solving systems of equations involving multiple right hand sides since the LU decomposition of the primary coefficient matrix only needs to be solved for once From that point on multiple right hand sides can be solved for using the decomposed matrix and simple back substitution LESXLU The LESXLU class is the complex number version of the LESLU class using the LU decomposition algorithm to solve a system of complex number linear equations LESGJ LESXGJ LESCholesky LESXCholesky LESQR Class Architecture 15 The LESGJ class uses the Gauss Jordan with partial pivoting algorithm to solve a system of real number linear equations It is a very stable efficient method of solving systems of equations The LESXGJ class is the complex number version of the LESGJ class using the Gauss Jordan with partial pivoting algorithm to solve a system of complex number linear equations The LESCholesky class uses the Cholesky decomposition algorithm to solve a system of real number linear equations The Choles
50. QCMatPack Matrix and Numerical Analysis Software for Java lol x GSlForm1 Solve Re Initialize ee Rank Solanan tec I Symmetric A I Diagonally Dominant A C OK a 9 1 2 0 0 1 2 0 0 19 0 47i 0 44 0 31i 0 72 0 87i 9 64 10 07i 0 0 85 0 19i 0 10 0 11i 0 45 0 05i 1 0 57 0 04i 0 70 0 42i 0 23 0 59i 1 71 0 93i 1 0 87 0 07i 0 05 0 22i 0 14 0 54i 2 0 09 0 75i 0 05 0 491 0 23 0 40i 1 33 0 65i 2 0 80 0 20 1 00 0 00 0 78 0 22i 3 0 29 0 90 0 16 0 291 0 43 0 531 0 48 1 38i 3 0 77 0 121 0 48 0 18 0 30 0 06i 4 0 34 0 25i 1 00 0 27i 0 67 0 42i 1 30 0 83i 4 0 73 0 091 0 13 0 32 0 28 0 26i 5 0 26 0 98i 0 82 0 43i 0 39 0 05i 0 71 1 55i 5 0 93 0 19i 0 55 0 82i 0 59 0 65i 6 0 80 0 87i 0 83 0 03i 0 58 0 22i 0 83 1 35i 6 0 83 0 11i 0 62 0 74i 0 52 0 16i 7 0 04 0 57i 0 69 0 591 0 62 0 68i 1 04 0 42i 7 0 90 0 13i 0 76 0 331 0 33 0 14i 8 0 58 0 50i 0 01 0 10i 0 66 0 43i 1 51 0 54i 8 0 86 0 04i 0 23 0 15i 0 05 0 40i 9 0 42 0 22i 0 00 0 08i 0 07 0 24i 0 27 1 18i g 0 79 0 09 0 18 0 67i 0 33 0 53i See Complex Unsymmetric A lee FE Complex Eigenvalues Complex Eigenvectors Izle Contact Information Company Web Site http www quinn curtis com Electronic mail General Information info quinn curtis com Sales sales quinn curtis com Technical
51. SLU LESGJ LESCholesky LESQR LESXBase LESXLU LESXGJ LESXCholesky LESXQR EigenBase EigenQR EigenQL EigenQR2 EigenXBase EigenXLR MulReg StepwiseMulReg SplitMulReg CurveFit CubicSplines DSP TCLin BMTimer DTriplet XTriplet MatError MatSupport RegStats IVect Class Architecture 21 3 Complex Numbers The linear algebra solution of linear systems and eigenvalue eigenvector algorithms in this software come in versions that apply to both real and complex numbers While the double numeric type is what we use for real numbers Java does not have a standard equivalent numeric type for complex numbers To that end we have created our own complex class DComplex for use in our algorithms and your programs DComplex In mathematics a complex number is a number of the form a bi where a and b are real numbers and i is the imaginary unit with the property i 1 The real number a is called the real part of the complex number and the real number b is the imaginary part Real numbers may be considered to be complex numbers with an imaginary part of zero that is the real number a is equivalent to the complex number a 0i source Wikipedia http en wikipedia org wiki Complex_number The DComplex class represents the real and imaginary parts of a complex number using two double values It includes overrides of the most common math operators making it very simple and logical to write mathematical express
52. Support Forum http www quinn curtis com ForumFrame htm Revision Date 11 24 2007 Rev 1 0 QCMatPack for Java Documentation and Software Copyright Quinn Curtis Inc 2007 Quinn Curtis Inc QCMatPack Tools Java END USER LICENSE AGREEMENT IMPORTANT READ CAREFULLY This Software End User License Agreement EULA is a legal agreement between you either an individual or a single entity and Quinn Curtis Inc for the Quinn Curtis Inc SOFTWARE identified above which includes the Quinn Curtis Inc QCMatPack Matrix and Numerical Analysis Software By installing copying or otherwise using the SOFTWARE you agree to be bound by the terms of this EULA If you do not agree to the terms of this EULA do not install or use the SOFTWARE If the SOFTWARE was mailed to you return the media envelope UNOPENED along with the rest of the package to the location where you obtained it within 30 days from purchase 1 The SOFTWARE is licensed not sold 2 GRANT OF LICENSE A Developer License After you have purchased the license for SOFTWARE and have received the file containing the licensed copy you are licensed to copy the SOFTWARE only into the memory of the number of computers corresponding to the number of licenses purchased The primary user of the computer on which each licensed copy of the SOFTWARE is installed may make a second copy for his or her exclusive use on a portable computer Under no other circumstances may the SOFTWARE be oper
53. UT will be excluded from of the model Setup up the regression the problem using the StepwiseMulReg constructor StepwiseMulReg constructor Set up a stepwise multiple regression including all of the independent variables in the final solution public StepwiseMulReg DMatBase iv DMatBase dv Set up a stepwise multiple regression including only the independent variables flagged by the consid vector in the final solution public StepwiseMulReg DMatBase iv DMatBase dv IVect consid iv A matrix of the independent variable x values dv A matrix of the dependent variable y values consid A vector contains consideration values one for each independent variable i the iv array plus the constant Element 0 contains the constant flag the other elements contain the flags for the iv array Initialize the array elements using one of the consider constants REG_VARIN REG_VAROUT REG VARCONSIDER If an element is REG_VARIN it is always included in the final regression equation If an element is REG_VAROUT it is excluded from the final regression equation If an element is REG_VARCONSIDER the corresponding variable available for selection into the final regression equation Multiple Regression 87 Backward Elimination The backward elimination algorithm starts with a model that includes all of the variables marked with REG_VARIN and REG_VARCONSIDER The basic algorithm is as follows Step 1 Solve for the current regression equ
54. VPLUGINUG pS html The complete source to the HTML pages are found in the LESTest examples directory file LESTestApplet1 htm and LESTestApplet2 htm Note the following The main applet class com quinn curtis matpackjava examples LESTest Applet1 class is specified using the code lt PARAM gt The required jar files are specified using the archive lt PARAM gt The directory that the applet is in is specified using the codebase lt PARAM gt If the applet is in the same directory as the calling HTML page the codebase lt PARAM is not necessary The HEIGHT and WIDTH attribute specify the pixel dimensions of the window the applet will display in The rest of the tags and lt PARAM gt values support the automatic download of the Sun Java plug in if the client viewing the applet needs it 148 Using QCMatPack to Create Windows Applications e The lt EMBED gt section implements an equivalent block of HTML code that is only called if a Netscape browser is used to view the applet Selected Bibliography Afifi A A and Clark Virginia Computer Aided Multivariate Analysis Lifetime Learning Publications 1984 Atkinson L V and Harley P J An Introduction to Numerical Methods with Pascal Addison Wesley Publishing Company 1983 Brigham E Oran The Fast Fourier Transform and It s Applications Prentice Hall Englewood Cliffs 1988 Chapra S C and Canale R P Numerical Method
55. actualY getElement i Concatenate interpolateX and interpolateY to display in a single matrix viewer DDMat interpolateXY new DDMat DDMat colConcat interpolateX interpolateY interpolateXY DDMat colConcat interpolateXY actualY interpolateXY DDMat colConcat interpolateXY residualY interpolateXY 104 Curve Fitting dMatViewer2 new DMatViewer dMatViewer2 setViewMatrix interpolateXY 20 4 dMatViewer2 setPosition 330 10 400 500 dMatViewer2 setMatName Data dMatViewer2 setDecimals 3 dMatViewer2 getColumnHeads setElement 0 X dMatViewer2 getColumnHeads setElement 1 Y Interpolate dMatViewer2 getColumnHeads setElement 2 Y Actual dMatViewer2 getColumnHeads setElement 3 Y Residual this add dMatViewer2 dMatViewer2 updateDraw 9 Digital Signal Processing DSP FFT Basics A large class of numerical analysis tools involves use of Fourier analysis Applications for the Fourier transform include speech image radar and general signal processing Implementations of the Fourier transform using sampled data on computers are generally referred to as DFT Discrete Fourier Transform The equation below computes the DFT X of the input signal xn N 1 27 p AT X Y men k 0 N 1 n O A complex summation of N complex multiplications is required for each of N samples This adds up to N complex multiplications and N complex additions to compute an
56. algorithm Since the Cholesky L matrix does not need to be recalculated for each new right hand side it is even faster if you are solving for multiple right hand sides Systems of Equations 67 Unlike the Gauss Jordan algorithm the Cholesky algorithm does not produce a matrix inverse as a by product If you call the Inverse method the inverse is calculated by solving the Ax b equation using b b is a square matrix in this case with the same dimensions as A as the identity matrix If Ax b and b is the identity matrix then x must be the inverse of A Solving for x produces the inverse of A In the complex version of the algorithm LESXCholesky the coefficient matrix A must be Hermitian positive definite since Hermitian is the complex equivalent of the real symmetric Since L is complex L is the conjugate transpose of L LESCholesky LESXCholesky Constructors LESCholesky public LESCholesky DMatBase a LESXCholesky public LESCholesky XMatBase a Parameters a The coefficient matrix A for the Ax b system of equations Example The code in the Standard Interface for the Solution of Linear Equations section is extracted from the SimpleLES and SimpleLESX example programs The class is also used in the LESTest and LESXTest example programs Selected LESGJ LESXGJ Methods LESGJ solve LESXGJ solve Methods Solve the square non singular system of equations represented by the matrix equation Ax b using the G
57. ampled at a rate of 480 Hz The sampled waveform is contained in the array w w 6 0 1 85 2 59 0 765 4 0 0 765 5 41 1 85 2 0 1 85 5 41 0 765 4 0 0 765 2 59 1 85 The FFT function returns a complex number for each positive and negative harmonic in the waveform Assume that the real parts are returned in the vector va and the imaginary parts in vector vb The complex number representing the value for harmonic 3 would be va 3 vb 3 1 The complex number for the highest harmonic in a 16 point FFT would be va 8 vb 8 i Many questions arise regarding interpretation of the FFT results The absolute values of a FFT are proportional to the number of data points used in the calculation Some FFT routines normalize the results of the FFT by dividing the calculated values by N or N 2 while the majority do not We do not normalize the results because this would make our algorithms different from the majority of popular FFT algorithms described in the literature This can be confusing to people who have never calculated an FFT before If you are looking for the relative magnitude of one harmonic versus the other you can ignore the normalization factor If you need to recover the exact value of a given harmonic normalize the results of the FFT by N 2 if you intend to use the positive harmonics and N if you plan to use both positive and negative harmonics The frequency magnitude and phase shift for each harmonic index are calculate
58. and_X The n values of A that satisfy the equation are the eigenvalues of matrix A and the corresponding values of X are its right eigenvectors For example A z X 541 11 fO 0 ro 4511 0 0 9 ti aal feat ara A Cio o 18 2 1 The eigenvalue A in this case is 2 and the eigenvector associated with A is 70 Eigenvalues and Eigenvectors To find the eigenvalues of a matrix A rewrite the equation 4X AX as AX AIX AX AIX 0 A AI X 0 We want to find the numbers for A for which the equation above has non zero solutions X which is the same as finding the numbers for that make det A A 0 This is referred to as the characteristic equation of A The left hand side of the equation is the characteristic polynomial of A For the matrix 5411 4 511 IETA 1 1 2 4 5 4 1 1172 0 001 ea amp 1 la 5 11 0 20 0 2 324 4 Ae Nay ae eG aa a 1 4 a 2 tro 4 10002 3 1 2 4 a The characteristic equation det 4 A7 0 can be written as follows es and Eigenvectors 71 Solving for the determinant will yield the following eigenvalues and their associated eigenvectors 1 0 05 l 0 05 l PY 0 1 0 1 l io __ Eigensystem problems are divided into different categories depending on the symmetry of the original matrix and whether or not the original matrix contains complex values The class you should use Source Matrix Real Symmetr
59. ated at the same time on more than the number of computers for which you have paid a separate license fee You may not duplicate the SOFTWARE in whole or in part except that you may make one copy of the SOFTWARE for backup or archival purposes You may terminate this license at any time by destroying the original and all copies of the SOFTWARE in whatever form B 30 Day Trial License You may download and use the SOFTWARE without charge on an evaluation basis for thirty 30 days from the day that you DOWNLOAD the trial version of the SOFTWARE The termination date of the trial SOFTWARE is embedded in the downloaded SOFTWARE and cannot be changed You must pay the license fee for a Developer License of the SOFTWARE to continue to use the SOFTWARE after the thirty 30 days If you continue to use the SOFTWARE after the thirty 30 days without paying the license fee you will be using the SOFTWARE on an unlicensed basis Redistribution of 30 Day Trial Copy Bear in mind that the 30 Day Trial version of the SOFTWARE becomes invalid 30 days after downloaded from our web site or one of our sponsor s web sites If you wish to redistribute the 30 day trial version of the SOFTWARE you should arrange to have it redistributed directly from our web site If you are using SOFTWARE on an evaluation basis you may make copies of the evaluation SOFTWARE as you wish give exact copies of the original evaluation SOFTWARE to anyone and distribute the evaluation SOFTW
60. ation Step 2 Calculate the partial F statistic for every variable in the model Step 3 Find the variable with the smallest partial F statistic and using that value compare the probability that the variable is significant to the value of the property FToLeave If the probability that the variable is significant is less than FToLeave the variable is removed from the model Step 4 Repeat starting at Step 1 until the significance test in Step 3 finds no variable to remove The current solution then becomes the final solution of the regression equation Method StepwiseMulReg backwardSolve public int BackwardSolve DDMat regcoef RegStats regstats Parameters regcoef Returns the calculated regression coefficients regstats Returns regression statistics The RegStats class is a simple class that encapsulates several related summary statistics for multiple regression See the description under the MulReg class section Forward Selection The forward selection algorithm starts with a model that includes only the variables marked with REG_VARIN If there are no variables marked with REG_VARIN then the model starts with the variable that has the highest correlation with the dependent variable Step 1 Select the starting variable s This is either the variables marked with REG_VARIN or the variable with the highest correlation with the dependent variable Step 2 Calculate the partial F statistic for every other variable marked
61. atpackjava examples LESTest EB com quinncurtis matpackjava examples LESXTest 1 8 com quinncurtis matpackjava examples MatErrorTest EB com quinncurtis matpackjava examples MatrixMath E com quinncurtis matpackjava examples SimpleLESX E com quinncurtis matpackjava examples SplitRegTest E com quinncurtis matpackjava examples StepRegTest EB com quinncurtis matpackjava examples TCLinTest BA JRE System Library jre1 6 0_03 pa qematyiewjava jar C Quinn Curtishjavatlib eye qematpackjava jar C Quinn Curtis javallib Expand one of the example program folders and locate the Application java file Right click on that file and select Run Run Java Application Using QCMatPack to Create Windows Applications 143 EAE aioix File Edit Source Refactor Navigate Search Project Run Window Help Ci O Q BHoe e 7 7 Fy l ava OR An outline is not available B el Eclipse_QCMatPackJavaExamples EB com quinncurtis matpackjava examples ComplexMath E com quinncurtis matpackjava examples CubicSplinesTest amp E com quinncurtis matpackjava examples CurveFitTest E E com quinncurtis matpackjava examples EigenComplex E com quinncurtis matpackjava examples EigenRealSym E com quinncurtis matpackjava examples EigenRealUnSym 1 f i com quinncurtis matpackjava examples FFTTest i 3 com quinncurtis matpackjava examples LESTest F TADA J Applett java A A 2 Framei java amp T LESTe
