nag_dorgql (f08cfc) (PDF version)
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NAG 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:
(note that the array a must have at least m columns) or its trailing p columns by:
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 orthogonalize 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:
or its trailing k columns by:

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
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 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.
  • 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

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, m=value.
Constraint: m0.
On entry, pda=value.
Constraint: pda>0.
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.
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  Parallelism and Performance

nag_dorgql (f08cfc) is not threaded by NAG in any implementation.
nag_dorgql (f08cfc) 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 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).

10  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 .

10.1  Program Text

Program Text (f08cfce.c)

10.2  Program Data

Program Data (f08cfce.d)

10.3  Program Results

Program Results (f08cfce.r)

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

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