nag_dggqrf (f08zec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dggqrf (f08zec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dggqrf (f08zec) computes a generalized QR factorization of a real matrix pair A,B, where A is an n by m matrix and B is an n by p matrix.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dggqrf (Nag_OrderType order, Integer n, Integer m, Integer p, double a[], Integer pda, double taua[], double b[], Integer pdb, double taub[], NagError *fail)

3  Description

nag_dggqrf (f08zec) forms the generalized QR factorization of an n by m matrix A and an n by p matrix B 
A =QR ,   B=QTZ ,
where Q is an n by n orthogonal matrix, Z is a p by p orthogonal matrix and R and T are of the form
R = mm(R11) n-m 0 ,   if ​nm; nm-nn(R11R12) ,   if ​n<m,
with R11 upper triangular,
T = p-nnn(0T12) ,   if ​np, pn-p(T11) p T21 ,   if ​n>p,
with T12 or T21 upper triangular.
In particular, if B is square and nonsingular, the generalized QR factorization of A and B implicitly gives the QR factorization of B-1A as
B-1A= ZT T-1 R .

4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Anderson E, Bai Z and Dongarra J (1992) Generalized QR factorization and its applications Linear Algebra Appl. (Volume 162–164) 243–271
Hammarling S (1987) The numerical solution of the general Gauss-Markov linear model Mathematics in Signal Processing (eds T S Durrani, J B Abbiss, J E Hudson, R N Madan, J G McWhirter and T A Moore) 441–456 Oxford University Press
Paige C C (1990) Some aspects of generalized QR factorizations . In Reliable Numerical Computation (eds M G Cox and S Hammarling) 73–91 Oxford University Press

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     nIntegerInput
On entry: n, the number of rows of the matrices A and B.
Constraint: n0.
3:     mIntegerInput
On entry: m, the number of columns of the matrix A.
Constraint: m0.
4:     pIntegerInput
On entry: p, the number of columns of the matrix B.
Constraint: p0.
5:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least
  • max1,pda×m when order=Nag_ColMajor;
  • max1,n×pda when order=Nag_RowMajor.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the n by m matrix A.
On exit: the elements on and above the diagonal of the array contain the minn,m by m upper trapezoidal matrix R (R is upper triangular if nm); the elements below the diagonal, with the array taua, represent the orthogonal matrix Q as a product of minn,m elementary reflectors (see Section 3.3.6 in the f08 Chapter Introduction).
6:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor, pdamax1,n;
  • if order=Nag_RowMajor, pdamax1,m.
7:     taua[minn,m]doubleOutput
On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix Q.
8:     b[dim]doubleInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×p when order=Nag_ColMajor;
  • max1,n×pdb when order=Nag_RowMajor.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the n by p matrix B.
On exit: if np, the upper triangle of the subarray B 1:n , p-n+1:p  contains the n by n upper triangular matrix T12.
If n>p, the elements on and above the n-pth subdiagonal contain the n by p upper trapezoidal matrix T; the remaining elements, with the array taub, represent the orthogonal matrix Z as a product of elementary reflectors (see Section 3.3.6 in the f08 Chapter Introduction).
9:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,p.
10:   taub[minn,p]doubleOutput
On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix Z.
11:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, p=value.
Constraint: p0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pdamax1,m.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and p=value.
Constraint: pdbmax1,p.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7  Accuracy

The computed generalized QR factorization is the exact factorization for nearby matrices A+E and B+F, where
E2 = Oε A2   and   F2= Oε B2 ,
and ε is the machine precision.

8  Further Comments

The orthogonal matrices Q and Z may be formed explicitly by calls to nag_dorgqr (f08afc) and nag_dorgrq (f08cjc) respectively. nag_dormqr (f08agc) may be used to multiply Q by another matrix and nag_dormrq (f08ckc) may be used to multiply Z by another matrix.
The complex analogue of this function is nag_zggqrf (f08zsc).

9  Example

This example solves the general Gauss–Markov linear model problem
minx y2   subject to   d=Ax+By
where
A = -0.57 -1.28 -0.39 -1.93 1.08 -0.31 2.30 0.24 -0.40 -0.02 1.03 -1.43 ,   B= 0.5 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 5.0   and   d= 1.32 -4.00 5.52 3.24 .
The solution is obtained by first computing a generalized QR factorization of the matrix pair A,B. The example illustrates the general solution process, although the above data corresponds to a simple weighted least squares problem.

9.1  Program Text

Program Text (f08zece.c)

9.2  Program Data

Program Data (f08zece.d)

9.3  Program Results

Program Results (f08zece.r)


nag_dggqrf (f08zec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012