62. auss Jordan with partial pivoting algorithm If the right hand side matrix b has more than one column a solution for each column is returned in the columns of the solution matrix x LESGJ public int gJ Solve DMatBase b DMatBase x LESXGJ public int gJ Solve XMatBase b XMatBase x 68 Systems of Equations Parameters b The right hand side matrix b for the system of equations Ax b The right hand side values of b are stored as a column vector If b has more then one column representing multiple right hand sides the solution x will have more than one column one column for each solution x Returns the solution s x to the system Ax b If b has more then one column representing multiple right hand sides the solution x will have more than one column one column for each solution Example The code in the Standard Interface for the Solution of Linear Equations section is extracted from the SimpleLES and SimpleLESX example programs The class is also used in the LESTest and LESXTest example programs Chapter 6 Eigenvalue and Eigenvector Solutions EigenBase EigenQR EigenQL EigenQR2 EigenXBase EigenXLR Eigenvalues and eigenvectors provide very important tools for analysis of many physical systems The eigenvalue problem is to find the non trivial solutions of the equation AX AX where 4 is an x n matrix X is a vector of length n and A is a scalar This equation involves two unknowns A
63. c int calcEigenValuesAndVectors XMatBase eigval XMatBase eigvect Parameters eigval Returns the eigenvalues of the source matrix A eigvect Returns the eigenvectors of the source matrix A Return Value Returns an error code Method EigenXBase checkEigenVectors es and Eigenvectors 79 The eigenvalues and eigenvectors of a matrix must satisfy the following equation Ax Lx where A is the original matrix L is an eigenvalue of A and x is the eigenvector of A corresponding to eigenvalue L Given the original matrix A and the eigenvalues and eigenvectors of A this method calculates Ax Lx for each eigenvalue eigenvector pair and compares the result to a threshold residual returning true if the actual residual is less than the test threshold residual This routine assumes the matrices use base zero 0 N 1 indices public static boolean checkEigenVectors XMatBase A XMatBase evalues XMatBase evectors XMatBase residuals double resid Parameters A The original complex matrix evalues A vector of the eigenvalues for the matrix A evectors A matrix of the eigenvectors for the matrix A residuals Returns a vector containing the actual residuals of Ax Lx resid A reference number if all residuals of ABS Ax Lx is less than resid 0 the method returns true Return Value Returns true if the all of the residuals of Ax Lx are less than the resid 0 value Example Code listing extracted from the EigenCompl
64. cated in the Quinn Curtis java lib folder The gematpackjava jar file contains the com quinncurtis matpackjava package reference in the import com quinncurtis matpackjava statements found in all of the example program source files that contain calls to the charting routines The qcmatviewjava jar file contains the com quinncurtis matviewjava package referenced in the import com quinncurtis matviewjava 136 Using QCMatPack to Create Windows Applications Select Finish to complete the creation of the basic project settings Say Yes ifit asks you for the Perspective Switch The Package Explorer should look like Using QCMatPack to Create Windows Applications 137 amp Properties for Eclipse_QCChart3DJ type filter text fm Info ky Builders E Java Code Style aiani EEE Fre I Java Compiler gechart3djava jar C Quinn Curtis javallib Javadoc Location E Source attachment None H Project References ZAE Javadoc location C Quinn Curtis java lib Native library location None Sif Access rules No restrictions 8A JRE System Library jre1 5 0_10 Ecinse_qcchatabievatxencls Bou To add the QCMatPack Javadoc documentation to the project expand the Eclipse_ QCMatPackJavaExamples node and right click on QCMatPackJava jar and select Properties 138 Using QCMatPack to Create Windows Applications W Eclipse_QCMatPackJavaExamples EB JRE Syste
65. cessing modes Chapter 13 is a tutorial that describes how to use QCMatPack to create Java Applications and Applets 2 Class Architecture of QCMatPack Major Design Considerations This chapter presents an overview of the QCMatPack for Java class architecture It discusses the major design considerations of the architecture QCMatPack is 100 managed code using only the Java core numeric routines in the matrix and numerical analysis classes The software makes NO calls to external non Java based libraries Simple to use matrix and numerical analysis classes for the most common problems in science engineering industry and business There are no limits regarding the size of vectors and matrices other than the amount of free memory on the host computer There are no limits regarding the size of the vectors and matrices used in the solution of general linear systems solution of eigenvalue problems multiple regression curve fitting and DSP FFT routines The chapter also summarizes the classes in the QCMatPack for Java library There are four primary features of the overall architecture of the QCMatPack for Java classes These features address major shortcomings in existing numerical software for use with both Java and other computer languages First QCMatPack implements a complex number class DComplex that provides a data structure functions and operators for manipulating complex numbers Operator overloading of the
66. ck for Java and summarizes all of the classes found in the software Chapter 3 describes complex number class used throughout the software Chapter 4 describes the real and complex number matrix classes that you will use to get data in and out of the numerical analysis routines Chapter 5 describes the classes that solve systems of linear equations for real and complex numbers Algorithms include Cholesky LU Decomposition QR Factorization and Gauss Jordan Chapter 6 describes the general eigenvalue and eigenvector problem Algorithms are presented for matrices that are real symmetric QL and QR real unsymmetric QR and complex LR 6 Introduction Chapter 7 describes a least squares multiple regression class that handles multiple regression and stepwise multiple regression along with related regression summary statistics Chapters 8 describes a curve fitting class that handles polynomial exponential rational polynomial and mixed curve fitting It also describes a cubic splines class Chapters 9 describes a digital signal processing class DSP that solves for real and complex FFT s power spectrums and related parameters Chapters 10 describes the matrix viewer controls used to display real and complex matrices in a Windows form Chapters 11 describes a thermocouple linearization class based on the most current standards NIST ITS 90 Chapter 12 discussed errors that the software can generate and the various error pro
67. d Green Blue components Class java awt font This class provides routines for defining and manipulating character fonts Package java util This package provides classes for tree list and vector management date and time facilities internationalization and miscellaneous utility classes a string tokenizer a random number generator and a bit array Introduction 3 Package java text This package provides classes and interfaces for handling text dates numbers and messages in a manner independent of natural languages Package java io This package provides classes for file i o and serialization Package javax swing event This package provides for the MouseInputListener events fired by Swing components Package javax swing This package provides classes of lightweight all Java language components that in theory if not practice work the same on all platforms Directory Structure The QCMatPack for Java class library uses the following directory structure Drive Quinn Curtis Root directory java Quinn Curtis Java based products directory RedistributableFiles contains the qcmatpackjava jar file that you should use when redistributing applications that use this library It does not contain the Javadoc documentation is much smaller than the version of the same file found in the lib folder docs Quinn Curtis Java related documentation directory lib Quinn Curtis Java related compiled librarie
68. d according to the formulas below It is assumed that results of the FFT have been returned in vector s va real parts and vb imaginary parts DSP Windowing Windowing functions are frequently used in digital signal processing One of the common applications is to reduce the undesirable effects related to spectral leakage Spectral leakage is the result of discontinuities in the sampled waveform Since the waveform is sampled for a finite length of time discontinuities occur at the end points of the waveform Windows are weighting functions applied to the raw data to reduce the spectral leakage associated with the finite observation interval A window function typically multiplies the data in the sample interval by a function which has a value of one 108 Digital Signal Processing at its center and tapers to zero or near zero at both end points The most common DSP windows are High and moderate resolution windows Rectangular Gauss Parzen Hann Hanning Hamming Bartlett Bartlett Hann Blackman Low dynamic range low resolution windows Nuttall Blackman Harris Blackman Nuttall Other Exponential Kaiser Bessel 512 Bartlett Bartlett Hann Nuttall Blackman Parzen 512 Blackman Nuttall N Flat Top Blackman Harris 1 0 tu 05 te isisnseiseiit 0 0 Welch Triangular Flat top Triangular TO AOSA 05 0 0 Rectangular 512 Exponenitial WO De ee ee po ee 0 5 S
69. d interface for solving for eigenvalues and eigenvectors for complex symmetric Hermitian and unsymmetric un Hermitian matrices There are four concrete classes derived from EigenBase and EigenXBase that solve for all general categories of the initial eigensystem matrix real symmetric real unsymmetric complex symmetric Hermitian actually and complex unsymmetric non Hermitian EigenQL Use this class to solve for the eigenvalues and eigenvectors of real symmetric matrices It uses the implicit QL algorithm and can solve real symmetric eigensystem problems which generate real only eigenvalue and eigenvector solutions Class Architecture 17 EigenQR Use this class to solve for the real and or complex eigenvalues of general not necessarily symmetric real matrices The QR algorithm can solve general eigensystem problems that generate both real and complex eigenvalue and eigenvector solutions EigenQR2 Use this class to solve for the real and or complex eigenvalues and eigenvectors of general real matrices It uses the QR algorithm and can solve general symmetric and non symmetric eigensystem problems which generate complex eigenvalue and eigenvector solutions EigenXLR Use this class to solve for the eigenvalues and eigenvectors of general complex matrices It uses the LR algorithm and can solve general Hermitian and non Hermitian eigensystem problems which generate complex eigenvalue and eigenvector solutions
70. de to an HTML page viewable from your web site Using Eclipse you would select the File Export option Select the files and resources in the LESTest directory and any other data files and resources you are using Using QCMatPack to Create Windows Applications 145 Se JAR Package Specification Define which resources to package into which JAR J Select the resources to export a com quinncurtis chart2djava E oe com quinncurtis chart2djava examples E oe com quinncurtis chart2djava examples bigchartdemo oe com quinncurtis chart3djava E oe com quinncurtis chart3djava examples E oe com quinncurtis chart3djava qcchart3djavademo i oe com quinncurtis matpackjava B be com quinncurtis matpackjava examples Be src O amp com quinncurtis matpackjaya examples ComplexMath O amp com quinncurtis matpackjava examples CubicSplinesTest o th com quinncurtis matpackjava examples CurveFitTest O amp com quinncurtis matpackjava examples EigenComplex O amp com quinncurtis matpackjaya examples EigenRealSym O amp com quinncurtis matpackjaya examples EigenRealUnsym O amp com quinncurtis matpackjaya examples FFTTest com quinncurtis matpackjava examples LESTest oO amp com quinncurtis matpackjava examples LESXTest im amp com quinncurtis matpackjava examples MatErrorTest U amp com quinncurtis matpackjaya examples Matrix Math nn 5 s Le i im y 2 Appleti java T Applicationi ja
71. deciding which algorithm to use Each algorithm is supplied in versions for both real and complex numbers Algorithm Strengths Weaknesses Gauss Jordan Stable for all but the most ill The slowest of the four algorithms conditioned of matrices Produces presented here Since it does not the inverse of the coefficient produce an intermediate matrix matrix A as a by product factorization it is not the Produces a solution in finite time recommended algorithm if you need on the order of 2n 3 operations to be able to solve multiple right hand sides b for the same coefficient matrix A Requires a square coefficient matrix A LU Stable for all but the most ill Not as fast as Cholesky conditioned of matrices Faster decomposition in solving symmetric than Gauss Jordan when used to positive definite matrices Requires solve for one right hand side a square coefficient matrix A Produces an intermediate matrix factorization that can be used for high speed solution of multiple right hand sides using back substitution Produces a solution in finite time on the order of 2n 3 operations QR Stable for all but the most ill Not as fast as Cholesky conditioned of matrices Faster decomposition in solving symmetric than Gauss Jordan when used to Systems of Equations 61 solve for one right hand side Produces an intermediate matrix factorization that can be used for high speed solution
72. e Overloaded Set a sub range of a column in the current matrix setSubCol Set the current matrix error setError 58 Matrix Classes Overloaded Set the sub range of a row or a column setSubRC Overloaded Set a sub range of a row in the current matrix setSubRow Set the values of a matrix vector using a double array setVector Set an element of the vector setVectorElement Returns the sum of squares of all elements in the matrix sSQ Returns the sum of all elements in the matrix sum Returns the sum of the ABS of all elements in the matrix sumAbs Swap a pair of columns of the current matrix swapCols Swap a pair of rows of the current matrix swapRows Returns the transpose of the current matrix T Returns a String that represents the current Object toString Transpose the current matrix transp Chapter 5 Solution of Linear Systems of Equations LESBase LESLU LESGJ LESCholesky LESQR LESXBase LESXLU LESXGJ LESXCholesky LESXQR The central problem of linear algebra and one of the most common numerical problems encountered is the solution of a set of linear equations Typically this takes the form of equations that when fully written out look something like the equations below 32 2 zta 1 2 2 4 4 lr ta 0 The coefficients of the left hand side are the numbers in front of the x1 x2 and x3 variables and are usually referred to as the coefficient matrix variable
73. e SOFTWARE Use of Support Services is governed by the Quinn Curtis Inc polices and programs described in the user manual in online documentation and or other Quinn Curtis Inc provided materials as they may be modified from time to time Any supplemental SOFTWARE code provided to you as part of the Support Services shall be considered part of the SOFTWARE and subject to the terms and conditions of this EULA With respect to technical information you provide to Quinn Curtis Inc as part of the Support Services Quinn Curtis Inc may use such information for its business purposes including for product support and development Quinn Curtis Inc will not utilize such technical information in a form that personally identifies you 5 TERMINATION Without prejudice to any other rights Quinn Curtis Inc may terminate this EULA if you fail to comply with the terms and conditions of this EULA In such event you must destroy all copies of the SOFTWARE 6 COPYRIGHT The SOFTWARE is protected by United States copyright law and international treaty provisions You acknowledge that no title to the intellectual property in the SOFTWARE is transferred to you You further acknowledge that title and full ownership rights to the SOFTWARE will remain the exclusive property of Quinn Curtis Inc and you will not acquire any rights to the SOFTWARE except as expressly set forth in this license You agree that any copies of the SOFTWARE will contain the same propr
74. e The MatBase abstract base class is the base class for all matrix types It defines the dimensions of a matrix usually DMatBase XMatBase DDMat DXMat Class Architecture 13 referred to as m x n representing the number ofrows and columns of a matrix It also contains numeric constants commonly used in matrix math routines The DMatBase class is the abstract representation of a real matrix It contains a large number of methods specific to real matrices At this time the only matrix class that derives from DMatBase is the concrete matrix class DDMat Future versions of the software may define one or more real sparse matrix classes derived from DMatBase The XMatBase class is the abstract representation of a complex matrix It contains a large number of methods specific to complex matrices At this time the only matrix class that derives from XMatBase is the concrete matrix class DXMat Future versions of the software may define one or more complex sparse matrix classes derived from XMatBase The DDMat class is the concrete representation of a real matrix using doubles for numeric storage It contains a large number of matrix manipulation methods optimized for dense double storage It also includes matrix math methods for the etc operators The DXMat class is the concrete representation of a complex matrix using DComplex objects for numeric storage It contains a large number of matrix manipulation methods op
