Integer, Intent (In)	::	m, p, n, k, l, lda, ldb, ldu, ldv, ldq
Integer, Intent (Out)	::	ncycle, 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)	::	alpha(n), beta(n), work(2*n)
Character (1), Intent (In)	::	jobu, jobv, jobq

C Header Interface

#include nagmk26.h

void

f08yef_ ( const char *jobu, const char *jobv, const char *jobq, const Integer *m, const Integer *p, const Integer *n, const Integer *k, const Integer *l, double a[], const Integer *lda, double b[], const Integer *ldb, const double *tola, const double *tolb, double alpha[], double beta[], double u[], const Integer *ldu, double v[], const Integer *ldv, double q[], const Integer *ldq, double work[], Integer *ncycle, Integer *info, const Charlen length_jobu, const Charlen length_jobv, const Charlen length_jobq)

The routine may be called by its LAPACK name dtgsja.

3

Description

f08yef (dtgsja) computes the GSVD of the matrices

A

and

B

which are assumed to have the form as returned by f08vef (dggsvp) or f08vgf (dggsvp3)

\begin{matrix} A = & \{\begin{matrix} \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{matrix} \\ B = & \begin{array}{rc} n - k - l & k & l \\ l & 0 & 0 & B_{13} \\ p - l & 0 & 0 & 0 \\ ( & ) \end{array}, \end{matrix}

where the

k

k

matrix

A_{12}

and the

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.

f08yef (dtgsja) computes orthogonal matrices

Q

U

and

V

, diagonal matrices

D_{1}

and

D_{2}

, and an upper triangular matrix

R

such that

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

Optionally

Q

U

and

V

may or may not be computed, or they may be premultiplied by matrices

Q_{1}

U_{1}

and

V_{1}

respectively.

(m - k - l) \geq 0

then

D_{1}

D_{2}

and

R

have the form

D_{1} = \begin{array}{rc} k & l \\ k & I & 0 \\ l & 0 & C \\ m - k - l & 0 & 0 \\ ( & ) \end{array},

D_{2} = \begin{array}{rc} k & l \\ l & 0 & S \\ p - l & 0 & 0 \\ ( & ) \end{array},

R = \begin{array}{rc} k & l \\ k & R_{11} & R_{12} \\ l & 0 & R_{22} \\ ( & ) \end{array},

where

C = diag (α_{k + 1},,, \dots,,, α_{k + l}), S = diag (β_{k + 1},,, \dots,,, β_{k + l})

(m - k - l) < 0

then

D_{1}

D_{2}

and

R

have the form

D_{1} = \begin{array}{rc} k & m - k & k + l - m \\ k & I & 0 & 0 \\ m - k & 0 & C & 0 \\ ( & ) \end{array},

D_{2} = \begin{array}{rc} k & m - k & k + l - m \\ m - k & 0 & S & 0 \\ k + l - m & 0 & 0 & I \\ p - l & 0 & 0 & 0 \\ ( & ) \end{array},

R = \begin{array}{rc} k & m - k & k + l - m \\ k & R_{11} & R_{12} & R_{13} \\ m - k & 0 & R_{22} & R_{23} \\ k + l - m & 0 & 0 & R_{33} \\ ( & ) \end{array},

where

C = diag (α_{k + 1},,, \dots,,, α_{m}), S = diag (β_{k + 1},,, \dots,,, β_{m})

In both cases the diagonal matrix

C

has non-negative diagonal elements, the diagonal matrix

S

has positive diagonal elements, so that

S

is nonsingular, and

C^{2} + S^{2} = 1

. See Section 2.3.5.3 of Anderson et al. (1999) for further information.

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

5

Arguments

1: $jobu$ – Character(1)Input

On entry: if

jobu ='U'

, u must contain an orthogonal matrix

U_{1}

on entry, and the product

U_{1} U

is returned.

jobu ='I'

, u is initialized to the unit matrix, and the orthogonal matrix

U

is returned.

jobu ='N'

U

is not computed.

Constraint:

jobu ='U'

'I'

'N'

2: $jobv$ – Character(1)Input

On entry: if

jobv ='V'

, v must contain an orthogonal matrix

V_{1}

on entry, and the product

V_{1} V

is returned.

jobv ='I'

, v is initialized to the unit matrix, and the orthogonal matrix

V

is returned.

jobv ='N'

V

is not computed.

Constraint:

jobv ='V'

'I'

'N'

3: $jobq$ – Character(1)Input

On entry: if

jobq ='Q'

, q must contain an orthogonal matrix

Q_{1}

on entry, and the product

Q_{1} Q

is returned.

jobq ='I'

, q is initialized to the unit matrix, and the orthogonal matrix

Q

is returned.

jobq ='N'

Q

is not computed.

Constraint:

jobq ='Q'

'I'

'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: $k$ – IntegerInput

8: $l$ – IntegerInput

On entry: k and l specify the sizes,

k

and

l

, of the subblocks of

A

and

B

, whose GSVD is to be computed by f08yef (dtgsja).

