nag_dggrqf (f08zfc) (PDF version)
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f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dggrqf (f08zfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dggrqf (f08zfc) computes a generalized RQ factorization of a real matrix pair A,B, where A is an m by n matrix and B is a p by n matrix.

2  Specification

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

3  Description

nag_dggrqf (f08zfc) forms the generalized RQ factorization of an m by n matrix A and a p by n matrix B 
A = RQ ,   B= ZTQ ,
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 = n-mmm(0R12) ;   if ​ mn , nm-n(R11) n R21 ;   if ​ m>n ,
with R12 or R21 upper triangular,
T = nn(T11) p-n 0 ;   if ​ pn , pn-pp(T11T12) ;   if ​ p<n ,
with T11 upper triangular.
In particular, if B is square and nonsingular, the generalized RQ factorization of A and B implicitly gives the RQ factorization of AB-1 as
AB-1= R T-1 ZT .

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:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3:     pIntegerInput
On entry: p, the number of rows of the matrix B.
Constraint: p0.
4:     nIntegerInput
On entry: n, the number of columns of the matrices A and B.
Constraint: n0.
5:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least
  • max1,pda×n when order=Nag_ColMajor;
  • max1,m×pda when order=Nag_RowMajor.
Where Ai,j appears in this document, it refers to the array element
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the m by n matrix A.
On exit: if mn, the upper triangle of the subarray A1:m,n-m+1:n contains the m by m upper triangular matrix R12.
If mn, the elements on and above the m-nth subdiagonal contain the m by n upper trapezoidal matrix R; the remaining elements, with the array taua, represent the orthogonal matrix Q as a product of minm,n 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,m;
  • if order=Nag_RowMajor, pdamax1,n.
7:     taua[minm,n]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×n when order=Nag_ColMajor;
  • max1,p×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 p by n matrix B.
On exit: the elements on and above the diagonal of the array contain the minp,n by n upper trapezoidal matrix T (T is upper triangular if pn); the elements below the diagonal, 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,p;
  • if order=Nag_RowMajor, pdbmax1,n.
10:   taub[minp,n]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 RQ 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_dggrqf (f08zfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dggrqf (f08zfc) 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_dorgrq (f08cjc) and nag_dorgqr (f08afc) respectively. nag_dormrq (f08ckc) may be used to multiply Q by another matrix and nag_dormqr (f08agc) may be used to multiply Z by another matrix.
The complex analogue of this function is nag_zggrqf (f08ztc).

10  Example

This example solves the linear equality constrained least squares problem
minx c-Ax2   subject to   Bx= d
A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 ,   B= 1 0 -1 0 0 1 0 -1 ,
c = -1.50 -2.14 1.23 -0.54 -1.68 0.82   and   d= 0 0 .
The constraints Bx=d correspond to x1=x3 and x2=x4.
The solution is obtained by first computing a generalized RQ factorization of the matrix pair B,A. The example illustrates the general solution process.

10.1  Program Text

Program Text (f08zfce.c)

10.2  Program Data

Program Data (f08zfce.d)

10.3  Program Results

Program Results (f08zfce.r)

nag_dggrqf (f08zfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

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