75. e an external DDMat matrix to match this getMatrix one Returns the number of non zero elements in the matrix getNumNonZeroEl Get a row or a column of the matrix getRC Get a row in the matrix getRow Get a sub range of a column in the matrix getSubCol Get a sub range of a row or a column in the matrix getSubRC Get a sub range of a row in the matrix getSubRow Returns the transpose of the current matrix getTransp Returns a matrix that is the transpose of this one getTranspose Get an array of triplets representing the non zero elements of the getTriplets matrix Returns a double vector of the data in the matrix getVector Set an element of the vector getVectorElement Solve a system of linear equations using Gauss Jordan in the gJ Solve standard form Ax b where A is the coefficient matrix b is the right hand side and x is the solution The A matrix must be symmetric positive definite matrix Gauss Jordan with partial pivoting is a very stable method of solving systems of equations Returns the inversion of the current matrix I Initialize the entire matrix to the specified value initialize 56 Matrix Classes Insert a column into the current matrix increasing the column insertCol count of the matrix by one Insert a row into the current matrix increasing the row count of insertRow the matrix by one Returns the inverse of the array Gauss Jordan is used inv Returns true if a matrix el
76. e new matrix Method DDMat makeRHSFromCoefs DXMat makeRHSFromCoefs Using the coefficient matrix A a rhs vector b is constructed so that the solution x of the linear system of equations Ax b is 1 2 3 rank DDMat static DDMat makeRHSFromCoefs DDMat A DXMat static DXMat makeRHSFromCoefs DXMat A Parameters A Returns the coefficient array A of the linear system of equations Matrix Classes 49 Return Value Returns the rhs vector Method DDMat product DXMat product Calculates the product of two matrices A B and returns the result as a third matrix DDMat static DDMat product DDMat A DDMat B lt DXMat static DXMat product DXMat A DXMat B _ Parameters A The first of two arguments in the matrix multiple A B B The second of two arguments in the matrix multiple A B Return Value Returns the product of two matrices A B Method DDMat R DXMat R Returns the specified row or a sub range ofthe row as a vector DDMat DDMat R int row DDMat R int row int start int num DXMat DXMat R int row DXMat R int row int start int num Parameters row specified column start specified column to start copying from start number to copy 50 Matrix Classes Return Value Returns the specified row or a sub range of the row as a vector Method DDMat setElement DXMat setElement Set a matrix element DDMat void
77. e overall F Stat value and the standard error and the F Stat value for each regression equation coefficient StepwiseMulReg The stepwise multiple regression algorithm includes forward backward and stepwise regression algorithms for automatic selection of regression variables Stepwise regression removes variables that fail an F statistic significance test usually leaving you with fewer variables and a simpler model than you started with SplitMulRe The SplitMulReg splits the original data into two develops p g p g Sp 8 a model using one dataset and uses that to predict the values in the second dataset Curve Fitting CurveFit CubicSplines Experimental data often takes the form of measurements of some kind temperature or volts for example taken at discrete and possibly periodic times In such cases a table of the measured variable versus time is produced The goal of the research is to uncover the underlying function f x which relates temperature to time Once the underlying function is discovered the solution is usually only an approximation of the true function the function can be used to calculate the values of f x temperature in this case for x values points in time which were not part of the original measured data The CurveFit algorithms create a single model that is the least squares fit for all the data points in the original dataset The CubicSplines class creates a family of cubic equations one cubic equation for
78. e the rows represent groups and the columns represent data values for each group ROW MAJOR Use the MatCSV setOrientation method to initialize the csv argument for the proper data orientation int load MatCSV csv java lang String filename int rowskip int columnskip int load java lang String filename Parameters csv An instance of a MatCSV object filename The name of the file rowskip Skip this many rows before starting the read operation columnskip For each new row skip this many columns before starting read 52 Matrix Classes Method DMatBase save XMatBase save This method will create a text file and output the dataset to that file ina CSV Comma Separated Value format A CSV file can be read by popular spreadsheet and word processing programs Some localization for different operating systems and locales can be handled by the modifying the default csv MatCSV object The file can be organized so that the columns represent groups and the rows represent data values for each group COLUMN MAJOR or the where the rows represent groups and the columns represent data values for each group ROW MAJOR Use the MatCSV setOrientation method to initialize the csv argument for the proper data orientation int save MatCSV csv java lang String filename int save MatCSV csv java lang String filename boolean append int save java lang String filename Parameters csv An instance of a MatCSV object filena
79. e voltage into a polynomial equation or by looking up the voltage in standard thermocouple tables published by the National Bureau of Standards NBS The thermocouple routines in this package are based on the reference tables summarized in NIST ITS 90 as published in the NIST Monograph 175 This standard defines 8 standard thermocouple types B E J K N R S T Each of the 8 thermocouple types have one or more polynomial equations which define thermocouple voltage as a function of temperature It is important to understand that the thermocouple temperature to voltage tables presented in NIST ITS 90 are not from direct measurements Thermocouple voltages were sampled at discrete intervals of temperature and equations fitted to these sampled 122 Thermocouple Linearization points The NIST ITS 90 tables were generated from the fitted equation and not actual measured values 2000 1800 1600 1400 1200 oOo oOo Degrees Cent O NIST ITS 90 Thermocouple Linearization 20 30 40 50 60 70 80 Millivolts The polynomial equations used to convert from volts to temperature and temperature to volts are those specified in the NIST ITS 90 tables The implicit errors for the voltage to temperature conversions are listed below Thermocouple Type Temperature Range Voltage Range Error Range Thermocouple Type Voltage Range Error Range Temperature Sub range 1 J 210 C to 0 C
80. each sample interval that can be used to interpolate y values based on x values not in the original dataset CurveFit The CurveFit algorithm calculates the coefficients for a single equation that fits all of the data points in the original CubicSplines Class Architecture 19 dataset It supports four different algorithms for curve fitting polynomial exponential rational fraction and mixed a combination of the first three methods The CubicSplines class creates a family of cubic equations one cubic equation for each sample interval that can be used to interpolate y values based on x values not in the original dataset Digital Signal Processing DSP A fast Fourier transform FFT is an efficient algorithm to compute the discrete Fourier transform DFT and its inverse FFTs are of great importance to a wide variety of applications from digital signal processing and solving partial differential equations to algorithms for quick multiplication of large integers The DSP class is built around the Cooley Tukey radix 2 FFT algorithm Support routines windowing magnitude frequency and phase calculations and popular variants power spectrum calculations real only FFTs are also included DSP The DSP class includes methods for DDMat real only and DXMat complex FFT calculations It also includes static FFT functions that accept double array arguments for maximum speed The raw signal can be processed with a DPS window
81. eal matrix base class 12 13 20 33 35 42 43 51 52 64 65 66 67 72 73 75 76 83 86 90 96 111 113 118 152 Index Real matrix class vii 2 7 8 10 12 13 19 20 33 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 55 62 72 74 77 84 85 87 88 89 91 96 97 98 99 102 103 109 111 112 113 114 115 118 119 131 Regression statistics 20 21 84 85 87 88 89 91 96 97 98 99 RegStats 20 21 84 85 87 88 89 91 96 97 98 99 Split regression 8 17 18 21 90 91 SplitMulReg 8 17 18 21 90 91 Stepwise regression 8 17 18 21 85 86 87 88 89 StepwiseMulReg 8 17 18 21 85 86 87 88 89 TCLin 21 121 124 125 126 127 Thermocouple linearization 21 121 124 125 126 127 XMatBase 12 13 20 33 35 42 43 51 52 64 65 66 67 73 75 76 77 78 79 111 112 113 114 118 XMatViewer 8 9 11 20 117 118 XTriplet 19 20 21 36
82. ear equations may become ill conditioned and produce erroneous results If there are many independent variables which are not correlated with the dependent one they will never less all be included in the model and the model may be unreliable as a result MulReg constructor Set up a multiple regression including all of the independent variables in the final solution public MulReg DMatBase iv DMatBase dv Set up a multiple regression including only the independent variables flagged by the consid vector in the final solution public MulReg DMatBase iv DMatBase dv IVect consid 84 Multiple Regression iv A matrix of the independent variable x values dv A matrix of the dependent variable y values consid A vector contains consideration values one for each independent variable i the iv array plus the constant Element 0 contains the constant flag the other elements contain the flags for the iv array Initialize the array elements using one of the consider constants REG_VARIN REG_VAROUT REG VARCONSIDER If an element is REG_VARIN or REG_VARCONSIDER the corresponding variable is included in the regression Use the solve method to solve the current multiple regression using the current parameters The methods returns the regression equation coefficients and summary regression statistics Method MulReg solve public int solve DDMat regcoef RegStats regstats Parameters regcoef Returns the calculated r
83. eferred to as the linear least squares solution to the problem LESQR LESXQR Constructors LESQR public LESOR DMatBase a LESXQR public LESQR XMatBase a Parameters a The coefficient matrix A for the Ax b system of equations Example The code in the Standard Interface for the Solution of Linear Equations section is extracted from the SimpleLES and SimpleLESX example programs The class is also used in the LESTest and LESXTest example programs Cholesky Decomposition Class LESCholesky LESXCholesky The LESCholesky and LESXCholesky classes implement the Cholesky decomposition algorithm Cholesky decomposition starts with the standard definition of a linear system of equations Ax b The coefficient matrix A must be symmetric positive definite in order for the algorithm to work The Cholesky algorithm factors the non singular matrix A into the matrix product of an upper and lower triangular matrix A LU much like the LU algorithm In this case L and U are specially factored so that it U is the transpose of L So the equation becomes A LL The resulting linear algebra equation LL x b is solved for x Most of the work is done in the factoring of the L matrix Once L is calculated for a given right hand side b the solution x can be quickly calculated using a combination of forward and back substitution For a single right hand side the Cholesky decomposition algorithm is two to three times faster than the LU
84. egression coefficients regstats Returns regression statistics The RegStats class is a simple class that encapsulates several related summary statistics for multiple regression RegStats class Public Instance Properties Get Set overall F statistic ovfst Get Set multiple correlation coefficient R r value Get Set coefficient of determination R rsq Squared Get Set Returns the Standard error of see estimate Get Set significance of regression flag sigfla Get Set sum of residuals squared sumressq Multiple regression example Extracted from the example program Mulregtest Multiple Regression 85 MulReg mulreg new MulReg DDMat A new DDMat Independent variable matrix DDMat RHS new DDMat Ddependent variable matrix Initialize A and RHS matrices mulreg new MulReg A RHS for int i 0 i lt mulreg getIndependentVariables numColumns 1 i mulreg setConsider i MulReg REG VARIN RegStats regstats new RegStats DDMat coefficients new DDMat mulreg solve coefficients regstats Other useful regression statistics can be calculated on demand These include the partial F statistics F Test statistics can take a lot of time to calculated since it involves as many separate multiple regressions as there are independent variables Call the method FTestAnalysis to calculate the F statistics Method MulReg FTestAnalysis public int FTestAnalysis DDMat fstat double conf_le
85. ement is non zero isNonZero Solve a system of linear equations in the standard form Ax b IES Solve where A is the coefficient matrix b is the right hand side and x is the solution If A is a square matrix then LU Decomposition is used to solve for x If A NumRows gt A NumColumns i e the system of equations is over determined the QR algorithm is used to solve for x Overloaded These methods will open a text file and read the load CSV Comma Separated Value formatted data values in that file A CSV file can be created by popular spreadsheet and word processing programs This version of Load assumes CSV values are organized ina ROW_MAJOR format Solve a system of linear equations using LU Decomposition in IU_Solve the standard form Ax b where A is the coefficient matrix b is the right hand side and x is the solution Make a Hilbert matrix of the specified rank makeHilbert Initializes the matrix to empty 0 rows and 0 columns matNull Calculate the a norm for the current matrix norm Calculate the 1 norm of the current matrix norm 1 Calculate the 2 norm of the current matrix norm2 Calculate the infinity norm of the current matrix normInf Returns the number of row or column elements in the matrix numEIRC Solve a system of linear equations using QR Factorization in qR_Solve the standard form Ax b where A is the coefficient matrix b is the right hand side and x is the solution Overloaded Returns the specified row
86. ether you look at the real parts the complex parts or the magnitude of the complex number Complex Math Functions The DComplex class also include many functions that need to be explicity calculated for complex numbers These include the complex conjugate magnitude same as absolute value and modulus normalization polar angle square root raise to a power and 28 Complex Numbers exponent e to a complex number Each function comes in two forms the first is a static function that operates on an argument that is an instance of a DComplex class The second is a DComplex member function that operates on its own instance variables DComplex conj Method The complex conjugate of the complex number z a bi is defined to be a bi Get the conjugate of a complex number DComplex conj static DComplex conj DComplex a DComplex cl new DComplex 1 0 2 0 DComplex c2 cl conj DComplex c3 DComplex conj c1 The complex conjugate of c1 is 1 0 2 01 DComplex abs Method The absolute value or modulus or magnitude z of a complex number a bi is jz V B Calculate the absolute value of a complex number double abs static double Jabs DComplex c DComplex cl new DComplex 3 0 4 0 double rl cl abs double r2 DComplex abs c1 The absolute value of cl is areal number DComplex Abs 3 41 5 DComplex normalize Method Complex Numbers 29 The normalize method scales the
87. etric or Hermitian for complex numbers matrix that has the added property that all of its eigenvalues are positive While it seems of little importance for the non mathematician the positive definite matrix shows up automatically in many linear algebra applications including the solution of least squares problems multiple regression and the solution of coupled differential equations Algorithms Cholesky factorization in particular have been developed explicitly to exploit the structure of positive definite matrices and these algorithms can be 2 4 times as fast as algorithms that work on more generalized matrices Orthogonal Matrix an orthogonal matrix is a square matrix Q whose transpose is its inverse FQ QQ f Base 0 vs Base 1 Matrices Originally most computer matrix math and linear algebra algorithms were written in FORTRAN and ALGOL Those computer languages define the index range of their built in array types dimension m x n as extending from 1 m Matrix Classes 35 1 n Matrices of this type use base 1 indexing In contrast C C C and Java all use base 0 indexing where the index range of an array with the dimensions m x n is 0 m 1 0 n 1 Algorithms written for base 1 indexing will not work and will generally crash programs written using compilers that require base 0 indexing This software assumes base 0 indexing All matrices dimensioned to the size m x n use array indexing in the range 0
