nag_dorgrq (f08cjc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_dorgrq (f08cjc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dorgrq (f08cjc) generates all or part of the real n by n orthogonal matrix Q from an RQ factorization computed by nag_dgerqf (f08chc).

2  Specification

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

3  Description

nag_dorgrq (f08cjc) is intended to be used following a call to nag_dgerqf (f08chc), which performs an RQ factorization of a real matrix A and represents the orthogonal matrix Q as a product of k elementary reflectors of order n.
This function may be used to generate Q explicitly as a square matrix, or to form only its trailing rows.
Usually Q is determined from the RQ factorization of a p by n matrix A with pn. The whole of Q may be computed by:
nag_dorgrq(order,n,n,p,a,pda,tau,info)
(note that the matrix A must have at least n rows), or its trailing p rows as:
nag_dorgrq(order,p,n,p,a,pda,tau,info)
The rows of Q returned by the last call form an orthonormal basis for the space spanned by the rows of A; thus nag_dgerqf (f08chc) followed by nag_dorgrq (f08cjc) can be used to orthogonalise the rows of A.
The information returned by nag_dgerqf (f08chc) also yields the RQ factorization of the trailing k rows of A, where k<p. The orthogonal matrix arising from this factorization can be computed by:
nag_dorgrq(order,n,n,k,a,pda,tau,info)
or its leading k columns by:
nag_dorgrq(order,k,n,k,a,pda,tau,info)

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 Q.
Constraint: m0.
3:     nIntegerInput
On entry: n, the number of columns of the matrix Q.
Constraint: nm.
4:     kIntegerInput
On entry: k, the number of elementary reflectors whose product defines the matrix Q.
Constraint: mk0.
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.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_dgerqf (f08chc).
On exit: the m by n matrix Q.
If order=Nag_ColMajor, the i,jth element of the matrix is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, the i,jth element of the matrix is stored in a[i-1×pda+j-1].
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,m;
  • if order=Nag_RowMajor, pdamax1,n.
7:     tau[dim]const doubleInput
Note: the dimension, dim, of the array tau must be at least max1,k.
On entry: tau[i-1] must contain the scalar factor of the elementary reflector Hi, as returned by nag_dgerqf (f08chc).
8:     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, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, m=value and k=value.
Constraint: mk0.
On entry, n=value and m=value.
Constraint: nm.
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 matrix Q differs from an exactly orthogonal matrix by a matrix E such that
E2 = Oε
and ε is the machine precision.

8  Further Comments

The total number of floating point operations is approximately 4mnk-2m+nk2+43k3; when m=k this becomes 23m23n-m.
The complex analogue of this function is nag_zungrq (f08cwc).

9  Example

This example generates the first four rows of the matrix Q of the RQ factorization of A as returned by nag_dgerqf (f08chc), where
A = -0.57 -1.93 2.30 -1.93 0.15 -0.02 -1.28 1.08 0.24 0.64 0.30 1.03 -0.39 -0.31 0.40 -0.66 0.15 -1.43 0.25 -2.14 -0.35 0.08 -2.13 0.50 .

9.1  Program Text

Program Text (f08cjce.c)

9.2  Program Data

Program Data (f08cjce.d)

9.3  Program Results

Program Results (f08cjce.r)


nag_dorgrq (f08cjc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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