NAG FL Interface
f08kff (dorgbr)

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1 Purpose

f08kff generates one of the real orthogonal matrices Q or PT which were determined by f08kef when reducing a real matrix to bidiagonal form.

2 Specification

Fortran Interface
Subroutine f08kff ( vect, m, n, k, a, lda, tau, work, lwork, info)
Integer, Intent (In) :: m, n, k, lda, lwork
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (In) :: tau(*)
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*)
Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
Character (1), Intent (In) :: vect
C Header Interface
#include <nag.h>
void  f08kff_ (const char *vect, const Integer *m, const Integer *n, const Integer *k, double a[], const Integer *lda, const double tau[], double work[], const Integer *lwork, Integer *info, const Charlen length_vect)
The routine may be called by the names f08kff, nagf_lapackeig_dorgbr or its LAPACK name dorgbr.

3 Description

f08kff is intended to be used after a call to f08kef, which reduces a real rectangular matrix A to bidiagonal form B by an orthogonal transformation: A=QBPT. f08kef 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×n matrix):
  1. 1.To form the full m×m matrix Q:
    Call dorgbr('Q',m,m,n,...)
    (note that the array a must have at least m columns).
  2. 2.If m>n, to form the n leading columns of Q:
    Call dorgbr('Q',m,n,n,...)
  3. 3.To form the full n×n matrix PT:
    Call dorgbr('P',n,n,m,...)
    (note that the array a must have at least n rows).
  4. 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.
Q is generated.
PT is generated.
Constraint: vect='Q' or 'P'.
2: m Integer Input
On entry: m, the number of rows of the orthogonal matrix Q or PT to be returned.
Constraint: m0.
3: n Integer Input
On entry: n, the number of columns of the orthogonal matrix Q or PT to be returned.
  • 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 Integer Input
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: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: details of the vectors which define the elementary reflectors, as returned by f08kef.
On exit: the orthogonal matrix Q or PT, or the leading rows or columns thereof, as specified by vect, m and n.
6: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08kff is called.
Constraint: ldamax(1,m).
7: tau(*) Real (Kind=nag_wp) array Input
Note: the dimension of the array tau must be at least max(1,min(m,k)) if vect='Q' and at least max(1,min(n,k)) if vect='P'.
On entry: further details of the elementary reflectors, as returned by f08kef in its argument tauq if vect='Q', or in its argument taup if vect='P'.
8: work(max(1,lwork)) Real (Kind=nag_wp) array Workspace
On exit: if info=0, work(1) contains the minimum value of lwork required for optimal performance.
9: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08kff 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, lworkmin(m,n)×nb, where nb is the optimal block size.
Constraint: lwork max(1,min(m,n)) or lwork=−1.
10: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

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 = O(ε) ,  
where ε is the machine precision. A similar statement holds for the computed matrix PT.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08kff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kff 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. 1.To form the whole of Q:
    • 43n(3m2-3mn+n2) if m>n,
    • 43m3 if mn;
  2. 2.To form the n leading columns of Q when m>n:
    • 23 n2 (3m-n) ;
  3. 3.To form the whole of PT:
    • 43n3 if mn,
    • 43m(3n2-3mn+m2) if m<n;
  4. 4.To form the m leading rows of PT when m<n:
    • 23 m2 (3n-m) .
The complex analogue of this routine is f08ktf.

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 bidiagonal form by f08kef. The program then calls f08kff twice to form Q and PT, and passes these matrices to f08mef, 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)