88. ex example program DXMat A new DXMat Initialize complex matrix DXMat EValuesl new DXMat DXMat EVectorsl new DXMat EigenXLR eigensolvel new EigenXLR A eigensolvel calcEigenValuesAndVectors EValuesl EVectorsl DXMat residuals new DXMat double resid 1 0e 5 boolean okl EigenXLR checkEigenVectors A EValuesl 80 Eigenvalues and Eigenvectors EVectorsl residuals ref resid DXMat residuals new DXMat residuals returned here Double refnum 1 0e 4 if any residual gt this value solution invalid returns actual max error boolean ok EigenXBase checkEigenVectors A EValues EVectors residuals refnum Chapter 7 Multiple Regression Regression analysis is a statistical tool for the evaluation of the relationship of one or more independent variables X X2 Xx to a single dependent variable Y for a fixed number of observations The primary goal of regression analysis is to obtain predictions of Y using the known values of X These predictions are made using a regression equation They are subject to error and are only estimates of the true values of Y There can be other goals of regression analysis for example to determine which of several independent variables are important and which are not for describing and predicting a dependent variable The simplest case of regression is known as simple linear regression and it calculates the relationship between
89. f identical dimensions m and n can be added Given m by n matrices A and B their sum A B is the m by n matrix computed by adding corresponding matrix elements double aData 1 2 3 4 5 6 7 8 9 double Hoata 5 6 7 e 19 u Il DDMat A new DDMat aData DDMat B new DDMat bData DDMat C DDMat add A B Matrix Subtraction static subtraction DDMat a DDMat d static DXMat subtraction DXMat a DXMat d 38 Matrix Classes Two or more matrices of identical dimensions m and n can be subtracted Given m by n matrices A and B their difference A B is the m by n matrix computed by subtracting corresponding matrix elements double double DDMat A DDMat B DDMat C aData 1 2 3 4 5 6 7 8 9 bData 5 6 7 9 7 8 3 2 1 new DDMat aData new DDMat bData DDMat subtraction A B Matrix Multiplication static static static static static static static static DDMat DDMat DDMat DXMat DXMat DXMat DXMat DXMat multiply DDMat a DDMat b matrix matrix operator overload multiply DDMat a double b matrix scalar operator overload multiply double b DDMat a scalar matrix operator overload multiply DComplex b DXMat a scalar matrix operator overload multiply double b DXMat a scalar matrix operator overload multiply DXMat a DComplex
90. f the actual residual is less than the test threshold residual This routine assumes the matrices use base zero 0 N 1 indices public static boolean checkEigenVectors DMatBase A DMatBase evalues DMatBase evectors DMatBase residuals double resid public static boolean checkEigenVectors DMatBase A XMatBase evalues XMatBase evectors XMatBase residuals double resid Parameters A The original real matrix evalues A vector of the eigenvalues for the matrix A 74 Eigenvalues and Eigenvectors evectors A matrix of the eigenvectors for the matrix A residuals Returns a vector containing the actual residuals of Ax Lx resid A reference number if all residuals of ABS Ax Lx are less than resid the method returns true The maximum error is returned in resid 0 Return Value Returns true if the all of the residuals of Ax Lx are less than the resid 0 value Example Code listing extracted from the EigenRealSym example program DDMat A new DDMat Matrix A needs to be initialized results returned in EValues and EVectors DDMat EValues new DDMat DDMat EVectors new DDMat EigenQL eigensolve new EBigenQL A eigensolve calcEigenValuesAndVectors EValues EVectors DDMat residuals new DDMat residuals returned here double refnum 1 0e 4 if any residual gt this value solution invalid returns actual max error boolean ok EigenBase checkEigenVectors
91. flagged by the consid vector in the final solution public SplitMulReg DMatBase iv DMatBase dv IVect consid Multiple Regression 91 iv A matrix of the independent variable x values dv A matrix of the dependent variable y values consid A vector contains consideration values one for each independent variable i the iv array plus the constant Element 0 contains the constant flag the other elements contain the flags for the iv array Initialize the array elements using one of the consider constants REG_VARIN REG_VAROUT REG VARCONSIDER If an element is REG_VARIN it is always included in the final regression equation If an element is REG_VAROUT it is excluded from the final regression equation If an element is REG_VARCONSIDER the corresponding variable available for selection into the final regression equation Use the Solve method to solve the current multiple regression using the current parameters The methods returns the regression equation coefficients and summary regression statistics Method SplitMulReg solve public int solve DDMat regcoef RegStats regstats Parameters regcoef Returns the calculated regression coefficients regstats Returns regression statistics The RegStats class is a simple class that encapsulates several related summary statistics for multiple regression 8 Curve Fitting CurveFit CubicSplines Experimental data often takes the form of measurements of some kind temperature or
92. fthe complex number using the member variables r and i or the setRe getRe setIm getIm methods DComplex c new DComplex 1 0 2 0 eo 6 07 Ck 107 double r2 c getRe double i2 c getIm Complex Math Operators Complex numbers are added subtracted multiplied and divided in a manner analagous to that of real numbers following the associative commutative and distributive laws of algebra together with the equation i 1 Since the standard Java arithmetic operators do not automatically work with user defined classes the DComplex class includes these functions so that you can use a mix of DComplex and double values in math expressions Special note Java does not have the ability to process overloaded operators As a result complex math operators are implemented as Java static method calls Complex Numbers 25 tic DComplex add DComplex a DComplex b tic DComplex add DComplex a double d tic DComplex add double d DComplex a a bi c di a c 4 b4 d i DComplex cl new DComplex 1 0 2 0 DComplex c2 new DComplex 3 0 4 0 double rl 5 0 o3 cl c2 rl 6 0 DComplex c3 DComplex add DComplex add cl c2 rl 6 0 lex subtract DComplex a DComplex b lex subtract DComplex a double d lex subtract double d DComplex a a bi c di a c b dyi DComplex cl new DComplex 1 0 2 0 DComplex c2 new DComplex
93. gital Signal Processing Power Spectrum Calculations This function calculates the power spectrum periodogram of a sampled data set The power spectral density of each frequency bin is returned along with the exact frequency in Hz for each frequency bin Method If N data points of a waveform w t are collected at equal intervals then the FFT of the waveform can be denoted as W t The periodogram estimate of the power spectrum P k is defined as 1 2 Py a k 1 2 N 2 1 l 2 2 ryan ll l P W z N where k is defined for zero and positive frequencies Method DSP powerSpectrumCalc Calculates the power spectrum periodogram of a sampled dataset public static DDMat powerSpectrum XMatBase source double rInterval DDMat freqvalues Real only version of the power spectrum periodogram of a sampled dataset public static DDMat powerSpectrum DDMat source double rInterval DDMat freqvalues Digital Signal Processing 115 Parameters source Raw data The power spectrum routine takes the FFT of the data pass in raw data rInterval The sample period between adjacent data points in the seconds freqvalues Returns the frequency values associated with each harmonic index Return Value Returns the power spectrum of the original signal Example Extracted from the example FFTTest int n 4096 double xR new double n double iR new double n for int 10 i lt m i DDMat
94. gression summary statistics are found in the MulReg StepwiseMulReg and SplitMulReg classes Class Architecture 9 Curve fitting The CurveFit class includes support for polynomial exponential rational polynomial and mixed curve fitting The CubicSplines class uses a cubic splines algorithm for discrete function evaluation Digital signal processing The DSP class includes routines for real and complex FFT s and power spectrum calculations Auxiliary routines calculate FFT magnitudes phases and frequencies Matrix viewer classes The DMatViewer and XMatViewer classes are UserControl subclasses that display real and complex number matrices respectively Thermocouple linearization The thermocouple linearization routines will convert from millivolts to temperature and temperature to millivolts for B E J K N R S T thermocouple types Calculations are is based on the NIST ITS 90 standard as published in the NIST Monograph 175 A summary of each category appears in the following section Matrix Viewer Classes System Windows Forms UserControl MatView DMatViewer XMatViewer The matrix viewer classes display a matrix in a Java JPanel derived object As such it can be placed and resized anywhere on a JPanel derived form If the matrix is larger than the size you have created the view scrollbars give the option of scrolling the rows and columns of the matrix The DMatViewer class displays real matrix while the XMatViewer disp
95. hat it works with both square and an over determined where the number of rows is greater than the number of columns m gt n coefficient matrices Like the LU and Cholesky algorithms it is efficient at solving multiple right hand sides since the QR factorization of the primary coefficient matrix only needs to be solved for once From 16 Class Architecture that point on multiple right hand sides can be solved for using the factored matrix and simple back substitution LESXQR The LESXQR class is the complex number version of the LESQR class using the QR factorization algorithm to solve a system of complex number linear equations Solution of Eigensystems EigenBase EigenQL EigenQR EigenQR2 EigenXBase EigenXLR The classes above implement algorithms for the solution of real and complex eigensystems The EigenBase class is the abstract representation of an eigensystem solver that starts with a coefficient matrix of real numbers The EigenXBase class is the abstract representation of an eigensystem solver that starts with a coefficient matrix of complex numbers EigenBase This class is the abstract base class for eigensystem solvers that start with real coefficient matrix It defines a standard interface for solving for eigenvalues and eigenvectors for real symmetric and unsymmetric matrices EigenXBase This class is the abstract base class for eigensystem solvers that start with complex coefficient matrix It defines a standar
96. he holdout group The original independent variables are replaced with their orthogonalized equivalents so that inclusion of a new variable does not affect the coefficients for those already included in the model At every step the independent variable with the smallest difference of coefficients determined independently on both data groups is chosen The algorithm does not involve the solution of a system of linear and is not sensitive to the colinearity of the independent variables Depending on the data the residual error on the original data may be higher than that using one of the other multiple regression algorithms but the regression equation is usually more simple and reliable The StepMulReg class allows the programmer to include and exclude independent variables from the final regression equation using the consider consid parameter of the StepMulReg constructor The selection mechanism will choose between variables marked REG_VARCONSIDER Those marked with REG_VARIN will be included in the model no matter how insignificant they are found to be Those marked with REG _VAROUT will be excluded from of the model Setup up the regression the problem using the SplitMulReg constructor SplitMulReg constructor Set up a split multiple regression including all of the independent variables in the final solution public SplitMulReg DMatBase iv DMatBase dv Set up a split multiple regression including only the independent variables
97. hods libraries for execution on super mainframe and even microprocessor based computers are written in FORTRAN These include EISPACK LINPACK LAPACK BLAS SLATEC and MINPACK In many cases algorithms were translated from ALGOL into FORTRAN often obscuring the logic and flow of the original algorithm FORTRAN versions of the BLAS and LAPACK libraries are still used today in compiled form with modern computer languages You are able to find C versions of these libraries that are nothing more than C wrappers around the original compiled FORTRAN code You will even find these compiled libraries used with Java and the Net languages C and VB passing data through an interface that permits mixing unmanaged compiled libraries with managed Net code The result though cannot be considered managed code This software implements what we consider the most useful numerical algorithms written entirely in Java with no calls to external libraries that do not utilize the Java 2 Introduction managed memory system That includes routines for the generalized solution of linear algebra problems and specific algorithms for solutions of linear equations eigensystems digital signal processing multiple regression and curve fitting Since the algorithms are implemented using 100 managed code they are the better choice for workstation and web based Java applications Many of the algorithms in science and engineering require the use of complex
98. ic Real Unsymmetric Complex Hermitian Complex Un Hermitian depends on the category Eigenvalues Class to Use solution numeric type EigenQL real EigenQR complex EigenXLR complex EigenXLR complex Eigenvectors Class to Use solution numeric type EigenQL real EigenQR2 complex EigenXLR complex EigenXLR complex 72 Eigenvalues and Eigenvectors If you have a real symmetric matrix use the EigenQL class for both eigenvalues and eigenvectors If you have a real unsymmetric matrix use EigenQR for eigenvalues only and EigenQR2 if you want eigenvalues and eigenvectors If you have a complex matrix of any type use the EigenXLR class for both eigenvalues and eigenvectors Standard Interface for Eigenvalue and Eigenvectors Solutions The eigen problem classes use the same interface with minor variants depending on whether or not the source matrix is real or complex and whether or not the eigenvalues and eigenvectors are real or complex You define the problem by passing the coefficient matrix A in the constructor of the class you are using Real matrices of type DDMat use the EigenQL EigenQR and EigenQR2 classes Complex matrices of type DXMat use the EigenXLR class If you need only the eigenvalues of a matrix use the calcEigenValues method If you need the eigenvalues and eigenvectors of a matrix use the calcEigenValuesAndVectors The checkEigenVectors method will check the accuracy of
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100. ints defining a curve The cubic splines algorithm implementation creates a set of equations n 1 x n 1 based on the constraints that the y value and the first and second derivatives at each of the interior nodes must match for each interval in the range xo xn Because each segment is only linked to its nearest neighbors the resulting system of equations is tridiagonal A tridiagonal system of equations can be solved using a specialized algorithm that is orders Curve Fitting 101 of magnitudes faster than generalized methods such as Gauss Jordan and LU decomposition The solution is used to created the table n x 4 of cubic equation coefficients that are used to interpolate the f x values for x in the range Xo Xn Cubic splines has several limitation that you must be aware of before using it The raw data must be monotonic in x This means that the data must be sorted in increasing order by x and that the x values must always be increasing You cannot have exactly the same x value repeated twice The data cannot reflect a function that has a discontinuous first or second derivative A good example of a discontinuous second derivative is a step change in the function This also violates the previous limitation since in order to implement a true step change a given f x value is associated with two x values For example the line segments represented by the data points Point 1 0 0 Point 2 1 0 Point 3 1 1 Point 4 2 1
