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 orthogonalize 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:     order Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     m IntegerInput
On entry: m, the number of rows of the matrix Q.
Constraint: m0.
3:     n IntegerInput
On entry: n, the number of columns of the matrix Q.
Constraint: nm.
4:     k IntegerInput
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:     pda IntegerInput
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:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

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
Parallelism and Performance

nag_dorgrq (f08cjc) 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 x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also 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-2m+nk2+43k3; when m=k this becomes 23m23n-m.
The complex analogue of this function is nag_zungrq (f08cwc).

10
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 .  

10.1
Program Text

Program Text (f08cjce.c)

10.2
Program Data

Program Data (f08cjce.d)

10.3
Program Results

Program Results (f08cjce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017