nag_dorgqr (f08afc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dorgqr (f08afc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dorgqr (f08afc) generates all or part of the real orthogonal matrix Q from a QR factorization computed by nag_dgeqrf (f08aec), nag_dgeqpf (f08bec) or nag_dgeqp3 (f08bfc).

2  Specification

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

3  Description

nag_dorgqr (f08afc) is intended to be used after a call to nag_dgeqrf (f08aec), nag_dgeqpf (f08bec) or nag_dgeqp3 (f08bfc). which perform a QR 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 leading columns.
Usually Q is determined from the QR factorization of an m by p matrix A with mp. The whole of Q may be computed by:
nag_dorgqr(order,m,m,p,&a,pda,tau,&fail)
(note that the array a must have at least m columns) or its leading p columns by:
nag_dorgqr(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_dgeqrf (f08aec) followed by nag_dorgqr (f08afc) can be used to orthogonalise the columns of A.
The information returned by the QR factorization functions also yields the QR factorization of the leading k columns of A, where k<p. The orthogonal matrix arising from this factorization can be computed by:
nag_dorgqr(order,m,m,k,&a,pda,tau,&fail)
or its leading k columns by:
nag_dorgqr(order,m,k,k,&a,pda,tau,&fail)

4  References

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 order of the orthogonal 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_dgeqrf (f08aec), nag_dgeqpf (f08bec) or nag_dgeqp3 (f08bfc).
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_dgeqrf (f08aec), nag_dgeqpf (f08bec) or nag_dgeqp3 (f08bfc).
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_zungqr (f08atc).

9  Example

This example forms the leading 4 columns of the orthogonal matrix Q from the QR factorization of the matrix A, 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 .
The columns of Q form an orthonormal basis for the space spanned by the columns of A.

9.1  Program Text

Program Text (f08afce.c)

9.2  Program Data

Program Data (f08afce.d)

9.3  Program Results

Program Results (f08afce.r)


nag_dorgqr (f08afc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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