101. ion or disclosure by the Government is subject to restrictions as set forth in subparagraph c 1 ii of The Rights in Technical Data and Computer SOFTWARE clause of DFARS 252 227 7013 or subparagraphs c i and 2 of the Commercial Computer SOFTWARE Restricted Rights at 48 CFR 52 227 19 as applicable Manufacturer is Quinn Curtis Inc 18 Hearthstone Dr Medfield MA 02052 USA 11 MISCELLANEOUS If you acquired the SOFTWARE in the United States this EULA is governed by the laws of the state of Massachusetts If you acquired the SOFTWARE outside of the United States then local laws may apply Should you have any questions concerning this EULA or if you desire to contact Quinn Curtis Inc for any reason please contact Quinn Curtis Inc by mail at Quinn Curtis Inc 18 Hearthstone Dr Medfield MA 02052 USA or by telephone at 508 359 6639 or by electronic mail at support Quinn Curtis com iv Table of Contents Table of Contents Table of Content sa ae aaa vii 1 Introduction to CCM ALP aC ees 2 sac adade ee l Tilorals nis ansehe 1 OCMatPack for Java Background uuuuussessenise na 1 QCMatPack for Java Dependeneiss nenne a caedbe dab edyeataaseaaedens 2 Directory Struct re weitesten lassen 3 Chapter S mmary sai ieee eier iR ls lau 5 DG lass Archit ct re ot OC Wat PACK ea 7 Major Design Considerations nee nee 7 QCMatPack for Java Class Summary dan 8 Matix Viewer lasses hs Een 9 Complex number class ne ran 11
102. ions using complex numbers For example the following line in bold is a valid expression involving DComplex complex number arguments DComplex cl new DComplex 1 0 2 0 DComplex c2 new DComplex 3 0 4 0 double r1 5 0 ria 63 cl g2 ri 6 0 DComplex c3 DComplex add DComplex multiply DComplex multiply cl c2 rl 6 0 The DComplex class also includes a large number of mathematical functions used in conjunction with complex numbers Conjugate Magnitude Abs Sqrt Exp Pow and Arg The DXMat class is the main class used to represent vectors and matrices of DComplex complex numbers The linear equations classes LESXLU LESXGJ LESXCholesky and LESXQR are specific to the DComplex complex number type The eigensystem classes EigenQR and EigenQR2 EigenXLR also uses the DComplex number type DComplex constructors 24 Complex Numbers Create a DComplex object using a pair of doubles a single double or another DComplex object DComplex DComplex c Constructor for a complex number DComplex double re Constructor for a complex number DComplex double re double im Constructor for a complex number re The real part of the complex number im The imaginary part of the complex number You can also specify a complex number using the polar coordinates convention using the FromPolar static method public static DComplex fromPolar double mag double angle Access the real and imaginary part o
103. ky algorithm requires that the coefficient matrix is a symmetrical where the elements below the diagonal are the transpose of the elements above the diagonal positive definite matrix It turns out that this class of matrices are quite common in systems of equations so this algorithm should be given special consideration It is up to twice as fast as LU decomposition and four times as fast as Gauss Jordan Like LU decomposition it is particularly useful for solving systems of equations involving multiple right hand sides since the Cholesky decomposition of the primary coefficient matrix only needs to be solved for once From that point on multiple right hand sides can be solved for using the decomposed matrix and simple back substitution The LESXCholesky class is the complex number version of the LESCholesky class using the Cholesky decomposition algorithm to solve a system of complex number linear equations The complex version of the Cholesky algorithm requires that the source coefficient matrix is Hermitian the complex version of a symmetric matrix where the elements below the diagonal are the complex conjugate of the elements above the diagonal and the diagonal is real positive and definite The LESQR class uses the QR factorization algorithm to solve a system of real number linear equations It is a very stable efficient method of solving systems of equations It has one major advantage over the other equation solvers in t
104. lays a complex matrix The matrix viewers include options to edit matrix values change font characteristics the numeric format of the matrix element values and the number of displayed rows and columns 10 Class Architecture m DMatViewer displays matrix elements of a DDMat real matrix 0 70 0 73i 0 52 0 50i 0 25 0 88i Class Architecture 11 a 0 42 0 57i 0 90 0 98i 0 49 0 77i 0 89 0 49i 0 08 0 58i 0 68 0 12i 0 43 0 72i 0 72 0 74i 0 87 0 51i 0 48 0 89i 0 39 0 74 0 32 0 76i 0 93 0 67i 0 85 0 17i 0 07 0 95i 0 14 0 36i 0 05 0 89i 0 54 0 49i 0 55 0 23i 0 73 0 24 0 07 0 66i 0 81 0 22i 0 28 0 34 0 47 0 60i 0 45 0 82i 0 44 0 80 0 83 0 17i EN PO a XMatViewer displays matrix elements of a DXMat complex matrix Complex number class DComplex In mathematics a complex number is a number of the form a bi where a and b are real numbers and i is the imaginary unit with the property i 1 The real number a is called the real part of the complex number and the real number 5 is the imaginary part Real numbers may be considered to be complex numbers with an imaginary part of zero that is the real number a is equivalent to the complex number a 0i source Wikipedia 12 Class Architecture This software represents a complex number using the DComplex class It represents the real and imaginary parts of acomplex number using two double
105. le n xR i L3 Math cos f 31 Math cos 2 Add in a fixed offset and Randomize it a bit xR i 4 0 0 5 DMatBase getRandomDouble iR i 0 DDMat origData new DDMat xR DXMat FFTResult new DXMat Since iR i is all zeros can use real only version of DSP DSP fftobj new DSP origData FFTResult fftobj fFTCalc false Method DSP fFTMagnitude Calculates the FFT magnitude associated with a given harmonic index public static double fFTMagnitude XMatBase source int i Parameters source Source vector contains the results of an FFT calculation i The index of the harmonic you are calculating the magnitude for Return Value Returns the FFT magnitude associated with a given harmonic index Example DDMat magdata new DDMat FFTResult getLength 112 Digital Signal Processing for int i 0 i lt FFTResult getLength i magdata setElement i DSP fFTMagnitude FFTResult i Calculate all of the FFT magnitudes associated with an FFT results matrix public static DDMat fFTMagnitude XMatBase source he Parameters source Source vector contains the results of an FFT calculation Return Value Returns a DDMat matrix of the FFT magnitudes associated with the FFT data Example Extracted from the example FFTTest DDMat magdata DSP fFTMagnitude FFTResult Method DSP fFTPhase Calculates the FFT phase associated with a given harmonic index public
106. le Linearization 127 This would be replaced by your own routine that reads uses a voltmeter or A D converter to read the thermocouple voltage double getThermocoupleMilliVolts double tcmv 10 1 millivolts return tcmv This would be replaced by your own routine that reads uses a voltmeter or A D converter to read the cjc temperature double getCJCTemperature double cjctemp 16 3 degrees Cent return cjctemp public double tCLinearizeWithCJC int tci ink tot TCLiA TC TYPE K TCLin tc new TCLin tct You would need to implement the routine GetThermocoupleMilliVolts double tcMilliVolts getThermocoupleMilliVolts You would need to implement the routine GetCJCTemperature double cjcTemp getCJCTemperature Convert terminal block temperature to equivalent thermocouple millivolts double cjcEquivMilliVolts tc tempToMilliVolts cjcTemp Add the cjc millivolts to the thermocouple millivolts tcMilliVolts cjcEquivMilliVolts Convert the cjc compensated millivolts to temperature double tcTemperature tc milliVoltsToTemp tcMilliVolts return tcTemperature 12 Error Handling The matrix math classes and numerical algorithm classes can generate a limited number of runtime errors Any time an error is detected an object MatError object is created and that object processes the error The MatError object can process the error in several different ways I
107. ll namespace of the main applet class com quinn curtis gematpack examples LESTest Appletl class e The ARCHIVE attribute specifies two jars one that contains the code of the applet LESTest jar and the other the QCMatPack libraries qematpackjava jar and gematviewjava jar e The CODEBASE attribute tells the browser in which directory to find the applet If the applet is in the same directory as the calling HTML page the CODEBASE attribute is not necessary e The HEIGHT and WIDTH attribute specify the pixel dimensions of the window the applet will display in This is the simple version of enabling an applet in a web page The more complicated way and the way recommended by Sun is to use the HTML lt OBJECT gt tag Apparently the lt APPLET gt tag has been deprecated by the lt OBJECT gt tag in the HTML standard and Sun is encouraging web designers to use it You can read all about it from the source at http docsun cites uiuc edu sun_docs C solaris_9 SUNWaadm JVPLUGINUG p5 html They have a program that will convert lt APPLET gt code to lt OBJECT gt code and forces the browser to download a Java plug in if one is not already present on the computer It also uses the EMBED tag instead of OBJECT if the Netscape browser is used The Netscape trick involves the use of the lt COMMENT gt tag Read about that in the Sun article hyperlinked above The converted version of the HTML applet code looks like lt OBJECT classid c1lsid 8A
108. m Library jre1 6 0_03 Fide gematviewjava jar C Quinn Curtis javatlib H wa jar C Ouinn Curtis java lit Description Select Javadoc Location Select the Javadoc in Archive radio button and browse to qcmatpackjava jar file as the Archive Path then select doc as the Path within Archive Using QCMatPack to Create Windows Applications 139 Properties for C Quinn Curtis java lib qcmatpackjava jar type filter text x Javadoc Location Orr Javadoc Location Java Source Attachment specify the location URL of the documentation generated by Javadoc The Javadoc location will contain a file called package list Javadoc URL e g http www sample url org doc or file c myworkspace myproject doc J yadot location pathy Browseis validate Javadoc in archive Archive path C Quinn Curtisijava libigacmatpackjava jar Browse Path within archive doc F Validate Restore Defaults Apply OK Cancel From that point on Eclipse F1 and Shift F2 help should be available on the QCMatPack Java classes The project is still without the source files of the QCMatPack Java examples and these are added next Right click the Eclipse QCMatPackJavaExamples project in the Package Explorer and select Import The Into Folder field should say QCMatPackJavaExamples Select File System and Browse to the Quinn Curtis java directory
109. matrix element column The column of the matrix element Return Value Returns the value of the matrix element Method DDMat getMatrix DXMat getMatrix Initialize an external matrix to match this one DDMat void getMatrix DDMat dest DXMat void getMatrix DXMat dest Parameters dest destination array Method DDMat getTranspose DXMat getTranspose Returns the transpose of a matrix A complex transpose converts the complex numbers to their complex conjugate i e a bi gt a bi DXMat DDMat getTranspose DXMat DDMat getTranspose Return Value Returns the transpose of the matrix 46 Matrix Classes Method DDMat getVector DXMat getVector Returns a vector of the data in the matrix DDMat double getVector DXMat DComplex getVector Return Value Returns a vector of the data in the matrix Method DDMat getVectorElement DXMat getVectorElement Get an element of a vector DDMat double getVectorElement int element DXMat DComplex getVectorElement int element Parameters element The index of the vector element Return Value The value of the vector element Method DDMat I DXMat I Returns the inversion of the current matrix DDMat DDMat I Matrix Classes 47 DXMat DXMat I Return Value Returns the inversion of the current matrix Method DDMat invert DXMat invert Returns the inversion of the specified matrix DDMat static intlin
110. mber class The DComplex class implements a complex number class using a pair of doubles to hold the real and imaginary parts of a complex number Extensive support for support of complex numeric functions and operator overloading including conjugate magnitude power exponent and square root functions Matrix classes Two matrix classes DDMat for real numbers and DXMat for complex numbers simplify data manipulation and the input output of data to from the numerical analysis routines Solution of linear systems There is a symmetrical group of eight classes for the solution of real and complex linear systems These classes correspond to real and complex versions of Cholesky factorization LESCholesky LESXCholesky LU Decomposition LESLU LESXLU QR Factorization LESQR LESXQR and Gauss Jordan LESGJ LESXGJ Each class provides for the solution of one right hand side multiple right hand sides and matrix inverse Solution of eigensystems The eigensystem problem is divided into three categories real symmetric real unsymmetric and complex both symmetric Hermitian and complex unsymmetric The real symmetric case is handled using the QL class EigenQL The EigenQR and EigenQR2 classes solve the real unsymmetric case And the EigenXLR class will solve general complex eigenvalue problems Multiple regression General multiple regression forward regression backward regression stepwise multiple and split regression and re
111. me The name of the file append True and if the file exists data is appended to the file A Summary of all DDMat DXMat Methods Public Static Shared Methods Overloaded Returns the inversion and determinant of the determinant specified matrix Returns the inversion of the specified matrix invert Make a pair of matrices that form the A and b matrices of a linear set of equations Ax b The matrices are constructed so that the solution x of Ax b is always a vector with the values 1 2 3 rank Make a random square matrix of the specified rank symmetry and diagonal dominance makeRandomMatrix Using the coefficient matrix A a rhs vector b is constructed so that the solution x of the linear system of equations Ax b is 1 makeRHSFromCoefs j2 3 rank Calculates the product of two matrices A B and returns the product result as a third matrix Matrix Classes 53 Public Instance Constructors Overloaded Initializes a new instance of the DDMat class DDMat Public Instance Properties Returns a reference to the Array object that represents internal getDataBuffer data buffer This reference will remain valid only as long as the array is not forced to resize forcing a reallocation of the internal buffer Returns true ifthe matrix is square numRows numColumns isSquare Returns true ifthe matrix is symmetric A Transpose A isSymmetric Returns the number of rows column
112. ment of the matrix A by the scalar c double DI abata 112 EEE TI FF fF os 2 DDMat A new DDMat aData double c 3 1415 DDMat B DDMat divide A c Mix all of the operators in complex math expressions c double p adata 11 2 Brb 71 18 Tr 7 Cow 4 double y boata 15 9 780 She 9r 6 41 13 2 993 N 77 4 x 5 double Ip amp Data 115 6 1 891 19 748 841 12 1 0911 77 2 5 DDMat A new DDMat aData DDMat B new DDMat bData DDMat C new DDMat cData DDMat D A B 3 1415 3x5 VB Dim aData As Daublet ily 27 3z 4 4z Se The 7z u Sy DOP 3 x 4 Dim bData As Double 115 u Be Ble f9 Vy Se She foe 2 Tp GG Ohy t3y te Op 4y 7J 44 x 5 Din Data As Double 11 Gy Te By She 9r Fe Be Ge the fay 2 Te Ge OF 22 5 Dim A As New DDMat aData Dim B As New DDMat bData Dim C As New DDMat cData Matrix Classes 41 Dim D As DDMat A B 3 1415 3 x 5 VB 2005 and greater only Dim D As DDMat DDMat Add DDMat Multiply A B DDMat Multiply C 3 1415 Matrix Methods Many methods are available in the DDMat and DXMat classes that manipulate the rows and columns of the matrix Other methods calculate important characteristics of the matrix Since the DDMat and DXMat classes include the same routines they are documented in parallel A subset of the available methods follows A complete description is found in the XX XXX help file Method DDMat
113. mocouple temperature in degrees Centigrade 126 Thermocouple Linearization Return Value Returns the thermocouple voltage in millivolts There are also simple methods getMinMilliVolts getMaxMilliVolts getMinTemp and getMaxTemp that return the minimum and maximum voltages and temperatures associated with a given thermocouple type Example The example program TCLinTest divides each thermocouple voltage range into 10 000 equal segments and calculates the milliVoltsToTemp conversion for each point It then feeds the resulting temperature back to the tempToMilliVolts method and compares the resulting millivolt measurement to the original and remembers the maximum error This represents a good test of the overall accuracy of the linearization routines since the milliVoltsToTemp and tempToMilliVolts methods used different linearization polynomials int toi TCL IC TYPE J TCLin tc new TCLin tci double mV tc getMinMilliVolts mv2 0 0 double temp 0 0 int count 10000 double voltageInc tc getMaxMilliVolts tc getMinMilliVolts count double maxerror 0 0 error 0 for int 1 05 2 lt count i temp tc milliVoltsToTemp mV mv2 tc tempToMilliVolts temp error Math abs mV mv2 if error gt maxerror MaxeError error if error gt 0 1 break mV voltagelnc A more practical method would simulate an actual thermocouple measurement with cold junction compensation Thermocoup
