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

C Header Interface

#include nagmk26.h

void

f08vcf_ ( 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[], const Integer *lwork, Integer iwork[], Integer *info, const Charlen length_jobu, const Charlen length_jobv, const Charlen length_jobq)

The routine may be called by its LAPACK name dggsvd3.

3

Description

Given an

m

n

real matrix

A

and a

p

n

real matrix

B

, 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

l

be the effective numerical rank of

B

and

(k + l)

be the effective numerical rank of the matrix

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

, then the first

k

generalized singular values are infinite and the remaining

l

are finite.

R

is a

(k + l)

(k + l)

nonsingular upper triangular matrix,

D_{1}

and

D_{2}

are

m

(k + l)

and

p

(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

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 setting

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

A two stage process is used to compute the GSVD of the matrix pair

(A, B)

. The pair is first reduced to upper triangular form by orthogonal transformations using f08vgf (dggsvp3). The GSVD of the resulting upper triangular matrix pair is then performed by f08yef (dtgsja) which uses a variant of the Kogbetliantz algorithm (a cyclic Jacobi method).

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: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrices

A

and

B

Constraint:

n \geq 0

6: $p$ – IntegerInput

On entry:

p

, the number of rows of the matrix

B

Constraint:

p \geq 0

7: $k$ – IntegerOutput

8: $l$ – 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 \\ B \end{matrix})

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: contains the triangular matrix

R

, or part of

R

. See Section 3 for details.

10: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08vcf (dggsvd3) 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: contains the triangular matrix

R

m - k - l < 0

. See Section 3 for details.

12: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f08vcf (dggsvd3) is called.

Constraint:

ldb \geq \max (1, p)

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

On exit: see the description of beta.

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

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) 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

m

m

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 f08vcf (dggsvd3) 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

p

p

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 f08vcf (dggsvd3) 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

n

n

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 f08vcf (dggsvd3) is called.

Constraints:

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

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

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

22: $lwork$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08vcf (dggsvd3) 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 $jobv ='V'$ , $lwork \geq \max (3 * n + 1, m, p)$ ;
if $jobv ='N'$ , $lwork \geq \max (3 * n + 1, m)$ .

23: $iwork (n)$ – Integer arrayOutput

On exit: stores the sorting information. More precisely, if

I

is the ordered set of indices of alpha containing

C

(denote as

alpha (I)

, see beta), then the corresponding elements

iwork (I)

contain the swap pivots,

J

, that sorts

I

such that

alpha (I)

is in descending numerical order.

The following pseudocode sorts the set

I

\begin{array}{l} for ​ i \in I \\ j = J_{i} \\ swap ​ I_{i} ​ and ​ I_{j} \\ end \end{array}

24: $info$ – 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 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

f08vcf (dggsvd3) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08vcf (dggsvd3) 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 f08vaf (dggsvd) which used an unblocked algorithm and therefore did not make best use of level 3 BLAS routines.

The complex analogue of this routine is f08vqf (zggsvd3).

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.

NAG Library Routine Document

f08vcf (dggsvd3)

▸▿ 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