NAG FL Interface
f08vgf (dggsvp3)

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1 Purpose

f08vgf uses orthogonal transformations to simultaneously reduce the m×n matrix A and the p×n matrix B to upper triangular form. This factorization is usually used as a preprocessing step for computing the generalized singular value decomposition (GSVD). For sufficiently large problems, a blocked algorithm is used to make best use of Level 3 BLAS.

2 Specification

Fortran Interface
Subroutine f08vgf ( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola, tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, tau, work, lwork, info)
Integer, Intent (In) :: m, p, n, lda, ldb, ldu, ldv, ldq, lwork
Integer, Intent (Out) :: k, l, iwork(n), info
Real (Kind=nag_wp), Intent (In) :: tola, tolb
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), u(ldu,*), v(ldv,*), q(ldq,*)
Real (Kind=nag_wp), Intent (Out) :: tau(n), work(max(1,lwork))
Character (1), Intent (In) :: jobu, jobv, jobq
C Header Interface
#include <nag.h>
void  f08vgf_ (const char *jobu, const char *jobv, const char *jobq, const Integer *m, const Integer *p, const Integer *n, double a[], const Integer *lda, double b[], const Integer *ldb, const double *tola, const double *tolb, Integer *k, Integer *l, double u[], const Integer *ldu, double v[], const Integer *ldv, double q[], const Integer *ldq, Integer iwork[], double tau[], double work[], const Integer *lwork, Integer *info, const Charlen length_jobu, const Charlen length_jobv, const Charlen length_jobq)
The routine may be called by the names f08vgf, nagf_lapackeig_dggsvp3 or its LAPACK name dggsvp3.

3 Description

f08vgf computes orthogonal matrices U, V and Q such that
UTAQ= { n-k-lklk0A12A13l00A23m-k-l000() , if ​m-k-l0; n-k-lklk0A12A13m-k00A23() , if ​m-k-l<0;   VTBQ= n-k-lkll00B13p-l000()  
where the k×k matrix A12 and l×l matrix B13 are nonsingular upper triangular; A23 is l×l upper triangular if m-k-l0 and is (m-k)×l upper trapezoidal otherwise. (k+l) is the effective numerical rank of the (m+p)×n matrix (ATBT)T.
This decomposition is usually used as the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see routine f08yef; the two steps are combined in f08vcf.

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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: jobu Character(1) Input
On entry: if jobu='U', the orthogonal matrix U is computed.
If jobu='N', U is not computed.
Constraint: jobu='U' or 'N'.
2: jobv Character(1) Input
On entry: if jobv='V', the orthogonal matrix V is computed.
If jobv='N', V is not computed.
Constraint: jobv='V' or 'N'.
3: jobq Character(1) Input
On entry: if jobq='Q', the orthogonal matrix Q is computed.
If jobq='N', Q is not computed.
Constraint: jobq='Q' or 'N'.
4: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
5: p Integer Input
On entry: p, the number of rows of the matrix B.
Constraint: p0.
6: n Integer Input
On entry: n, the number of columns of the matrices A and B.
Constraint: n0.
7: 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: the m×n matrix A.
On exit: contains the triangular (or trapezoidal) matrix described in Section 3.
8: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08vgf is called.
Constraint: ldamax(1,m).
9: b(ldb,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least max(1,n).
On entry: the p×n matrix B.
On exit: contains the triangular matrix described in Section 3.
10: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f08vgf is called.
Constraint: ldbmax(1,p).
11: tola Real (Kind=nag_wp) Input
12: tolb Real (Kind=nag_wp) Input
On entry: tola and tolb are the thresholds to determine the effective numerical rank of matrix B and a subblock of A. Generally, they are set to
tola=max(m,n)Aε, tolb=max(p,n)Bε,  
where ε is the machine precision.
The size of tola and tolb may affect the size of backward errors of the decomposition.
13: k Integer Output
14: l Integer Output
On exit: k and l specify the dimension of the subblocks k and l as described in Section 3; (k+l) is the effective numerical rank of (aTbT)T.
15: u(ldu,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array u must be at least max(1,m) if jobu='U', and at least 1 otherwise.
On exit: if jobu='U', u contains the orthogonal matrix U.
If jobu='N', u is not referenced.
16: ldu Integer Input
On entry: the first dimension of the array u as declared in the (sub)program from which f08vgf is called.
Constraints:
  • if jobu='U', ldu max(1,m) ;
  • otherwise ldu1.
17: v(ldv,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array v must be at least max(1,p) if jobv='V', and at least 1 otherwise.
On exit: if jobv='V', v contains the orthogonal matrix V.
If jobv='N', v is not referenced.
18: ldv Integer Input
On entry: the first dimension of the array v as declared in the (sub)program from which f08vgf is called.
Constraints:
  • if jobv='V', ldv max(1,p) ;
  • otherwise ldv1.
19: q(ldq,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array q must be at least max(1,n) if jobq='Q', and at least 1 otherwise.
On exit: if jobq='Q', q contains the orthogonal matrix Q.
If jobq='N', q is not referenced.
20: ldq Integer Input
On entry: the first dimension of the array q as declared in the (sub)program from which f08vgf is called.
Constraints:
  • if jobq='Q', ldq max(1,n) ;
  • otherwise ldq1.
21: iwork(n) Integer array Workspace
22: tau(n) Real (Kind=nag_wp) array Workspace
23: 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.
24: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08vgf 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, lwork must generally be larger than the minimum; increase workspace by, say, nb×(n+1), where nb is the optimal block size
Constraints:
  • if jobv='V', lworkmax(2*n+1,p,m);
  • if jobv='N', lworkmax(2*n+1,m).
25: info Integer Output
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 factorization is nearly the exact factorization for nearby matrices (A+E) and (B+F), where
E2 = O(ε)A2   and   F2= O(ε)B2,  
and ε is the machine precision.

8 Parallelism and Performance

f08vgf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08vgf 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

This routine replaces the deprecated routine f08vef which used an unblocked algorithm and, therefore, did not make best use of Level 3 BLAS routines.
The complex analogue of this routine is f08vuf.

10 Example

This example finds the generalized factorization
A = UΣ1 ( 0 S ) QT ,   B= VΣ2 ( 0 T ) QT ,  
of the matrix pair (AB), where
A = ( 123 321 456 788 )   and   B= ( −2−33 465 ) .  

10.1 Program Text

Program Text (f08vgfe.f90)

10.2 Program Data

Program Data (f08vgfe.d)

10.3 Program Results

Program Results (f08vgfe.r)