nag_dorgql (f08cfc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_dorgql (f08cfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dorgql (f08cfc) generates all or part of the real m by m orthogonal matrix Q from a QL factorization computed by nag_dgeqlf (f08cec).

2  Specification

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

3  Description

nag_dorgql (f08cfc) is intended to be used after a call to nag_dgeqlf (f08cec), which performs a QL factorization of a real matrix A. The orthogonal matrix Q is represented as a product of elementary reflectors.
This function may be used to generate Q explicitly as a square matrix, or to form only its trailing columns.
Usually Q is determined from the QL factorization of an m by p matrix A with mp. The whole of Q may be computed by:
nag_dorgql(order,m,m,p,&a,pda,tau,&fail)
(note that the array a must have at least m columns) or its trailing p columns by:
nag_dorgql(order,m,p,p,&a,pda,tau,&fail)
The columns of Q returned by the last call form an orthonormal basis for the space spanned by the columns of A; thus nag_dgeqlf (f08cec) followed by nag_dorgql (f08cfc) can be used to orthogonalise the columns of A.
The information returned by nag_dgeqlf (f08cec) also yields the QL factorization of the trailing k columns of A, where k<p. The orthogonal matrix arising from this factorization can be computed by:
nag_dorgql(order,m,m,k,&a,pda,tau,&fail)
or its trailing k columns by:
nag_dorgql(order,m,k,k,&a,pda,tau,&fail)

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: mn0.
4:     kIntegerInput
On entry: k, the number of elementary reflectors whose product defines the matrix Q.
Constraint: nk0.
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_dgeqlf (f08cec).
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: further details of the elementary reflectors as returned by nag_dgeqlf (f08cec).
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 n=value.
Constraint: mn0.
On entry, n=value and k=value.
Constraint: nk0.
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 ,
where ε is the machine precision.

8  Further Comments

The total number of floating point operations is approximately 4mnk-2 m+n k2 + 43 k3 ; when n=k, the number is approximately 23 n2 3m-n .
The complex analogue of this function is nag_zungql (f08ctc).

9  Example

This example generates the first four columns of the matrix Q of the QL factorization of A as returned by nag_dgeqlf (f08cec), where
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 .

9.1  Program Text

Program Text (f08cfce.c)

9.2  Program Data

Program Data (f08cfce.d)

9.3  Program Results

Program Results (f08cfce.r)


nag_dorgql (f08cfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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