nag_dggqrf (f08zec) (PDF version)
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f08 Chapter Introduction
NAG 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
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 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.
  • 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.
Where Bi,j appears in this document, it refers to the array element
  • 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 B1: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.
  • 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

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
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.
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.
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  Parallelism and Performance

nag_dggqrf (f08zec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dggqrf (f08zec) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the Users' Note for your implementation for any additional implementation-specific information.

9  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).

10  Example

This example solves the general Gauss–Markov linear model problem
minx y2   subject to   d=Ax+By
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.

10.1  Program Text

Program Text (f08zece.c)

10.2  Program Data

Program Data (f08zece.d)

10.3  Program Results

Program Results (f08zece.r)

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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014