Integer, Intent (In)	::	m, n, p, lda, ldb, ldu, ldv, ldq
Integer, Intent (Out)	::	k, l, iwork(n), info
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(max(3*n,m,p)+n)
Character (1), Intent (In)	::	jobu, jobv, jobq

C Header Interface

#include <nag.h>

void

f08vaf_ (const char *jobu, const char *jobv, const char *jobq, const Integer *m, const Integer *n, const Integer *p, Integer *k, Integer *l, double a[], const Integer *lda, double b[], const Integer *ldb, double alpha[], double beta[], double u[], const Integer *ldu, double v[], const Integer *ldv, double q[], const Integer *ldq, double work[], Integer iwork[], Integer *info, const Charlen length_jobu, const Charlen length_jobv, const Charlen length_jobq)

The routine may be called by the names f08vaf, nagf_lapackeig_dggsvd or its LAPACK name dggsvd.

3 Description

The generalized singular value decomposition is given by

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

where

U

V

and

Q

are orthogonal matrices. Let

(k + l)

be the effective numerical rank of the matrix

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

, then

R

is a

(k + l) \times (k + l)

nonsingular upper triangular matrix,

D_{1}

and

D_{2}

are

m \times (k + l)

and

p \times (k + l)

‘diagonal’ matrices structured as follows:

m - k - l \geq 0

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}

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

where

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

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

and

C^{2} + S^{2} = I .

R

is stored as a submatrix of

A

with elements

R_{i j}

stored as

A_{i, n - k - l + j}

on exit.

m - k - l < 0

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}

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

where

C = diag (α_{k + 1}, \dots, α_{m}),

S = diag (β_{k + 1}, \dots, β_{m}),

and

C^{2} + S^{2} = I .

(\begin{matrix} R_{11} & R_{12} & R_{13} \\ 0 & R_{22} & R_{23} \end{matrix})

is stored as a submatrix of

A

with

R_{i j}

stored as

A_{i, n - k - l + j}

, and

R_{33}

is stored as a submatrix of

B

with

{(R_{33})}_{i j}

stored as

B_{m - k + i, n + m - k - l + j}

The routine computes

C

S

R

and, optionally, the orthogonal transformation matrices

U

V

and

Q

In particular, if

B

is an

n \times n

nonsingular matrix, then the GSVD of

A

and

B

implicitly gives the SVD of

A B^{- 1}

A B^{- 1} = U (D_{1} D_{2}^{- 1}) V^{T} .

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

has orthonormal columns, then the GSVD of

A

and

B

is also equal to the CS decomposition of

A

and

B

. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:

A^{T} A x = λ B^{T} B x .

In some literature, the GSVD of

A

and

B

is presented in the form

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

where

U

and

V

are orthogonal and

X

is nonsingular, and

D_{1}

and

D_{2}

are ‘diagonal’. The former GSVD form can be converted to the latter form by taking the nonsingular matrix

X

X = Q (\begin{matrix} I & 0 \\ 0 & R^{- 1} \end{matrix}) .

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: $n$ – Integer Input

On entry:

n

, the number of columns of the matrices

A

and

B

Constraint:

n \geq 0

6: $p$ – Integer Input

On entry:

p

, the number of rows of the matrix

B

Constraint:

p \geq 0

7: $k$ – Integer Output

8: $l$ – 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 \\ B \end{matrix})

9: $a (lda, *)$ – 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 matrix

R

, or part of

R

. See Section 3 for details.

10: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08vaf is called.

Constraint:

lda \geq \max (1, m)

11: $b (ldb, *)$ – 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

R

m - k - l < 0

. See Section 3 for details.

12: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f08vaf is called.

Constraint:

ldb \geq \max (1, p)

13: $alpha (n)$ – Real (Kind=nag_wp) array Output

On exit: see the description of beta.

14: $beta (n)$ – Real (Kind=nag_wp) array Output

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

A

and

B

α_{i}

and

β_{i}

;

$alpha (1 : k) = 1$ ,
$beta (1 : k) = 0$ ,

and if

m - k - l \geq 0

$alpha (k + 1 : k + l) = C$ ,
$beta (k + 1 : k + l) = S$ ,

or if

m - k - l < 0

$alpha (k + 1 : m) = C$ ,
$alpha (m + 1 : k + l) = 0$ ,
$beta (k + 1 : m) = S$ ,
$beta (m + 1 : k + l) = 1$ , and
$alpha (k + l + 1 : n) = 0$ ,
$beta (k + l + 1 : n) = 0$ .

The notation

alpha (k : n)

above refers to consecutive elements

alpha (i)

, for

i = k, \dots, n

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'

On exit: if

jobu ='U'

, u contains the

m \times m

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 f08vaf 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'

On exit: if

jobv ='V'

, v contains the

p \times p

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 f08vaf 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'

On exit: if

jobq ='Q'

, q contains the

n \times n

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 f08vaf is called.

Constraints:

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

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

22: $iwork (n)$ – Integer array Output

On exit: stores the sorting information. More precisely, the following loop will sort alpha

for i=k+1, min(m,k+l)
  swap alpha(i) and alpha(iwork(i))
endfor

such that

alpha (1) \geq alpha (2) \geq \dots \geq alpha (n)

23: $info$ – 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.

$info = 1$: The Jacobi-type procedure failed to converge.

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

f08vaf 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 f08vnf.

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},

where

A = (\begin{matrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 4 & 5 & 6 \\ 7 & 8 & 8 \end{matrix}) and B = (\begin{array}{r} - 2 & - 3 & 3 \\ 4 & 6 & 5 \end{array}),

together with estimates for the condition number of

R

and the error bound for the computed generalized singular values.

The example program assumes that

m \geq n

, and would need slight modification if this is not the case.

f08va: FL CL CPP AD

NAG FL Interfacef08vaf (dggsvd)

▸▿ 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
f08vaf (dggsvd)