numal.highprecision
Class Linear_algebra

java.lang.Object
  numal.highprecision.Linear_algebra

public class Linear_algebra
extends Object

Constructor Summary

Linear_algebra()

Method Summary

static void	hshreabid(double[][] a, int m, int n, double[] d, double[] b, double[] em) Reduces an m x n matrix a to a bidiagonal form B.
static void	pretfmmat(double[][] a, int m, int n, double[] d)
static void	psttfmmat(double[][] a, int n, double[][] v, double[] b)
static int	qrisngvaldec(double[][] a, int m, int n, double[] val, double[][] v, double[] em) Calculates the singular value decomposition UDV^T of a given matrix A.
static int	qrisngvaldecbid(double[] d, double[] b, int m, int n, double[][] u, double[][] v, double[] em)

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

Linear_algebra

public Linear_algebra()

Method Detail

qrisngvaldec

public static int qrisngvaldec(double[][] a,
                               int m,
                               int n,
                               double[] val,
                               double[][] v,
                               double[] em)

Calculates the singular value decomposition UDV^T of a given matrix A. The matrix is first transformed to bidiagonal form by calling hshreabid(double[][], int, int, double[], double[], double[]), the two transforming matrices are calculated by calling psttfmmat(double[][], int, double[][], double[]) and pretfmmat(double[][], int, int, double[]), and finally the singular value decomposition is calculated by qrisngvaldecbid(double[], double[], int, int, double[][], double[][], double[]). Dependencies: hshreabid(double[][], int, int, double[], double[], double[]), psttfmmat(double[][], int, double[][], double[]), pretfmmat(double[][], int, int, double[]), qrisngvaldecbid(double[], double[], int, int, double[][], double[][], double[]).

• input:
• em[0] : the machine precision
• em[2] : the relative precision in the singular values
• em[4] : the maximal number of iterations to be performed
• em[6] : the minimal nonneglectable singular value
• output:
• em[1] : the infinity norm of the matrix
• em[3] : the maximal neglected superdiagonal element
• em[5] : the number of iterations performed
• em[7] : the numerical rank of the matrix; i.e. the number of singular values greater than or equal to em[6]

Parameters:

a - input: the given mxn matrix output: the matrix U in the singular value decomposition UDV^T

m - the number of rows of the matrix a

n - the number of columns of a, precondition: n ≤ m

val - output: the singular values 1 to n

v - output: the transpose of matrix V in the singular value decomposition

em - input and output arameters mentioned in the list above

Returns:

number of singular values not found, i.e., a number not equal to zero if the number of iterations exceeds em[4]

hshreabid

public static void hshreabid(double[][] a,
                             int m,
                             int n,
                             double[] d,
                             double[] b,
                             double[] em)

Reduces an m x n matrix a to a bidiagonal form B. Method originally from class numal.LinearAlgebra - Lau (2004) section 3.12.1.A Other transformations : To bidiagonal form - real matrices p.212. Dependencies: tammat, mattam, elmcol, elmrow

Parameters:

a - input: m by n matrix output: data concerning the premultiplying and postmultiplying matrices

m - number of rows

n - number of columns

d - output: vector of the diagonal of the bidiagonal matrix B

b - output: vector of the super diagonal of the bidiagonal matrix B

em - output:

psttfmmat

public static void psttfmmat(double[][] a,
                             int n,
                             double[][] v,
                             double[] b)

pretfmmat

public static void pretfmmat(double[][] a,
                             int m,
                             int n,
                             double[] d)

qrisngvaldecbid

public static int qrisngvaldecbid(double[] d,
                                  double[] b,
                                  int m,
                                  int n,
                                  double[][] u,
                                  double[][] v,
                                  double[] em)