F08KFF (DORGBR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08KFF (DORGBR)

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

1  Purpose

F08KFF (DORGBR) generates one of the real orthogonal matrices Q or PT which were determined by F08KEF (DGEBRD) when reducing a real matrix to bidiagonal form.

2  Specification

SUBROUTINE F08KFF ( VECT, M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
INTEGER  M, N, K, LDA, LWORK, INFO
REAL (KIND=nag_wp)  A(LDA,*), TAU(*), WORK(max(1,LWORK))
CHARACTER(1)  VECT
The routine may be called by its LAPACK name dorgbr.

3  Description

F08KFF (DORGBR) is intended to be used after a call to F08KEF (DGEBRD), which reduces a real rectangular matrix A to bidiagonal form B by an orthogonal transformation: A=QBPT. F08KEF (DGEBRD) represents the matrices Q and PT as products of elementary reflectors.
This routine may be used to generate Q or PT explicitly as square matrices, or in some cases just the leading columns of Q or the leading rows of PT.
The various possibilities are specified by the arguments VECT, M, N and K. The appropriate values to cover the most likely cases are as follows (assuming that A was an m by n matrix):
1. To form the full m by m matrix Q:
CALL DORGBR('Q',m,m,n,...)
(note that the array A must have at least m columns).
2. If m>n, to form the n leading columns of Q:
CALL DORGBR('Q',m,n,n,...)
3. To form the full n by n matrix PT:
CALL DORGBR('P',n,n,m,...)
(note that the array A must have at least n rows).
4. If m<n, to form the m leading rows of PT:
CALL DORGBR('P',m,n,m,...)

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Arguments

1:     VECT – CHARACTER(1)Input
On entry: indicates whether the orthogonal matrix Q or PT is generated.
VECT='Q'
Q is generated.
VECT='P'
PT is generated.
Constraint: VECT='Q' or 'P'.
2:     M – INTEGERInput
On entry: m, the number of rows of the orthogonal matrix Q or PT to be returned.
Constraint: M0.
3:     N – INTEGERInput
On entry: n, the number of columns of the orthogonal matrix Q or PT to be returned.
Constraints:
  • N0;
  • if VECT='Q' and M>K, MNK;
  • if VECT='Q' and MK, M=N;
  • if VECT='P' and N>K, NMK;
  • if VECT='P' and NK, N=M.
4:     K – INTEGERInput
On entry: if VECT='Q', the number of columns in the original matrix A.
If VECT='P', the number of rows in the original matrix A.
Constraint: K0.
5:     ALDA* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
On entry: details of the vectors which define the elementary reflectors, as returned by F08KEF (DGEBRD).
On exit: the orthogonal matrix Q or PT, or the leading rows or columns thereof, as specified by VECT, M and N.
6:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08KFF (DORGBR) is called.
Constraint: LDAmax1,M.
7:     TAU* – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array TAU must be at least max1,minM,K if VECT='Q' and at least max1,minN,K if VECT='P'.
On entry: further details of the elementary reflectors, as returned by F08KEF (DGEBRD) in its argument TAUQ if VECT='Q', or in its argument TAUP if VECT='P'.
8:     WORKmax1,LWORK – REAL (KIND=nag_wp) arrayWorkspace
On exit: if INFO=0, WORK1 contains the minimum value of LWORK required for optimal performance.
9:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08KFF (DORGBR) 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, LWORKminM,N×nb, where nb is the optimal block size.
Constraint: LWORK max1,minM,N or LWORK=-1.
10:   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. An explanatory message is output, and execution of the program is terminated.

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. A similar statement holds for the computed matrix PT.

8  Parallelism and Performance

F08KFF (DORGBR) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08KFF (DORGBR) 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 for the cases listed in Section 3 are approximately as follows:
1. To form the whole of Q:
  • 43n3m2-3mn+n2 if m>n,
  • 43m3 if mn;
2. To form the n leading columns of Q when m>n:
  • 23 n2 3m-n ;
3. To form the whole of PT:
  • 43n3 if mn,
  • 43m3n2-3mn+m2 if m<n;
4. To form the m leading rows of PT when m<n:
  • 23 m2 3n-m .
The complex analogue of this routine is F08KTF (ZUNGBR).

10  Example

For this routine two examples are presented, both of which involve computing the singular value decomposition of a 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  
in the first example and
A = -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50  
in the second. A must first be reduced to tridiagonal form by F08KEF (DGEBRD). The program then calls F08KFF (DORGBR) twice to form Q and PT, and passes these matrices to F08MEF (DBDSQR), which computes the singular value decomposition of A.

10.1  Program Text

Program Text (f08kffe.f90)

10.2  Program Data

Program Data (f08kffe.d)

10.3  Program Results

Program Results (f08kffe.r)


F08KFF (DORGBR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

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