nag_zggrqf (f08ztc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zggrqf (f08ztc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zggrqf (f08ztc) computes a generalized RQ factorization of a complex 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_zggrqf (Nag_OrderType order, Integer m, Integer p, Integer n, Complex a[], Integer pda, Complex taua[], Complex b[], Integer pdb, Complex taub[], NagError *fail)

3  Description

nag_zggrqf (f08ztc) 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 unitary matrix, Z is a p by p unitary 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 ZH .

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]ComplexInput/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 unitary 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]ComplexOutput
On exit: the scalar factors of the elementary reflectors which represent the unitary matrix Q.
8:     b[dim]ComplexInput/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 unitary 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]ComplexOutput
On exit: the scalar factors of the elementary reflectors which represent the unitary 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  Further Comments

The unitary matrices Q and Z may be formed explicitly by calls to nag_zungrq (f08cwc) and nag_zungqr (f08atc) respectively. nag_zunmrq (f08cxc) may be used to multiply Q by another matrix and nag_zunmqr (f08auc) may be used to multiply Z by another matrix.
The real analogue of this function is nag_dggrqf (f08zfc).

9  Example

This example solves the general Gauss–Markov linear model problem
minx y2   subject to   d= Ax+By
A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i 0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i ,  
B = 1 0 -1 0 0 1 0 -1 ,   c= -2.54+0.09i 1.65-2.26i -2.11-3.96i 1.82+3.30i -6.41+3.77i 2.07+0.66i   and   d= 0 0 .
The constraints Bx=d correspond to x1=x3 and x2=x4.
The solution is obtained by first obtaining 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 (f08ztce.c)

9.2  Program Data

Program Data (f08ztce.d)

9.3  Program Results

Program Results (f08ztce.r)

nag_zggrqf (f08ztc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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