nag_zgerqf (f08cvc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zgerqf (f08cvc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgerqf (f08cvc) computes an RQ factorization of a complex m by n matrix A.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zgerqf (Nag_OrderType order, Integer m, Integer n, Complex a[], Integer pda, Complex tau[], NagError *fail)

3  Description

nag_zgerqf (f08cvc) forms the RQ factorization of an arbitrary rectangular real m by n matrix. If mn, the factorization is given by
A = 0 R Q ,
where R is an m by m lower triangular matrix and Q is an n by n unitary matrix. If m>n the factorization is given by
A =RQ ,
where R is an m by n upper trapezoidal matrix and Q is again an n by n unitary matrix. In the case where m<n the factorization can be expressed as
A = 0 R Q1 Q2 =RQ2 ,
where Q1 consists of the first n-m rows of Q and Q2 the remaining m rows.
The matrix Q is not formed explicitly, but is represented as a product of minm,n elementary reflectors (see the f08 Chapter Introduction for details). Functions are provided to work with Q in this representation (see Section 8).

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3:     nIntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
4:     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 R.
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 tau, represent the unitary matrix Q as a product of minm,n elementary reflectors (see Section 3.3.6 in the f08 Chapter Introduction).
5:     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,m;
  • if order=Nag_RowMajor, pdamax1,n.
6:     tau[dim]ComplexOutput
Note: the dimension, dim, of the array tau must be at least max1,minm,n.
On exit: the scalar factors of the elementary reflectors.
7:     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, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pdamax1,m.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
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 factorization is the exact factorization of a nearby matrix A+E, where
E2 = Oε A2
and ε is the machine precision.

8  Further Comments

The total number of floating point operations is approximately 23m23n-m if mn, or 23n23m-n if m>n.
To form the unitary matrix Q nag_zgerqf (f08cvc) may be followed by a call to nag_zungrq (f08cwc):
nag_zungrq(order, n, n, minmn, a, pda, tau, &fail)  
where minmn=minm,n, but note that the first dimension of the array a must be at least n, which may be larger than was required by nag_zgerqf (f08cvc). When mn, it is often only the first m rows of Q that are required and they may be formed by the call:
nag_zungrq(order, m, n, m, a, pda, tau, c, pdc, &fail)
To apply Q to an arbitrary real rectangular matrix C, nag_zgerqf (f08cvc) may be followed by a call to nag_zunmrq (f08cxc). For example:
nag_zunmrq(Nag_LeftSide, Nag_ConjTrans, n, p, minmn, a, pda, tau, c, pdc, &fail)
forms C=QHC, where C is n by p.
The real analogue of this function is nag_dgerqf (f08chc).

9  Example

This example finds the minimum norm solution to the underdetermined equations
Ax=b
where
A = 0.28-0.36i 0.50-0.86i -0.77-0.48i 1.58+0.66i -0.50-1.10i -1.21+0.76i -0.32-0.24i -0.27-1.15i 0.36-0.51i -0.07+1.33i -0.75+0.47i -0.08+1.01i
and
b = -1.35+0.19i 9.41-3.56i -7.57+6.93i .
The solution is obtained by first obtaining an RQ factorization of the matrix A.

9.1  Program Text

Program Text (f08cvce.c)

9.2  Program Data

Program Data (f08cvce.d)

9.3  Program Results

Program Results (f08cvce.r)


nag_zgerqf (f08cvc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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