to upper triangular form. This factorization is usually used as a preprocessing step for computing the generalized singular value decomposition (GSVD). f08vef is marked as deprecated by LAPACK; the replacement routine is f08vgf which makes better use of Level 3 BLAS.

2 Specification

Fortran Interface

Subroutine f08vef (

jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola, tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, tau, work, info)

Integer, Intent (In)	::	m, p, n, lda, ldb, ldu, ldv, ldq
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(3*n,m,p))
Character (1), Intent (In)	::	jobu, jobv, jobq

C Header Interface

#include <nag.h>

void

f08vef_ (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[], Integer *info, const Charlen length_jobu, const Charlen length_jobv, const Charlen length_jobq)

The routine may be called by the names f08vef, nagf_lapackeig_dggsvp or its LAPACK name dggsvp.

3 Description

f08vef computes orthogonal matrices

U

V

and

Q

such that

\begin{array}{l} U^{T} A Q = {\begin{array}{r} \begin{array}{rc} n - k - l & k & l \\ k & 0 & A_{12} & A_{13} \\ l & 0 & 0 & A_{23} \\ m - k - l & 0 & 0 & 0 \\ ( & ) \end{array}, if ​ m - k - l \geq 0; \\ \begin{array}{rc} n - k - l & k & l \\ k & 0 & A_{12} & A_{13} \\ m - k & 0 & 0 & A_{23} \\ ( & ) \end{array}, if ​ m - k - l < 0; \end{array} \\ V^{T} B Q = \begin{array}{rc} n - k - l & k & l \\ l & 0 & 0 & B_{13} \\ p - l & 0 & 0 & 0 \\ ( & ) \end{array} \end{array}

where the

k \times k

matrix

A_{12}

and

l \times l

matrix

B_{13}

are nonsingular upper triangular;

A_{23}

l \times l

upper triangular if

m - k - l \geq 0

and is

(m - k) \times l

upper trapezoidal otherwise.

(k + l)

is the effective numerical rank of the

(m + p) \times n

matrix

{(\begin{matrix} A^{T} & B^{T} \end{matrix})}^{T}

This decomposition is usually used as the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see routine f08vaf.

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 (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: $jobu$ – Character(1) Input

On entry: if

jobu ='U'

, the orthogonal matrix

U

is computed.

jobu ='N'

U

is not computed.

Constraint:

jobu ='U'

'N'

2: $jobv$ – Character(1) Input

On entry: if

jobv ='V'

, the orthogonal matrix

V

is computed.

jobv ='N'

V

is not computed.

Constraint:

jobv ='V'

'N'

3: $jobq$ – Character(1) Input

On entry: if

jobq ='Q'

, the orthogonal matrix

Q

is computed.

jobq ='N'

Q

is not computed.

Constraint:

jobq ='Q'

'N'

4: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

5: $p$ – Integer Input

On entry:

p

, the number of rows of the matrix

B

Constraint:

p \geq 0

6: $n$ – Integer Input

On entry:

n

, the number of columns of the matrices

A

and

B

Constraint:

n \geq 0

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 \times 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 f08vef is called.

Constraint:

lda \geq \max (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 \times 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 f08vef is called.

Constraint:

ldb \geq \max (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

\begin{matrix} tola = \max (m, n) ‖ A ‖ ε, \\ tolb = \max (p, n) ‖ B ‖ ε, \end{matrix}

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

{(\begin{matrix} a^{T} & b^{T} \end{matrix})}^{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)

jobu ='U'

, and at least

1

otherwise.

On exit: if

jobu ='U'

, u contains the orthogonal matrix

U

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 f08vef is called.

Constraints:

if $jobu ='U'$ , $ldu \geq \max (1, m)$ ;
otherwise $ldu \geq 1$ .

17: $v (ldv, *)$ – Real (Kind=nag_wp) array Output

Note: the second dimension of the array v must be at least

\max (1, p)

jobv ='V'

, and at least

1

otherwise.

On exit: if

jobv ='V'

, v contains the orthogonal matrix

V

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 f08vef is called.

Constraints:

if $jobv ='V'$ , $ldv \geq \max (1, p)$ ;
otherwise $ldv \geq 1$ .

19: $q (ldq, *)$ – Real (Kind=nag_wp) array Output

Note: the second dimension of the array q must be at least

\max (1, n)

jobq ='Q'

, and at least

1

otherwise.

On exit: if

jobq ='Q'

, q contains the orthogonal matrix

Q

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 f08vef is called.

Constraints:

if $jobq ='Q'$ , $ldq \geq \max (1, n)$ ;
otherwise $ldq \geq 1$ .

21: $iwork (n)$ – Integer array Workspace

22: $tau (n)$ – Real (Kind=nag_wp) array Workspace

23: $work (\max (3 \times n, m, p))$ – Real (Kind=nag_wp) array Workspace

24: $info$ – 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

{‖ E ‖}_{2} = O (ε) {‖ A ‖}_{2} and {‖ F ‖}_{2} = O (ε) {‖ B ‖}_{2},

and

ε

is the machine precision.

8 Parallelism and Performance

f08vef 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 complex analogue of this routine is f08vsf.

10 Example

This example finds the generalized factorization

A = U Σ_{1} (\begin{matrix} 0 & S \end{matrix}) Q^{T}, B = V Σ_{2} (\begin{matrix} 0 & T \end{matrix}) Q^{T},

of the matrix pair

(\begin{matrix} A & B \end{matrix})

, where

A = (\begin{matrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 4 & 5 & 6 \\ 7 & 8 & 8 \end{matrix}) and B = (\begin{matrix} −2 & −3 & 3 \\ 4 & 6 & 5 \end{matrix}) .

f08ve: FL CL CPP AD

NAG FL Interfacef08vef (dggsvp)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f08vef (dggsvp)