114. mon applications of polynomial curve fitting is thermocouple linearization where the voltage present at a thermocouple junction is transformed into temperature by applying a polynomial equation which has been solved for using these types of curve fitting techniques Polynomial curve fitting is actually a special case of least squares multiple regression Eqn 1 above is transformed into the standard multiple regression equation Eqn 2 Y C C X C X G X CaP Xa using the transforms 94 Curve Fitting X X RER EX X X A system with one dependent variable Y and one independent variable X is turned into a system of one dependent variable Y and as many independent variables X1 X2 Xn as the order of the curve fit Each new independent variable is the original independent variable raised to some power Once the transform is carried out the multiple regression function is called to calculate the solution coefficients Co through Cn One of the most important measures of the quality of the polynomial fit is known as the correlation coefficient The correlation coefficient is usually abbreviated as the r value R of the fit The closer this value is to 1 the better the fit Another measure of the fit is called the coefficient of determination usually referred to as the r squared R value This value is equal to the square of the correlation coefficient Again the closer this value is to 1 the better the fit The standard er
115. numbers in the form a bi where a and b are real numbers and i represents the square root of 1 The Java runtime at this time does not include a numeric type or object class that supports complex numbers and math therefore one DComplex is included with this software Building on the Java standard double numeric type and our DComplex complex number class two generalized matrix and linear algebra classes DDMat and DXMat were created The numerical algorithms were then implemented using those matrix classes for input and output QCMatPack for Java Dependencies The com quinncurtis matpackjava package is self contained It uses only standard classes that ship with the Java 1 2 API In addition all components referenced in the software are lightweight components The software uses the major Java packages listed below Package java awt The java awt package is the core of Java AWT Abstract Windowing Toolkit It contains all of the classes that create user interfaces and draw graphics and images The java awt classes most often used in the Quinn Curtis MatViewJava classes are the graphics graphics2D geom color font and image Class Java awt graphics The graphics class is the abstract base class for all graphics It contains the basic line drawing text area fill and color control functions needed to output a graph to an output device Class java awt color Provides a class to define colors in terms of their individual RGB Re
116. of equations x number of unknowns with m gt n The algorithm factors the non singular matrix A into the matrix product A QR of an orthogonal matrix Q and upper triangular matrix R The resulting linear algebra equation QRx b is solved for x Most of the work is done in the factoring or A into the Q and R matrices Once Q and R are calculated for a given right hand side b the solution x can be quickly calculated using a combination of forward and back substitution For a single right hand side the QR decomposition algorithm is two to three times faster than Gauss Jordan algorithm Since the QR matrices do not need to be recalculated for each new right hand side it is even faster if you are solving for multiple right hand sides Unlike the Gauss Jordan algorithm the QR algorithm does not produce a matrix inverse as a by product If you call the Inverse method the inverse is calculated by solving the Ax b equation using b b is a square matrix in this case with the A dimensions n x n as the identity matrix If Ax b and b is the identity matrix then x must be the inverse of A Solving for x produces the inverse of A 66 Systems of Equations When the QR algorithm is solving for the over determined case where the number of equations exceeds the number of unknowns the result is the same as a multiple regression problem run on the same data The solution are the values that minimize the sum of the residual errors squared usually r
117. oltage now a sum of the thermocouple voltage and the equivalent junction block voltage is what you actually convert back to temperature using our milliVoltsToTemp method If you are not familiar with cold junction compensation here is a good reference for you to study http www maxim ic com appnotes cfm an_pk 4026 Class TCLin The TCLin class provides methods to convert thermocouple millivolts into equivalent temperatures and thermocouple temperatures into equivalent millivolts The later conversion temperature to millivolts can be used in place of the Seebeck coefficient for use in the thermocouples cold junction compensation calculations TCLin constructor This constructor initializes the class for a specific thermocouple type Use one of the thermocouple TCLin TC_TYPE type constants TC_TYPE J TC_TYPE B TC TYPE E TC_TYPE K TC TYPE N TC TYPE R TC TYPE S TC_TYPE T public TCLin int tect Parameters tct Thermocouple type Use one of the thermocouple TCLin TC_TYPE type constants TC TYPE J TC TYPE B TC TYPE E TC TYPE K TC TYPE N TC TYPE R TC TYPE S TC TYPE T Once the class is instantiated for a specific thermocouple call the milliVoltsToTemp method to convert thermocouple millivolts into a temperature degrees Centigrade and tempToMilliVolts to convert a thermocouple temperature into millivolts used in cold junction compensation Method TCLin milliVoltsToTemp Convert the
118. origData new DDMat xR Since iR i is all zeros can use real only version of DSP DSP fftobj new DSP origData DDMat freqdata new DDMat This routine takes the raw data pre RealFFT sample interval for 1000 0 Hz is 0 001 seconds DDMat powerSpectrum fftobj powerSpectrum 0 001 freqdata 10 Matrix Viewers com quinncurtis matviewjava MatViewer MatViewerBase DMatViewer XMatViewer The QCMatViewJava DLL contains a couple of simple Java controls DMatViewer and XMatViewer that display real and complex matrix data on a form in a scrollable matrix format These controls are for use in Java JPanel based applets and applications The matrix viewer size number of rows number of columns fonts and numeric format are all user programmable The matrix viewers can display a number of rows and columns only limited the amount of free memory accessible by the program A matrix viewer is read only by default but it can be made editable using the enableEdit method DMatViewer XMatViewer XMatViewer a r a 0 1 2 3 4 0 10 11 12 o 0 84 047 013 032 0 20 0 9 69 0 00i o 0 03 0 28i 0 03 0 28i 0 01 0 58i 1 0 25 0 22 0 39 0 10 0 83 1 0 89 0 88i 1 0 48 0 53i 0 48 0 53i 0 56 0 18i 2 0 86 0 04 0 90 0 58 0 13 2 0 89 0 88i 2 1 00 0 00i 1 00 0 00i 0 43 0 59i 3 0 86 038 046 047 039 3 1 08 0 00i 3 0 63 0 51i 0 63 0 51i 0 16 0 02i 4 0 85 0 73 0 31 0 64 0 81 4 0 80
119. pose A Matrix must be symmetric ie java lang IllegalArgumentException POSDEFINITE MATRIX REQUIRED Matrix must be positive definite java lang IllegalArgumentException HERMITIAN MATRIX REQUIRED Matrix must be Hermitian java lang IllegalArgumentException NO_DSPACE Insufficient Disk Space java lang RuntimeException FILE ERROR File Read Write Error java lang RuntimeException INVALID FILENAME Invalid file name java lang RuntimeException SINGULAR Matrix Is Singular Does not throw and exception NO ACC Solution Is Not Accurate Does not throw and exception Select the error processing method that you want by setting the static MatError setReportMode method to one of the report mode constants public final static Int public final statie int ErrorReportModes DO NOTHING 0 ErrorReportModes MESSAGE BOX 1 Error Handling 131 public final static int ErrorReportModes SYSTEM CONSOLE 2 public final static int ErrorReportModes_THROW_EXCEPTION 3 public final static int ErrorReportModes_EXIT 4 The default is error reporting mode is MatError ErrorReportModes MESSAGE BOX The example below is extracted from the MatErrorTest example program and demonstrates the use of runtime exception error processing using try catch blocks public void matIndexErrorWithException try int m 2 n 20 DDMat A new DDMat m n for int i 0
120. quares or SEE The class includes in the final regression equation all of the independent variables except for those explicitly excluded by the consideration property or method In matrix math form the algorithm is as follows X matrix of independent variable values Y vector of dependent variable values B vector of regression coefficients to solve for Multiple Regression 83 The objective of the regression is to solve for B given X and Y First the data matrix X is converted into the square matrix A by multiplying it by its own transpose A X X A constant vector C is calculated by multiplying the transposed data matrix X by the dependent variable Y C xX y The solution vector B is calculated by solving the linear system of equations remember Ax b AB C for the unknown B using one of the equation solving algorithms Gauss Jordan LU decomposition Cholesky decomposition or QR factorization The Cholesky decomposition algorithm can be used because the A matrix XTX will always be symmetric positive definite B is a vector that returns the coefficients of the least squares multiple regression fit to the original data This method works well if the independent variables are not highly correlated with each other and if most of them contribute significantly to the dependent variable Otherwise it may encounter problems For example if two of the independent variables are highly or perfectly correlated the lin
121. re not selected as equation multipliers which can introduce round off errors that lower the numerical accuracy of the solution The algorithm produces the inverse of the original coefficient matrix as a by product LESGJ LESXGJ Constructors LESGJ public LESGd DMatBase a LESXGJ public LESGd XMatBase a Parameters a The coefficient matrix A for the Ax b system of equations Example The code in the Standard Interface for the Solution of Linear Equations section is extracted from the SimpleLES and SimpleLESX example programs The class is also used in the LESTest and LESXTest example programs LU Decomposition Class LESLU LESXLU The LESLU and LESXLU classes implement the Crout version of the LU decomposition algorithm LU decomposition starts with the standard definition of a linear system of equations Ax b where A is square n equations for n unknowns It factors the non singular matrix A into the matrix product A LU of a lower triangular matrix L and upper triangular matrix U with unit diagonal The resulting linear algebra equation LUx b is solved for x Most of the work is done in the factoring or A into the L and U matrices Once L and U are calculated for a given right hand side b the solution x can be quickly calculated using a combination of forward and back substitution For a single right hand side the LU decomposition algorithm is two to three times faster than Gauss Jordan algorithm Since
122. rent matrix determinant Dimension the matrix for size 0 rows 1 0 columns 1 dimMatrix Returns true if the matrix dimensioned to 0 0 dimZero Overloaded Determines whether the specified Object is equal to l the current Object Overloaded Determines whether the specified Object is equal to equals the current Object Initializes the matrix to empty 0 rows and 0 columns freeMat Initializes the matrix from the source matrix fromDMatBase Initialize the matrix using an array of triplets DTriplet fromTriplet Elements not initialized remain at their default 0 0 value Return a copy of the current matrix as a base one array getBaseOneMatrix Get a 1D array representing a copy of a column in this matrix co getCo Overloaded Get a matrix compatible with this one initialized with and dimensioned to the size of the source array getCompatibleMatrix Returns a reference to the internal double data buffer that getDataBuffer stores the matrix data Overloaded Get a vector representing a copy of the diagonal getDiagonal elements of this matrix Overloaded Returns a vector that is the diagonal of this one getDiagonal Matrix Classes 55 Returns the element of the matrix getElement Return the current matrix error getError Returns the number of rows columns in the matrix getLength Overloaded Get a 2D array representing a copy of the values in getMatrix this matrix Overloaded Initializ
123. rix classes you will use for the input and output of numeric data to the linear algebra linear systems eigenvalue eigenvector and FFT algorithms DDMat for real matrices and DXMat for complex matrices Using the matrix classes you can read and write to individual elements add delete swap copy scale read write rows and columns of a matrix The matrix classes also carry out fundamental linear algebra subroutines such as vector and matrix norms dot products and saxpy operations Matrices can be initialized an element at a time from standard Java one and two dimensional double arrays double and double from other matrices and from files The matrix classes also include matrix math methods that permit matrix math The matrix classes also double as vector classes a matrix with a column or row dimension of one is considered a column or row vector respectively In that case matrix elements can be accesses using 1 D vector brackets instead of the normal 2D matrix brackets The MatBase DMatBase and XMatBase classes are the base classes of the DDMat and DXMat classes See the hierarchy at beginning of the chapter Since these classes are abstract they can not be instantiated They do contain important static and instance properties and methods used in the derived DDMat and DXMat classes The only matrix classes you need to know how to instantiate are these two classes The DDMat and DXMat classes are very symmetrical The
124. rmocouple millivolts to temperature in degrees Centigrade public double milliVoltsToTemp double mv Parameters Thermocouple Linearization 125 mv Thermocouple voltage in millivolts Return Value Returns the thermocouple temperature in degrees Centigrade Method TCLin milliVoltsToTemplterative Convert thermocouple millivolts to temperature in degrees Centigrade This is the ultimate in voltage to temperature conversion for a thermocouple It first uses the MilliVoltsToTemp method to provide an initial estimate Then it backward iterates on the on the temperature to millivolts polynomials for which there is zero error using the Newton Raphson of root solving This ends up by returning a millivolts to temperature conversion accurate to at least the specified tolerance public double milliVoltsToTemplterative double mv double tolerance Ve Parameters mv Thermocouple voltage in millivolts tolerance Iterations are carried out until the tolerance level is reached Because the Newton Raphson method converges extremely fast typically the accuracy of the conversion is 3 6 orders of magnitude more accurate than the tolerance The minimum tolerance that you can specify is 10e 10 Return Value Returns the thermocouple temperature in degrees Centigrade Method TCLin tempToMilliVolts Convert thermocouple temperature to millivolts public double TempToMilliVolts double temp Parameters temp Ther
125. ror of the estimate usually abbreviated SEE is yet another measure of the quality of the fit It is a measure of the scatter of the actual data about the fitted line The smaller the SEE value the closer the actual data is to the theoretical polynomial equation Other equations such as rational functions exponentials etc also can be used for curve fitting These functions are handled in a manner analogous to polynomial curve fitting The original data is transformed in a manner which reduces a non linear curve fitting problem into linear curve fitting problem which can be solved using linear multiple regression Class CurveFit The CurveFit class will fit a polynomial equation to a set of data representing discrete samples of some function y f x It can also fit exponential rational polynomial and mixed equation to the same data A polynomial curve fit is represented by the equation y f x gt y ata xl a x a x a x2 The polynomial curve fit algorithm solves for the coefficients ag to an An exponential curve fit is represented by the equation Curve Fitting 95 y flx gt y ac The exponential curve fit algorithm solves for the coefficients a and b The rational polynomial is represented by the equation 96 Curve Fitting 1 2 3 starr a x ta x 8a x y fx gt y 1 b x b x b x b x The rational polynomial curve fit algorithm solves for the coefficients ag to an and
126. s r Contents Create new project in workspace Create project From existing source Directory C Quinn Curtis Java Development workspace Eclipse_OCMal BONSE JDK Compliance Use default compiler compliance Currently 1 4 igure default Use a project specific compliance 1 4 r Project layout Use project folder as root for sources and class files Create separate source and output folders Configure default lt Back Next gt Enis Cancel Select Next to define the Java build settings Select the Libraries tab and select Add External JARs and browse to the Quinn Curtis java lib directory and add the qcmatpackjava jar and qcmatviewjava jar files Using QCMatPack to Create Windows Applications 135 New Java Project xj Java Settings J Define the Java build settings ln 7 C Source gt Projects A Libraries Order and Export JARs and class Folders on the build path E En gcmatpackjava jar C Quinn Curtis javallib Add JARs Eny qcematviewjava jar C Quinn Curtis javallib GESE IRE System Library jre1 6 0_03 Add External JARs Add Yariable Add Library Add Class Folder Edit Remove Default output Folder Eclipse _QCMatPackJavaExamples Browse All of the example programs reference the QCMatPack and QCMatView for Java jar files gematpackjava jar and qcmatviewjava jar lo