9: $a (lda *)$ – 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: if

m - k - l \geq 0

a (1 : k + l, n - k - l + 1 : n)

contains the

(k + l)

(k + l)

upper triangular matrix

R

m - k - l < 0

a (1 : m, n - k - l + 1 : n)

contains the first

m

rows of the

(k + l)

(k + l)

upper triangular matrix

R

, and the submatrix

R_{33}

is returned in

b (m - k + 1 : l, n + m - k - l + 1 : n)

10: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08yef (dtgsja) is called.

Constraint:

lda \geq \max (1, m)

11: $b (ldb *)$ – 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: if

m - k - l < 0

b (m - k + 1 : l, n + m - k - l + 1 : n)

contains the submatrix

R_{33}

R

12: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f08yef (dtgsja) is called.

Constraint:

ldb \geq \max (1, p)

13: $tola$ – Real (Kind=nag_wp)Input

14: $tolb$ – Real (Kind=nag_wp)Input

On entry: tola and tolb are the convergence criteria for the Jacobi–Kogbetliantz iteration procedure. Generally, they should be the same as used in the preprocessing step performed by f08vef (dggsvp) or f08vgf (dggsvp3), say

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

where

ε

is the machine precision.

15: $alpha (n)$ – Real (Kind=nag_wp) arrayOutput

On exit: see the description of beta.

16: $beta (n)$ – Real (Kind=nag_wp) arrayOutput

On exit: alpha and beta contain the generalized singular value pairs of

A

and

B

;

$alpha (i) = 1$ , $beta (i) = 0$ , for $i = 1, 2, \dots, k$ , and
if $m - k - l \geq 0$ , $alpha (i) = α_{i}$ , $beta (i) = β_{i}$ , for $i = k + 1, \dots, k + l$ , or
if $m - k - l < 0$ , $alpha (i) = α_{i}$ , $beta (i) = β_{i}$ , for $i = k + 1, \dots, m$ and $alpha (i) = 0$ , $beta (i) = 1$ , for $i = m + 1, \dots, k + l$ .

Furthermore, if

k + l < n

alpha (i) = beta (i) = 0

, for

i = k + l + 1, \dots, n

17: $u (ldu *)$ – Real (Kind=nag_wp) arrayInput/Output

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

\max (1, m)

jobu ='U'

'I'

, and at least

1

otherwise.

On entry: if

jobu ='U'

, u must contain an

m

m

matrix

U_{1}

(usually the orthogonal matrix returned by f08vef (dggsvp) or f08vgf (dggsvp3)).

On exit: if

jobu ='U'

, u contains the product

U_{1} U

jobu ='I'

, u contains the orthogonal matrix

U

jobu ='N'

, u is not referenced.

18: $ldu$ – IntegerInput

On entry: the first dimension of the array u as declared in the (sub)program from which f08yef (dtgsja) is called.

Constraints:

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

19: $v (ldv *)$ – Real (Kind=nag_wp) arrayInput/Output

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

\max (1, p)

jobv ='V'

'I'

, and at least

1

otherwise.

On entry: if

jobv ='V'

, v must contain an

p

p

matrix

V_{1}

(usually the orthogonal matrix returned by f08vef (dggsvp) or f08vgf (dggsvp3)).

On exit: if

jobv ='I'

, v contains the orthogonal matrix

V

jobv ='V'

, v contains the product

V_{1} V

jobv ='N'

, v is not referenced.

20: $ldv$ – IntegerInput

On entry: the first dimension of the array v as declared in the (sub)program from which f08yef (dtgsja) is called.

Constraints:

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

21: $q (ldq *)$ – Real (Kind=nag_wp) arrayInput/Output

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

\max (1, n)

jobq ='Q'

'I'

, and at least

1

otherwise.

On entry: if

jobq ='Q'

, q must contain an

n

n

matrix

Q_{1}

(usually the orthogonal matrix returned by f08vef (dggsvp) or f08vgf (dggsvp3)).

On exit: if

jobq ='I'

, q contains the orthogonal matrix

Q

jobq ='Q'

, q contains the product

Q_{1} Q

jobq ='N'

, q is not referenced.

22: $ldq$ – IntegerInput

On entry: the first dimension of the array q as declared in the (sub)program from which f08yef (dtgsja) is called.

Constraints:

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

23: $work (2 \times n)$ – Real (Kind=nag_wp) arrayWorkspace

24: $ncycle$ – IntegerOutput

On exit: the number of cycles required for convergence.

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.

$info = 1$: The procedure does not converge after $40$ cycles.

7

Accuracy

The computed generalized singular value decomposition is nearly the exact generalized singular value decomposition 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. See Section 4.12 of Anderson et al. (1999) for further details.

8

Parallelism and Performance

f08yef (dtgsja) 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 f08ysf (ztgsja).

10

Example

This example finds the generalized singular value decomposition

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

of the matrix pair

(A, B)

, 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

f08yef (dtgsja)

▸▿ 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