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 nagmk26.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 its LAPACK name dggsvp3.

3

Description

f08vgf (dggsvp3) 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

k

matrix

A_{12}

and

l

l

matrix

B_{13}

are nonsingular upper triangular;

A_{23}

l

l

upper triangular if

m - k - l \geq 0

and is

(m - k)

l

upper trapezoidal otherwise.

(k + l)

is the effective numerical rank of the

(m + p)

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 f08yef (dtgsja); the two steps are combined in f08vcf (dggsvd3).

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 (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.

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: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

5: $p$ – IntegerInput

On entry:

p

, the number of rows of the matrix

B

Constraint:

p \geq 0

6: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrices

A

and

B

Constraint:

n \geq 0

7: $a (lda *)$ – Real (Kind=nag_wp) arrayInput/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: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08vgf (dggsvp3) is called.

Constraint:

lda \geq \max (1, m)

9: $b (ldb *)$ – Real (Kind=nag_wp) arrayInput/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: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f08vgf (dggsvp3) 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: $k$ – IntegerOutput

14: $l$ – IntegerOutput

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) arrayOutput

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: $ldu$ – IntegerInput

On entry: the first dimension of the array u as declared in the (sub)program from which f08vgf (dggsvp3) is called.

Constraints:

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

17: $v (ldv *)$ – Real (Kind=nag_wp) arrayOutput

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: $ldv$ – IntegerInput

On entry: the first dimension of the array v as declared in the (sub)program from which f08vgf (dggsvp3) is called.

Constraints:

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

19: $q (ldq *)$ – Real (Kind=nag_wp) arrayOutput

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: $ldq$ – IntegerInput

On entry: the first dimension of the array q as declared in the (sub)program from which f08vgf (dggsvp3) is called.

Constraints:

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

21: $iwork (n)$ – Integer arrayWorkspace

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

23: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

24: $lwork$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08vgf (dggsvp3) is called.

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 \times (n + 1)

, where

nb

is the optimal block size

Constraints:

if $jobv ='V'$ , $lwork \geq \max (2 * n + 1, p, m)$ ;
if $jobv ='N'$ , $lwork \geq \max (2 * n + 1, m)$ .

25: $info$ – 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 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

f08vgf (dggsvp3) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08vgf (dggsvp3) 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 (dggsvp) 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 (zggsvp3).

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}) .

NAG Library Routine Document

f08vgf (dggsvp3)

▸▿ 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

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