127. s jar and components directory The jar files in this folder contain Javadoc documentation for the classes in the jar file and are much larger than jar files of the same name in the RedistributableFiles folder com quinn curtis matpackjava 4 Introduction Examples Java examples directory ComplexMath A simple example demonstrating the use of the DComplex complex number class QCMatPackExamplel The tutorial example program EigenRealSym Eigenvalue Eigenvector solutions for real symmetric matrices using the Householder QL and Hessen QR algorithms EigenRealUnsym Eigenvalue Eigenvector solutions for real unsymmetric matrices using the Hessen QR algorithm EigenComplex Eigenvalue Eigenvector solutions for complex matrices using the Hessen LR algorithm SimpleLES A very simple example demonstrating the solution of systems of equations using real numbers LESTest Solving systems of equations real numbers using Cholesky LU Decomposition QR Factorization and Gauss Jordan algorithms SimpleLESX A very simple example demonstrating the solution of systems of equations using complex numbers XLESTest Solving systems of equations complex numbers using Cholesky LU Decomposition QR Factorization and Gauss Jordan algorithms MulRegTest A multiple regression test program StepRegTest A stepwise multiple regression test program CurveFitTest Curve fitting solutions using polynomial
128. s for Engineers McGraw Hill Book Company New York 1985 Golub Gene H and Van Loan Charles F Matrix Computations John Hopkins University Press 1989 Harris F J On the Use of Windows for Harmonic Analysis with the Fast Fourier Transform Proceedings of the IEEE Volume 66 No 1 January 1978 Kleinbaum David G et al Applied Regression Analysis and Other Multivariable Methods PWS Kent Publishing Company 1988 Lapin Lawrence L Statistics for Modern Business Decisions Harcourt Brace Jovanovich Inc 1978 Rabiner L R and Gold B Digital Signal Processing Prentice Hall Englewood Cliffs 1974 SLATEC Common Mathematical Library version 4 1 1993 SLATEC stands for Sandia Los Alamos Air Force Weapons Laboratory Technical Exchange Committee available at http gams nist gov serve cgi Package SLATEC Smith James T C For Scientists and Engineers Intertext Publications McGraw Hill 1991 Stoer J Bulirsch R Introduction to Numerical Analysis Springer 1992 Strang Gilbert Linear Algebra and Its Applications Harcourt Brace Jovanovich 1988 Temperature Electromotive Force Reference Functions and Tables for the Letter Designated Thermocouple Types Based on the ITS 90 Natl Inst Stand Technol Monograph 175 1993 http srdata nist gov its90 main 150 Appendix Trefethen Lloyd N and Bau David III Numerical Linear Algebra Society for Industrial and Applied Mathematics
129. s in the matrix getLength Returns the number of columns in the matrix getNumColumns Returns the number of rows in the matrix getNumRows Returns true if no error pending in matrix isOK Public Instance Methods Add a constant to a column of the current matrix addConstCol Add a constant to a row of the current matrix addConstRow le 7 Overloaded Returns the specified column as a vector Solve a system of linear equations using Cholesky cholesky Solve Decomposition in the standard form Ax b where A is the coefficient matrix b is the right hand side and x is the solution The A matrix must be symmetric positive definite matrix Cholesky is the fastest technique for solving linear systems of equations though it does require the coefficient matrix be symmetric positive definite Returns an object that is a clone of this matrix object clone Calculate square matrix condition number getCondNum Convert the current matrix from a base one matrix to a base zero matrix 54 Matrix Classes Convert a base zero matrix to a base one matrix copy lie Returns the determinant of the current matrix Delete a column from the current matrix decreasing the column deleteCol count of the matrix by one Delete a row from the current matrix decreasing the row count deleteRow of the matrix by one Returns the determinant of the current matrix Gauss Jordan is deter used Overloaded Returns the determinant of the cur
130. setElement int element double value void setElement int row int column double value DXMat void setElement int element DComplex value void setElement int row int column DComplex value Parameters row The row of the matrix element column The column of the matrix element value The value of the matrix element Method DDMat setMatrix DXMat setMatrix DDMat void setMatrix double source void setMatrix DDMat source DXMat void setMatrix DComplex source High speed routine for copying a matrix void setMatrix DXMat source initialize a matrix using a DXMat source Parameters source source array Matrix Classes 51 Method DDMat T DXMat T Returns the transpose of the current matrix A complex transpose converts the complex numbers to their complex conjugate i e a bi gt a bi DDMat DDMat T DXMat DXMat T Return Value Returns the transpose of the current matrix Method DMatBase load XMatBase load These method will open a text file and read the CSV Comma Separated Value formatted data values in that file A CSV file can be created by popular spreadsheet and word processing programs Some localization for different operating systems and locales can be handled by the modifying the default csv MatCSV object The file can be organized so that the columns represent groups and the rows represent data values for each group COLUMN MAJOR or the wher
131. snnnnnnnnnneennnnnnnnnn T2 QR Algorithm for Real Unsymmetric Matrices cessessessensennsensnesnnnsnnnnnennennenn 74 LR Algorithm for Complex Matrices a elek 71 Chapter 7 Multiple Regression za a RE I ldncaeiatanee ayaa aes 81 Least Squares Multiple Regression ccssssssteriitecasnigsietscy stra uiakiane tials 82 Automatic Selection of Regression Variables 0 ccccccccescceeseeesceeseeceeceeeeeeeeesseeesseenes 85 8 C rye FINS RE Re ge ee Sree re RE SORTER SOREN a a Sie een ere a tener eer 93 Polynomial Curve Rute 3 Nessie 93 Table of Contents Cubic SPIES nic nn SSR TERN SSR 100 9 Dig tal S ipnahPfotessne u ikea malen ea 105 PRT Basis er EE a i aE iaoi Va a A as 105 DSP ENV 1H Wy TASS ae seele 107 10 Matrix Viewer sn Bu Oa a a a a 117 11 Thermoconple Limeanzation aan 121 NIST TES 90 ze nr 121 Cold Junction Compensationv ale 123 127 Error Handlin as are e ee ee 129 13 Using QCMatPack for Java to Create Applications and Applets 133 Eclipse inen A E ch dg E A ER R RE ee eva ie 133 Java Applets i an a a e a a a a e 143 BEIECICU Bibliography eese A E E E ak ahd 149 INDEX eva shinies A E mel E EA 151 viii 1 Introduction to QCMatPack Tutorials A simple tutorial that describe how to get started with the QCMatPack for Java charting software are found in Chapter 12 Using OCMatPack for Java to Create Applications and Applets QCMatPack for Java Background Numerical analysis is the art of calculating sol
132. st java LESTestAppleti htm LESTest pplet2 htm 25 7 com quinncurtis matpackjava examples LESXTest iB com quinncurtis matpackjava examples MatErrorTest com quinncurtis matpackjava examples MatrixMath JB com quinncurtis matpackjava examples MulRegTest E com quinncurtis matpackjava examples QCMatPackExamplei EB com quinncurtis matpackjava examples SimpleLES E com quinncurtis matpackjava examples SimpleLESX 8 com quinncurtis matpackjava examples SplitRegTest feat com quinncurtis matpackjava examples StepRegTest EB com quinncurtis matpackjava examples TCLinTest Bh JRE System Library jre1 6 0_03 H gematviewjava jar C Quinn Curtistjavallib ge gematpackjava jar C Quinn Curtis javallib cag ca li F p E amp amp EE Problems Javadoc 53 Declaration Sa m com quinncurtis matpackjava examples LESTest Applicationi java Title LESTest example program Description Example program for QCMatPack for Java software Copyright Copyright c 2007 2008 Company Quinn Curtis Inc author Staff version ti iki com quinncurtis matpackjava examples LESTest Application java Eclipse_QCMatPackJavaExamples Java Applets While Java Applications are useful if you are creating a program to run as a standalone application on a desktop Java Applets are used to create Java programs that run in a web browser All of the example programs include both application and
133. static double fFTPhase XMatBase source int Parameters source Source vector contains the results of an FFT calculation i The index of the harmonic you are calculating the phase for Return Value Returns the FFT phase associated with a given harmonic index Example DDMat phasedata new DDMat FFTResult getLength for int i 0 i lt FFTResult Length i pasedata setElement I DSP fFTPhase FFTResult i Digital Signal Processing 113 Method DSP fFTPhase2 Calculates the FFT phases associated with already calculated fft data public static DMatBase fFTPhase2 XMatBase source Ve Parameters source Source vector contains the results of an FFT calculation Return Value Returns a DDMat matrix of the fft phases associated with the fft data Example Extracted from the example FFTTest C DDMat phasedata DSP fFTPhase2 FFTResult VB Dim phasedata As DDMat DSP fFTPhase2 FFTResult Method DSP fFTFrequency2 Calculates the FFT frequencies associated with already calculated FFT data public static DDMat fFTFrequency2 int iNumdat double rSampleFreq i Parameters INumdat The number of data points must be a power of 2 rSampleFreq The sample frequency of the raw data Return Value Returns a matrix of the FFT frequencies associated with the FFT data Example Extracted from the example FFTTest DDMat freqdata DSP fFTFrequency2 FFTResult Length 1000 0 114 Di
134. t can print an error message to a message box on the screen QCGMatLib Error xi ws Row index out of range error at DDMat print an error message to the system console ar DIFHIEISESE Application 101 Java Application C Program Files Javalidk1 5 0_03 bin javaw exe Nov 30 2007 2 56 53 PM Row index out of range error at com quinneurtis matpackjava DDNatl84cc09 a Application 101 Java Application C Program Files Javalidk1 5 0_03 bin javaw exe Nov 30 2007 3 09 20 PM ava lang ArrayIndexOutOfBoundsException Row index out of range error at DDMat at com quinncurtis matpackjava MatError throwException MatError java 393 at com quinncurtis matpackjava MatError reportError MatError java 351 at com quinncurtis matpackjava MatError lt init gt MatError java 306 at com quinncurtis matpackjava MatBase rowColValid MatBase java 221 at com quinncurtis matpackjava DDMat setElement DDMat java 500 at com quinncurtis matpackjava examples MatErrorTest MatErrorTest mat IndexErrorWithException MatErrorTest java 163 at com quinncurtis matpackjava examples MatErrorTest MatErrorTest generateError MatErrorTest java 291 at com quinncurtis matpackjava examples MatErrorTest MatErrorTest button4_Click MatErrorTest java 346 at com quinncurtis matpackiava examples MatErrorTest MatErrorTest access 4 MatErrorTest java 3421 ignore the error or exit the program The standard errors that the soft
135. the CalcEigenValues method next that accepts a complex matrix instead public int calcEigenValuesAndVectors DMatBase eigval DMatBase eigvect Calculate the complex eigenvalues and eigenvectors for the source matrix A public int calcEigenValuesAndVectors XMatBase eigval XMatBase eigvect 76 Eigenvalues and Eigenvectors Parameters eigval Returns the eigenvalues of the source matrix A eigvect Returns the eigenvectors of the source matrix A Return Value Returns an error code Example See code listing under EigenQR2 Class EigenQR2 The EigenOR class will calculate all of the eigenvalues and eigenvectors for a real symmetric matrix using the implicit QR method EigenQR2 Constructor public EigenQR2 DMatBase a Parameters a The source matrix for the eigensystem calculations The source matrix must be square but it does not have to be symmetric Method calcEigen Values Calculate the complex eigenvalues for the source matrix A public int calcEigenValues XMatBase eigval Parameters eigval Returns the eigenvalues of the source matrix A Return Value Returns an error code Method calcEigenValuesAndVectors Calculate the complex eigenvalues and eigenvectors for the source matrix A es and Eigenvectors 77 public int CalcEigenValuesAndVectors XMatBase eigval XMatBase eigvect Parameters eigval Returns the eigenvalues of the source matrix A eigvect Returns the eigenvectors of the so
136. the LU matrices do not need to be recalculated for each new right hand side it is even faster if you are solving for multiple right hand sides Unlike the Gauss Jordan algorithm the LU algorithm does not produce a matrix inverse as a by product If you call the Inverse method the inverse is calculated by solving the Ax b equation using b b is a square matrix in this case with the same dimensions as A as the identity matrix If Ax b and b is the identity matrix then x must be the inverse of A Solving for x produces the inverse of A Systems of Equations 65 LESLU LESXLU Constructors LESLU public LESLU DMatBase a LESXLU public LESLU XMatBase a Parameters a The coefficient matrix A for the Ax b system of equations Example The code in the Standard Interface for the Solution of Linear Equations section is extracted from the SimpleLES and SimpleLESX example programs The class is also used in the LESTest and LESXTest example programs QR Factorization Class LESQR LESXQR The LESQR and LESXQR classes implement the QR factorization by way of a Householder reflections algorithm QR factorization starts with the standard definition of a linear system of equations Ax b Unlike Gauss Jordan LU or Cholesky algorithms the A matrix does not have to be square The system of equations can be over determined where the number of equations is greater than the number of unknowns producing an m x n matrix number
137. the forward or inverse FFT of the raw data If the original data is real only no imaginary component a real only version of the FFT algorithm is called If the original data is complex a complex version of the FFT algorithm is called public DXMat fFTCalc boolean bInverse E Apply a FFT window to the raw data and then calculate the forward or inverse FFT of the raw data public DXMat fFTCalc boolean bInverse DSPWIN fftwin Parameters bInverse Set to true and an inverse FFT is calculated twin DSP window type Use one ofthe DSPWIN enumerated constants NOWIN RECTANGULAR GAUSS HAMMING HANN BARTLETT BARTLETT HANN NUTTALL BLACKMAN HARRIS BLACKMAN NUTTALL FLATTOP PARZEN WELCH BLACKMAN and TRIANGULAR KAISER BESSEL and EXPONENTIAL Return Value Returns the FFT of the raw data as a DXMat matrix A few of the windows have additional parameters that you may want to change Set the Gauss window sigma parameter using DSP setGaussSigma Set the Exponential window final value parameter using DPS setExpFinalValue Set the Kaiser Bessel window alpha parameter using DSP setKBAlphaValue Default values DSP setGuassSigma 0 45 DSP setExpFinalValue 0 1 DSP setKBAlphaValue 1 Digital Signal Processing 111 Example Extracted from the example FFTTest int n 4096 double xR new double n double iR new double n for int i 0 i lt m i double f 10 0 Math PI 2 double i doub
138. the solution by comparing the values AX AX QL Algorithm for Real Symmetric Matrices Class EigenQL The EigenQL class will calculate all of the eigenvalues and eigenvectors of a real symmetric matrix using the implicit QL method EigenQL Constructor public EigenQL DMatBase a Parameters a The source matrix for the eigensystem calculations The source matrix MUST be real symmetric Method calcEigenValues Calculate the real eigenvalues for the source matrix A public int calcEigenValues DMatBase eigval Parameters eigval Returns the real eigenvalues of the source matrix A es and Eigenvectors 73 Return Value Returns an error code Method calcEigenValuesAndVectors Calculate the real eigenvalues and eigenvectors for the source matrix A public int calcEigenValuesAndVectors DMatBase eigval DMatBase eigvect Parameters eigval Returns the real eigenvalues of the source matrix A eigvect Returns the real eigenvectors of the source matrix A Return Value Returns an error code Method checkEigenVectors The eigenvalues and eigenvectors of a matrix must satisfy the following equation Ax Lx where A is the original matrix L is an eigenvalue of A and x is the eigenvector of A corresponding to eigenvalue L Given the original matrix A and the eigenvalues and eigenvectors of A this method calculates Ax Lx for each eigenvalue eigenvector pair and compares the result to a threshold residual returning true i
