nag_dggsvd3 (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVType jobv, Nag_ComputeQType jobq, Integer m, Integer n, Integer p, Integer *k, Integer *l, double a[], Integer pda, double b[], Integer pdb, double alpha[], double beta[], double u[], Integer pdu, double v[], Integer pdv, double q[], Integer pdq, Integer iwork[], NagError *fail)

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 function 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 nag_dggsvp3 (f08vgc). The GSVD of the resulting upper triangular matrix pair is then performed by nag_dtgsja (f08yec) 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: $order$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $jobu$ – Nag_ComputeUTypeInput

On entry: if

jobu = Nag_AllU

, the orthogonal matrix

U

is computed.

jobu = Nag_NotU

U

is not computed.

Constraint:

jobu = Nag_AllU

Nag_NotU

3: $jobv$ – Nag_ComputeVTypeInput

On entry: if

jobv = Nag_ComputeV

, the orthogonal matrix

V

is computed.

jobv = Nag_NotV

V

is not computed.

Constraint:

jobv = Nag_ComputeV

Nag_NotV

4: $jobq$ – Nag_ComputeQTypeInput

On entry: if

jobq = Nag_ComputeQ

, the orthogonal matrix

Q

is computed.

jobq = Nag_NotQ

Q

is not computed.

Constraint:

jobq = Nag_ComputeQ

Nag_NotQ

5: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

6: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrices

A

and

B

Constraint:

n \geq 0

7: $p$ – IntegerInput

On entry:

p

, the number of rows of the matrix

B

Constraint:

p \geq 0

8: $k$ – Integer *Output

9: $l$ – 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})

10: $a [\dim]$ – doubleInput/Output

Note: the dimension, dim, of the array a must be at least

$\max (1, pda \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pda)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

m

n

matrix

A

On exit: contains the triangular matrix

R

, or part of

R

. See Section 3 for details.

11: $pda$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraints:

if $order = Nag_ColMajor$ , $pda \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

12: $b [\dim]$ – doubleInput/Output

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, p \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

p

n

matrix

B

On exit: contains the triangular matrix

R

m - k - l < 0

. See Section 3 for details.

13: $pdb$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, p)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, n)$ .

14: $alpha [n]$ – doubleOutput

On exit: see the description of beta.

15: $beta [n]$ – doubleOutput

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 - 1]

, for

i = k, \dots, n

16: $u [\dim]$ – doubleOutput

Note: the dimension, dim, of the array u must be at least

$\max (1, pdu \times m)$ when $jobu = Nag_AllU$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix

U

is stored in

$u [(j - 1) \times pdu + i - 1]$ when $order = Nag_ColMajor$ ;
$u [(i - 1) \times pdu + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

jobu = Nag_AllU

, u contains the

m

m

orthogonal matrix

U

jobu = Nag_NotU

, u is not referenced.

17: $pdu$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array u.

Constraints:

if $jobu = Nag_AllU$ , $pdu \geq \max (1, m)$ ;
otherwise $pdu \geq 1$ .

18: $v [\dim]$ – doubleOutput

Note: the dimension, dim, of the array v must be at least

$\max (1, pdv \times p)$ when $jobv = Nag_ComputeV$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix

V

is stored in

$v [(j - 1) \times pdv + i - 1]$ when $order = Nag_ColMajor$ ;
$v [(i - 1) \times pdv + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

jobv = Nag_ComputeV

, v contains the

p

p

orthogonal matrix

V

jobv = Nag_NotV

, v is not referenced.

19: $pdv$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array v.

Constraints:

if $jobv = Nag_ComputeV$ , $pdv \geq \max (1, p)$ ;
otherwise $pdv \geq 1$ .

20: $q [\dim]$ – doubleOutput

Note: the dimension, dim, of the array q must be at least

$\max (1, pdq \times n)$ when $jobq = Nag_ComputeQ$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix

Q

is stored in

$q [(j - 1) \times pdq + i - 1]$ when $order = Nag_ColMajor$ ;
$q [(i - 1) \times pdq + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

jobq = Nag_ComputeQ

, q contains the

n

n

orthogonal matrix

Q

jobq = Nag_NotQ

, q is not referenced.

21: $pdq$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array q.

Constraints:

if $jobq = Nag_ComputeQ$ , $pdq \geq \max (1, n)$ ;
otherwise $pdq \geq 1$ .

22: $iwork [n]$ – IntegerOutput

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] - 1

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}

23: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_CONVERGENCE: The Jacobi-type procedure failed to converge.
NE_ENUM_INT_2: On entry, $jobq = 〈value〉$ , $pdq = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $jobq = Nag_ComputeQ$ , $pdq \geq \max (1, n)$ ;
otherwise $pdq \geq 1$ .

On entry, $jobu = 〈value〉$ , $pdu = 〈value〉$ and $m = 〈value〉$ .
Constraint: if $jobu = Nag_AllU$ , $pdu \geq \max (1, m)$ ;
otherwise $pdu \geq 1$ .

On entry, $jobv = 〈value〉$ , $pdv = 〈value〉$ and $p = 〈value〉$ .
Constraint: if $jobv = Nag_ComputeV$ , $pdv \geq \max (1, p)$ ;
otherwise $pdv \geq 1$ .
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $p = 〈value〉$ .
Constraint: $p \geq 0$ .

On entry, $pda = 〈value〉$ .
Constraint: $pda > 0$ .

On entry, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .

On entry, $pdq = 〈value〉$ .
Constraint: $pdq > 0$ .

On entry, $pdu = 〈value〉$ .
Constraint: $pdu > 0$ .

On entry, $pdv = 〈value〉$ .
Constraint: $pdv > 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pda \geq \max (1, m)$ .

On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq \max (1, n)$ .

On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdb = 〈value〉$ and $p = 〈value〉$ .
Constraint: $pdb \geq \max (1, p)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

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

nag_dggsvd3 (f08vcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dggsvd3 (f08vcc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

This function replaces the deprecated function nag_dggsvd (f08vac) which used an unblocked algorithm and therefore did not make best use of level 3 BLAS functions.

The complex analogue of this function is nag_zggsvd3 (f08vqc).

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 Function Documentnag_dggsvd3 (f08vcc)

▸▿ 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 Library Function Document

nag_dggsvd3 (f08vcc)