#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include "include/hholder.h"
#include "include/qr.h"

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Functions
int	qr (double *A, int m, int n, double B, double X, double rnorm, double tau, double res, double *wws, double ws)
int	qr_decomp (double *a, int M, int N, double tau, double *cchain, double chain)
int	qr_solve (double *QR, int M, int N, double tau, double b, double x, double residual, double resNorm, double *cchain, double chain)
int	qr_QTvec (double *QR, double tau, double v, double help)
int	qr_Qvec (double *QR, double tau, double v, double help)
int	qr_invRvec (double *A, double X)
int	qr_weight (int N, int M, double *A, double b, double *weight)
Variables
int	qr_M
int	qr_N
int	qr_MNmin

Function Documentation

int qr	(	double **	A,
		int	m,
		int	n,
		double *	B,
		double *	X,
		double *	rnorm,
		double *	tau,
		double *	res,
		double **	wws,
		double *	ws
	)

Parameters:

A	On entry, a[m][n] contains the m by n matrix A. On exit, a[][] contains the QR factorization
m	Dimensions of matrix A are a[m][n]
n	Dimensions of matrix A are a[m][n]
B	B[] is an m-vector containing the right-hand side vector b.
X	On exit, x[] will contain the solution vector x.
rnorm	On exit, rnorm contains the Euclidean norm of the residual vector
tau	On exit, tau[] will contain the householder coefficients.
res	An m-array of working space, res[]. On output contains residual b - Ax
wws	m*n array of working space
ws	2m-array of working space

Definition at line 52 of file qr.c.

References qr_decomp(), and qr_solve().

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int qr_decomp	(	double **	a,
		int	M,
		int	N,
		double *	tau,
		double **	cchain,
		double *	chain
	)

Factorise a general M x N matrix A into

A = Q R

where Q is orthogonal (M x M) and R is upper triangular (M x N).

Q is stored as a packed set of Householder vectors in the strict lower triangular part of the input matrix A and a set of coefficients in vector tau.

R is stored in the diagonal and upper triangle of the input matrix.

The full matrix for Q can be obtained as the product

  Q = Q_1 Q_2 .. Q_k  and it's transform as the product

  Q^T = Q_k .. Q_2 Q_1

where k = min(M,N) and

  Q_i = (I - tau_i * h_i * h_i^T)

and where h_i is a Householder vector

  h_i = [1, A(i+1,i), A(i+2,i), ... , A(M,i)]

This storage scheme is the same as in LAPACK.

NOTICE! The calling program must take care that pointer tau is of size n.

Returns:: Function returns 0 if ok.

Parameters:

a	contains coefficient matrix A (m*n) as input and factorisation QR as output
M	nr of rows in matrix A
N	nr of columns in matrix A
tau	n-vector for householder coefficients
cchain	m*n array of working space
chain	2m array of working space

Definition at line 171 of file qr.c.

References householder_hm(), householder_transform(), M, qr_M, qr_MNmin, and qr_N.

Referenced by qr().

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int qr_invRvec	(	double **	A,
		double *	X
	)

Form the product R^-1 v. (R is saved in the upper triangel of QR matrix.)

Returns:: Returns 0 if ok.

Definition at line 409 of file qr.c.

References qr_N.

Referenced by qr_solve().

int qr_QTvec	(	double **	QR,
		double *	tau,
		double *	v,
		double *	help
	)

Form the product Q^T v from householder vectors saved in the lower triangel of QR matrix and householder coefficients saved in vector tau.

Returns:: Returns 0 if ok.

Definition at line 345 of file qr.c.

References householder_hv(), qr_M, and qr_MNmin.

Referenced by qr_solve().

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int qr_Qvec	(	double **	QR,
		double *	tau,
		double *	v,
		double *	help
	)

Form the product Q v from householder vectors saved in the lower triangel of QR matrix and householder coefficients saved in vector tau.

Returns:: Returns 0 if ok.

Definition at line 379 of file qr.c.

References householder_hv(), qr_M, and qr_MNmin.

Referenced by qr_solve().

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int qr_solve	(	double **	QR,
		int	M,
		int	N,
		double *	tau,
		double *	b,
		double *	x,
		double *	residual,
		double *	resNorm,
		double **	cchain,
		double *	chain
	)

Find the least squares solution to the overdetermined system

A x = b

for m >= n using the QR factorisation A = Q R. qr_decomp() must be used prior to this function in order to form the QR factorisation of A. Solution is formed in the following order: QR x = b => R x = Q^T b => x = R^-1 (Q^T b)

NOTICE! The calling program must take care that pointers b, x and residual are of the right size.

Returns:: Function returns 0 if ok.

Parameters:

QR	m*n matrix containing householder vectors of A
M	nr of rows in matrix A
N	nr of columns in matrix A
tau	vector containing householder coefficients tau
b	Contains m-vector b of A x = b
x	solution vector x of length n
residual	residual vector of length m
resNorm	norm^2 of the residual vector
cchain	m*n array of the working space
chain	2m lenght array of the working space