139. thms in this package require a number of data points that is a power of 2 You should round the result to the next power of 2 above the calculated value or 8192 The first harmonic of this sampled waveform becomes 1 8196 50KHz or 6 1 Hz that is as close as you can get to the original 8 Hz requirement without changing the sample rate The FFT routines use the original waveform as input This is passed to the function as an array vector or pointer of floating point numbers The output of the FFT function is one Digital Signal Processing 107 complex number a b 7 for each harmonic component in the original waveform The output is passed back as two arrays one of which holds real parts of the complex numbers and another imaginary parts The input to the FFT can be an array of real or complex numbers a b i Calculate an N point complex FFT and the primary results are returned in the first N 2 elements of the real and imaginary arrays These elements hold the harmonic values for the positive frequencies that make up the waveform The last half of the arrays hold the harmonic values for the negative frequencies that make up the waveform The negative frequencies are symmetrical to the positive frequencies about the folding frequency of the harmonic N 2 The folding frequency represented by the harmonic N 2 is also the Nyquist sample frequency for the original signal Continuing a previous example assume that 16 points of a signal are s
140. timized for dense DComplex object storage It also includes matrix math methods for the etc operators Solution of Linear Systems LESBase LESLU LESGJ LESCholesky LESQR LESXBase LESXLU LESXGJ LESXCholesky LESXQR 14 Class Architecture A system of linear equations is defined by the linear algebra equation Ax b where A x and b represent real complex matrices A is the known coefficient matrix b is the known right hand side and x is the unknown solution A is usually square though in the case of the QR algorithm A can be over determined where the number of rows is greater than the number of columns The matrices x and b are usually column matrices of one column though in the case of multiple right hand sides both x and b can have multiple columns This software provides a symmetrical group of classes for the solution of real and complex linear systems LESBase This class is the abstract base class for linear equation solvers that use real numbers It defines a standard interface for solving systems of equations that involve one or more right hand sides LESXBase This class is the abstract base class for linear equation solvers that use complex numbers It defines a standard interface for solving systems of equations that involve one or more right hand sides There are eight concrete classes derived from LESBase and LESXBase corresponding to real and complex versions of Cholesky factorization LESCholesky
141. trix with the dimension rows x columns and initialize it with triplets from the trips array DDMat DDMat DTriplet trips int rows int columns DXMat DXMat XTriplet trips int rows int columns Construct a vector initialized using the 1D source array The DXMat matrix has a constructor the initializes the real and imaginary parts of a complex matrix using the resource and imsource Java 1D arrays DDMat DDMat double source DXMat DXMat double resource DXMat double resource double imsource DXMat DComplex source Construct a matrix using the source Java 2D array The DXMat matrix has a constructor the initializes the real and imaginary parts of a complex matrix using the resource and imsource Java 2D arrays DDMat DDMat double source DXMat DXMat DComplex resource DXMat double resource double imsource DXMat double resource Matrix Classes 37 Construct a matrix as a vector with the dimension elements DDMat DDMat int elements DXMat DXMat int elements Construct a matrix as a vector dimension elements and initialize all elements to the value r DDMat DDMat int elements double r DXMat DXMat int elements DComplex r Matrix Operators Matrices are added subtracted multiplied and divided following the rules of linear algebra Matrix Addition add DDMat a DDMat d static add DXMat a DXMat d Two or more matrices o
142. uations x b A or more appropriatly Ax b static DXMat divide DXMat b DXMat a solves the linear system of equations x b A or more appropriatly Ax b Matrix division is a bit of a misnomer since the divide sign doesn t often appear in linear algebra forumlas Take the the matrix expression b A This can be rewritten as x b A where A is the matrix inverse of A In general you never want to solve for a matrix inverse then use that inverse in subsequent calculations It is not as accurate nor as fast as alternative methods When this software sees a divide sign in a matrix expression it redefines the original problem x b A as the matrix expression Ax b which is the standard format for solving for system of linear equations solves for x then replaces the expression b A with the matrix x In order for this to work properly A must be square and the number of rows in b must equal the number of columns in A double aData 11 2 3211 71 20 IT J7 14 x 4 double II bData 5 6 7 8 4 x 1 DDMat A new DDMat aData DDMat B new DDMat bData 40 Matrix Classes DDMat C DDMat divide A B Scalar Division static DDMat divide DDMat a double b matrix scalar static DXMat Idivide DXMat a DComplex b matrix scalar static DXMat Idivide DXMat a double b matrix scalar Given a matrix A and a scalar c the division of A c is calculated by dividing every ele
143. urce matrix A Return Value Returns an error code Example Code listing extracted from the EigenRealUnSym example program DDMat A new DDMat Matrix A needs to be initialized results returned in EValues and EVectors DXMat EValues new DXMat DXMat EVectors new DXMat EigenQR2 eigensolve new EigenQR2 A eigensolve calcEigenValuesAndVectors EValues EVectors DXMat residuals new DXMat residuals returned here double refnum 1 0e 4 if any residual gt this value solution invalid returns actual max error boolean ok EigenXBase checkEigenVectors A EValues EVectors residuals ref refnum LR Algorithm for Complex Matrices Class EigenXLR The EigenXLR class will calculate all of the eigenvalues and eigenvectors for a complex matrix using the LR method There are no symmetry constraints EigenXLR Constructor 78 Eigenvalues and Eigenvectors public EigenXLR XMatBase a Parameters a The source matrix for the eigensystem calculations The source matrix must be square but it does not have to be symmetric Method calcEigen Values Calculate the complex eigenvalues for the source matrix A public int calcEigenValues XMatBase eigval Parameters eigval Returns the eigenvalues of the source matrix A Return Value Returns an error code Method calcEigenValuesAndVectors Calculate the complex eigenvalues and eigenvectors for the source matrix A publi
144. utions to mathematical problems involving real and complex numbers Man has been trying to measure and calculate his physical world since at the least the beginning of recorded history Ancient Egypt and Summeria Babylonia are examples of cultures that required sophisticated numerical methods routines in order to support the construction religious and agricultural aspects of their societies The numerical algorithms in this software package all reside in the realm of continuous mathematics and use both real and in many cases complex numbers Real numbers in this software are approximated using 8 byte double numeric objects with complex numbers requiring two double objects It is the goal and art of numerical programming to minimize the negative effects and accumulated errors resulting from the imperfect representation of infinite precision real numbers by finite precision floating point numbers Numerical algorithms involving statistics the solution of equations eigenvalues and digital signal processing were among the first developed for use on computers Starting in the 1950 s these algorithms were primarily developed in FORTRAN short for Formula Translating System and starting in the 1960 s ALGOL short for ALGOrithmic Language The FORTRAN implementations seem to have won out even though ALGOL s structured programming elements made it the superior language for algorithm expression Most of the major well known numerical met
145. va P Framet java N LESTest java O GLestestApplet1 htm O LestestaApplet2 htm Pi IV Export generated class files and resources 17 Export all output folders for checked projects I Export java source files and resources Select the export destination JAR file C Quinn Curtis Java Development javallib LESTest jar Browse Options IV Compress the contents of the JAR file I Add directory entries IV Overwrite existing files without warning In this case the jar file is saved using the name LESTest jar This is one of the three jars you will need in order to run the application in a web browser The other jar files you will need are the gematpackjava jar and gematviewjava jar files found in the quinn curtis java lib directory These jar files are uploaded to your server The code that needs to be added to your web page will look like lt APPLET CODE com quinncurtis matpackjava examples LESTest Appletl class CODEBASE http quinn curtis com classes ARCHIVE LESTest jar qcmatpackjava jar qcmatviewjava jar WIDTH 900 HEIGHT 600 gt This browser does not support Java 1 2 or greater applets lt APPLET gt lt p gt This HTML page LESTestApplet htm can be loaded and the applet run at our web site using the following link 146 Using QCMatPack to Create Windows Applications http quinn curtis com LESTestApplet1 htm Note the following e The CODE attribute specifies the fu
146. variable with the highest correlation with the dependent variable Step 2 Calculate the partial F statistic for every other variable marked with REG VARCONSIDER Step 3 Find the variable with the largest partial F statistic and using that value compare the probability that the variable is significant to the value of the property FToEnter If the probability that the variable is significant is greater than FToEnter the variable is added to the model Step 4 Recalculate the model and find the variable with the smallest partial F statistic and using that value compare the probability that the variable is significant to the value of the property FToLeave If the probability that the variable is significant is less than FToLeave the variable is removed from the model Step 5 Repeat starting at Step 1 until the significance test in Step 3 finds no additional variables to add The current solution then becomes the final solution of the regression equation Method StepwiseMulReg stepwiseSolve Multiple Regression 89 public int StepwiseSolve DDMat regcoef RegStats regstats Parameters regcoef Returns the calculated regression coefficients regstats Returns regression statistics The RegStats class is a simple class that encapsulates several related summary statistics for multiple regression See the description under the MulReg class section Examples for Stepwise Regression The follow code fragments are extracted from the
147. vel DDMat coefsig Parameters fstat Returns the calculated f statistics one for each independent variable conf level A fractional value usually between 0 8 to 0 99 specifying the confidence level An independent variable is considered significant if the probability of it being significant exceeds this confidence level coefsig Returns an array of flags one for each independent variable If the value of an array element is 1 that means that the probably of the corresponding independent variable being significant exceeds the specified confidence level Automatic Selection of Regression Variables Class StepwiseMulReg 86 Multiple Regression The MulReg class allows the programmer to include and exclude independent variables from the final regression equation using the consider consid parameter of the MulReg constructor Another approach is to let the program decide what independent variables to include in the final regression equation The StepwiseMulReg class includes three algorithms for the automatic selection of the independent variables used in the regression model backward elimination forward selection and stepwise selection You can still force variables in and out using the consider parameter of the constructor The selection mechanism will choose between variables marked REG_VARCONSIDER Those marked with REG_VARIN will be included in the model no matter how insignificant they are found to be Those marked with REG_VARO
148. vert DDMat A DXMat static int invert DXMat A Parameters A Returns the inversion of the matrix overwriting the original matrix Return Value Returns an error code See the MatError class error code list Method DDMat makeRandomLEq DXMat makeRandomLEq Make a pair of matrices that form the A and b matrices of a linear set of equations Ax b The matrices are constructed so that the solution x of Ax b is always a vector with the values 1 2 3 rank DDMat static void makeRandomLEg DDMat A DDMat b int rank boolean posdefsym DXMat static void makeRandomLEg DXMat A DXMat b int rank boolean posdefsym 48 Matrix Classes Parameters A Returns the coefficient array A of the linear system of equations b Returns the right hand side array b of the linear system of equations rank The number of rows and columns of the coefficient A matrix posdefsym Set to true and the A matrix will be symmetric hermitian positive definite Method DDMat makeRandomMatrix DXMat makeRandomMatrix Make a random square matrix of the specified rank and symmetry DDMat static DDMat makeRandomMatrix int rank boolean posdefsym DXMat static DXMat ImakeRandomMatrix int rank boolean posdefsym Parameters rank The number of rows and columns of the generated matrix posdefsym Set to true and the A matrix will be symmetric hermitian for complex positive definite Return Value Returns th
149. ware generates are listed below 130 Error Handling case NOMEM throw new java lang OutOfMemoryError sMessageString case NULLMAT throw new java lang NullPointerException sMessageString case INVALID ARG DIM throw new java lang IllegalArgumentException sMessageString case INVALID MAT DIM throw new java lang IllegalArgumentException sMessageString case SYMMETRIC_MATRIX REQUIRED throw new java lang IllegalArgumentException sMessageString case POSDEFINITE_ MATRIX REQUIRED throw new java lang IllegalArgumentException sMessageString case HERMITIAN MATRIX REQUIRED throw new java lang IllegalArgumentException sMessageString Error Constant Description Exception if enabled NOERR No error MAT COL ERR Column index out of range error java lang ArrayIndexOutOfBoundsException MAT ROW ERR Row index out of range error java lang ArrayIndexOutOfBoundsException MAT ELEMENT ERR Array Element Error java lang ArrayIndexOutOfBoundsException INVALID_INDEX Array Index Exceeds java lang ArrayIndexOutOfBoundsException Dimension NOMEM Not Enough Memory java lang OutOfMemoryError NULLMAT Null matrix argument error Java lang NullPointerException INVALID ARG DIM Dimensions Do Not Match java lang IllegalArgumentException INVALID MAT DIM Matrix should have the same number ofrows and columns java lang IllegalArgumentException SYMMETRIC MATRIX REQUIRED A Trans
150. with REG VARCONSIDER Step 3 Find the variable with the largest partial F statistic and using that value compare the probability that the variable is significant to the value of the property FToEnter If the probability that the variable is significant is greater than FToEnter the variable is added to the model 88 Multiple Regression Step 4 Repeat starting at Step 1 until the significance test in Step 3 finds no additional variables to add The current solution then becomes the final solution of the regression equation Method StepwiseMulReg forwardSolve public int ForwardSolve DDMat regcoef RegStats regstats he Parameters regcoef Returns the calculated regression coefficients regstats Returns regression statistics The RegStats class is a simple class that encapsulates several related summary statistics for multiple regression See the description under the MulReg class section Stepwise Selection The stepwise selection algorithm is similar to the forward selection algorithm with an added step After a variable is found significant and added to the model the model is checked again to see if any of the partial F statistics fall below a predetermined minimum In this case the variable is removed from the model Variables might be removed in this fashion because of auto correlation between other variables in the model Step 1 Select the starting variable s This is either the variables marked with REG_VARIN or the
151. x row The number of display rows in the matrix viewer cols The number of display columns in the matrix viewer Set the decimal precision of the numeric values in the matrix viewer using the Decimals property If you want to be able to edit the individual cells of a matrix call the enableEdit true method If you make any changes to the properties of a matrix viewer call the updateDraw method to force an immediate update If the matrix viewer is edit enabled click on the matrix element you want to edit An edit box will appear in place of the element Make your edit and hit return or click anywhere else in the matrix viewer Matrix Viewers 119 Any edits made to a matrix using a matrix viewer will immediately update the associated matrix either a DDMat or DXMat object 0 1 2 0 1 6 8 7i 2 9 2 2 8 7 6 1i 1 5 6 7 4i 6 1 2 5i 7 4 3 4i 2 4 5 5 41 4 8 8 11 3 2 3 0i Above you see what the edit mode looks when the cell 2 1 is edited in real DDMat and complex DXMat matrices Example Extracted from the EigenRealUnSym example program dMatViewerl dMatViewerl dMatViewerl dMatViewerl new DMatViewer setPosition 40 100 300 296 setViewMatrix A 8 5 setDecimals 2 this add dMatViewerl xMatViewerl xMatViewer2 xMatViewerl xMatViewer2 xMatViewerl xMatViewer2 setViewMatrix EValues 10 1 setDecimals 2 setDecimals 2 updateDraw

QCMatPack Matrix and Numerical Analysis Software - Quinn

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