org.ejml.alg.dense.decompose.qr

Class QrHelperFunctions_CD64

java.lang.Object

org.ejml.alg.dense.decompose.qr.QrHelperFunctions_CD64

public class QrHelperFunctions_CD64
extends java.lang.Object

Contains different functions that are useful for computing the QR decomposition of a matrix.

Two different families of functions are provided for help in computing reflectors. Internally both of these functions switch between normalization by division or multiplication. Multiplication is most often significantly faster than division (2 or 3 times) but produces less accurate results on very small numbers. It checks to see if round off error is significant and decides which one it should do.

Tests were done using the stability benchmark in jmatbench and there doesn't seem to be any advantage to always dividing by the max instead of checking and deciding. The most noticeable difference between the two methods is with very small numbers.

Constructor Summary

Constructors

Constructor and Description
QrHelperFunctions_CD64()

Method Summary

Modifier and Type	Method and Description
static double	computeRowMax(org.ejml.data.CDenseMatrix64F A, int row, int col0, int col1) Finds the magnitude of the largest element in the row
static double	computeTauGammaAndDivide(int start, int stop, double[] x, double max, org.ejml.data.Complex64F tau) Performs the following operations:
static void	divideElements(int j, int numRows, double[] u, int startU, double realA, double imagA) Performs the following operation: u[(startU+j):(startU+numRows)] /= A were u and A are a complex
static double	extractColumnAndMax(org.ejml.data.CDenseMatrix64F A, int row0, int row1, int col, double[] u, int offsetU) Extracts the column of A and copies it into u while computing the magnitude of the largest element and returning it.
static void	extractHouseholderColumn(org.ejml.data.CDenseMatrix64F A, int row0, int row1, int col, double[] u, int offsetU) Extracts a house holder vector from the column of A and stores it in u
static void	extractHouseholderRow(org.ejml.data.CDenseMatrix64F A, int row, int col0, int col1, double[] u, int offsetU) Extracts a house holder vector from the rows of A and stores it in u
static double	findMax(double[] u, int startU, int length) Returns the maximum magnitude of the complex numbers
static void	rank1UpdateMultL(org.ejml.data.CDenseMatrix64F A, double[] u, int offsetU, double gammaR, int colA0, int w0, int w1) Performs a rank-1 update operation on the submatrix specified by w with the multiply on the left. A = A(I - γ*u*uH)
static void	rank1UpdateMultR(org.ejml.data.CDenseMatrix64F A, double[] u, int offsetU, double gamma, int colA0, int w0, int w1, double[] _temp) Performs a rank-1 update operation on the submatrix specified by w with the multiply on the right. A = (I - γ*u*uH)*A

Methods inherited from class java.lang.Object

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

Constructor Detail

QrHelperFunctions_CD64

public QrHelperFunctions_CD64()

Method Detail

findMax

public static double findMax(double[] u,
                             int startU,
                             int length)

Returns the maximum magnitude of the complex numbers

Parameters:

u - Array of complex numbers

startU - first index to consider in u

length - Number of complex numebrs to consider

Returns:

magnitude

divideElements

public static void divideElements(int j,
                                  int numRows,
                                  double[] u,
                                  int startU,
                                  double realA,
                                  double imagA)

Performs the following operation:
u[(startU+j):(startU+numRows)] /= A
were u and A are a complex

computeTauGammaAndDivide

public static double computeTauGammaAndDivide(int start,
                                              int stop,
                                              double[] x,
                                              double max,
                                              org.ejml.data.Complex64F tau)

Performs the following operations:

 x = x / max
 tau = x0*|x|/|xo|   adjust sign to avoid cancellation
 u = x; u0 = x0 + tau; u = u/u0  (x is not divided by x0)
 gamma = 2/|u|^2
 

Note that u is not explicitly computed here.

Parameters:

start - Element in 'u' that it starts at.

stop - Element in 'u' that it stops at.

x - Array

max - Max value in 'u' that is used to normalize it.

tau - Storage for tau

Returns:

Returns gamma

rank1UpdateMultR

public static void rank1UpdateMultR(org.ejml.data.CDenseMatrix64F A,
                                    double[] u,
                                    int offsetU,
                                    double gamma,
                                    int colA0,
                                    int w0,
                                    int w1,
                                    double[] _temp)

Performs a rank-1 update operation on the submatrix specified by w with the multiply on the right.

A = (I - γ*u*u^H)*A

Parameters:

A - matrix

u - vector

offsetU - offset added to w0 when indexing u. Multiplied by 2 since complex.

gamma - real component of gamma

colA0 - first column in A sub-matrix.

w0 - first index in sub-array in u and row sub-matrix in A

w1 - last index + 1 in sub-array in u and row sub-matrix in A

_temp - temporary storage. Same size as u.

rank1UpdateMultL

public static void rank1UpdateMultL(org.ejml.data.CDenseMatrix64F A,
                                    double[] u,
                                    int offsetU,
                                    double gammaR,
                                    int colA0,
                                    int w0,
                                    int w1)

Performs a rank-1 update operation on the submatrix specified by w with the multiply on the left.

A = A(I - γ*u*u^H)

The order that matrix multiplies are performed has been carefully selected to minimize the number of operations.

Before this can become a truly generic operation the submatrix specification needs to be made more generic.

extractHouseholderColumn

public static void extractHouseholderColumn(org.ejml.data.CDenseMatrix64F A,
                                            int row0,
                                            int row1,
                                            int col,
                                            double[] u,
                                            int offsetU)

Extracts a house holder vector from the column of A and stores it in u

Parameters:

A - Complex matrix with householder vectors stored in the lower left triangle

row0 - first row in A (implicitly assumed to be r + i0)

row1 - last row + 1 in A

col - Column in A

u - Output array storage

offsetU - first index in U

extractHouseholderRow

public static void extractHouseholderRow(org.ejml.data.CDenseMatrix64F A,
                                         int row,
                                         int col0,
                                         int col1,
                                         double[] u,
                                         int offsetU)

Extracts a house holder vector from the rows of A and stores it in u

Parameters:

A - Complex matrix with householder vectors stored in the upper right triangle

row - Row in A

col0 - first row in A (implicitly assumed to be r + i0)

col1 - last row +1 in A

u - Output array storage

offsetU - first index in U

extractColumnAndMax

public static double extractColumnAndMax(org.ejml.data.CDenseMatrix64F A,
                                         int row0,
                                         int row1,
                                         int col,
                                         double[] u,
                                         int offsetU)

Extracts the column of A and copies it into u while computing the magnitude of the largest element and returning it.

 u[ (offsetU+row0+i)*2    ] = A.getReal(row0+i,col)
 u[ (offsetU+row0+i)*2 + 1] = A.getImag(row0+i,col)
 

Parameters:

A - Complex matrix

row0 - First row in A to be copied

row1 - Last row in A + 1 to be copied

col - Column in A

u - Output array storage

offsetU - first index in U

Returns:

magnitude of largest element

computeRowMax

public static double computeRowMax(org.ejml.data.CDenseMatrix64F A,
                                   int row,
                                   int col0,
                                   int col1)

Finds the magnitude of the largest element in the row

Parameters:

A - Complex matrix

row - Row in A

col0 - First column in A to be copied

col1 - Last column in A + 1 to be copied

Returns:

magnitude of largest element