F08CKF (DORMRQ) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08CKF (DORMRQ)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

 Contents

    1  Purpose
    7  Accuracy
    10  Example

1  Purpose

F08CKF (DORMRQ) multiplies a general real m by n matrix C by the real orthogonal matrix Q from an RQ factorization computed by F08CHF (DGERQF).

2  Specification

SUBROUTINE F08CKF ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
INTEGER  M, N, K, LDA, LDC, LWORK, INFO
REAL (KIND=nag_wp)  A(LDA,*), TAU(*), C(LDC,*), WORK(max(1,LWORK))
CHARACTER(1)  SIDE, TRANS
The routine may be called by its LAPACK name dormrq.

3  Description

F08CKF (DORMRQ) is intended to be used following a call to F08CHF (DGERQF), which performs an RQ factorization of a real matrix A and represents the orthogonal matrix Q as a product of elementary reflectors.
This routine may be used to form one of the matrix products
QC ,   QTC ,   CQ ,   CQT ,  
overwriting the result on C, which may be any real rectangular m by n matrix.
A common application of this routine is in solving underdetermined linear least squares problems, as described in the F08 Chapter Introduction, and illustrated in Section 10 in F08CHF (DGERQF).

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:     SIDE – CHARACTER(1)Input
On entry: indicates how Q or QT is to be applied to C.
SIDE='L'
Q or QT is applied to C from the left.
SIDE='R'
Q or QT is applied to C from the right.
Constraint: SIDE='L' or 'R'.
2:     TRANS – CHARACTER(1)Input
On entry: indicates whether Q or QT is to be applied to C.
TRANS='N'
Q is applied to C.
TRANS='T'
QT is applied to C.
Constraint: TRANS='N' or 'T'.
3:     M – INTEGERInput
On entry: m, the number of rows of the matrix C.
Constraint: M0.
4:     N – INTEGERInput
On entry: n, the number of columns of the matrix C.
Constraint: N0.
5:     K – INTEGERInput
On entry: k, the number of elementary reflectors whose product defines the matrix Q.
Constraints:
  • if SIDE='L', M K 0 ;
  • if SIDE='R', N K 0 .
6:     ALDA* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,M if SIDE='L' and at least max1,N if SIDE='R'.
On entry: the ith row of A must contain the vector which defines the elementary reflector Hi, for i=1,2,,k, as returned by F08CHF (DGERQF).
On exit: is modified by F08CKF (DORMRQ) but restored on exit.
7:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08CKF (DORMRQ) is called.
Constraint: LDAmax1,K.
8:     TAU* – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array TAU must be at least max1,K.
On entry: TAUi must contain the scalar factor of the elementary reflector Hi, as returned by F08CHF (DGERQF).
9:     CLDC* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array C must be at least max1,N.
On entry: the m by n matrix C.
On exit: C is overwritten by QC or QTC or CQ or CQT as specified by SIDE and TRANS.
10:   LDC – INTEGERInput
On entry: the first dimension of the array C as declared in the (sub)program from which F08CKF (DORMRQ) is called.
Constraint: LDCmax1,M.
11:   WORKmax1,LWORK – REAL (KIND=nag_wp) arrayWorkspace
On exit: if INFO=0, WORK1 contains the minimum value of LWORK required for optimal performance.
12:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08CKF (DORMRQ) is called.
If LWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.
Suggested value: for optimal performance, LWORKN×nb if SIDE='L' and at least M×nb if SIDE='R', where nb is the optimal block size.
Constraints:
  • if SIDE='L', LWORKmax1,N or LWORK=-1;
  • if SIDE='R', LWORKmax1,M or LWORK=-1.
13:   INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

INFO<0
If INFO=-i, argument i had an illegal value.
If INFO=-999, dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
An explanatory message is output, and execution of the program is terminated.

7  Accuracy

The computed result differs from the exact result by a matrix E such that
E2 = Oε C2  
where ε is the machine precision.

8  Parallelism and Performance

F08CKF (DORMRQ) 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 routine. 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 2nk2m-k if SIDE='L' and 2mk2n-k if SIDE='R'.
The complex analogue of this routine is F08CXF (ZUNMRQ).

10  Example

See Section 10 in F08CHF (DGERQF).

F08CKF (DORMRQ) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016