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

3

Description

nag_ztgsja (f08ysc) computes the GSVD of the matrices

A

and

B

which are assumed to have the form as returned by nag_zggsvp (f08vsc) or nag_zggsvp3 (f08vuc)

\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.

nag_ztgsja (f08ysc) computes unitary matrices

Q

U

and

V

, diagonal matrices

D_{1}

and

D_{2}

, and an upper triangular matrix

R

such that

U^{H} A Q = D_{1} (\begin{matrix} 0 & R \end{matrix}), V^{H} 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 real non-negative diagonal elements, the diagonal matrix

S

has real 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: $order$ – 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 3.3.1.3 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

, u must contain a unitary matrix

U_{1}

on entry, and the product

U_{1} U

is returned.

jobu = Nag_InitU

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

U

is returned.

jobu = Nag_NotU

U

is not computed.

Constraint:

jobu = Nag_AllU

Nag_InitU

Nag_NotU

3: $jobv$ – Nag_ComputeVTypeInput

On entry: if

jobv = Nag_ComputeV

, v must contain a unitary matrix

V_{1}

on entry, and the product

V_{1} V

is returned.

jobv = Nag_InitV

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

V

is returned.

jobv = Nag_NotV

V

is not computed.

Constraint:

jobv = Nag_ComputeV

Nag_InitV

Nag_NotV

4: $jobq$ – Nag_ComputeQTypeInput

On entry: if

jobq = Nag_ComputeQ

, q must contain a unitary matrix

Q_{1}

on entry, and the product

Q_{1} Q

is returned.

jobq = Nag_InitQ

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

Q

is returned.

jobq = Nag_NotQ

Q

is not computed.

Constraint:

jobq = Nag_ComputeQ

Nag_InitQ

Nag_NotQ

5: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

6: $p$ – IntegerInput

On entry:

p

, the number of rows of the matrix

B

Constraint:

p \geq 0

7: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrices

A

and

B

Constraint:

n \geq 0

8: $k$ – IntegerInput

9: $l$ – 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 nag_ztgsja (f08ysc).

10: $a [\dim]$ – ComplexInput/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 $order = Nag_RowMajor$ .

Where

A (i, j)

appears in this document, it refers to the array element

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

On entry: the

m

n

matrix

A

On exit: 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)

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: $b [\dim]$ – ComplexInput/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 $order = Nag_RowMajor$ .

Where

B (i, j)

appears in this document, it refers to the array element

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

On entry: the

p

n

matrix

B

On exit: if

m - k - l < 0

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

contains the submatrix

R_{33}

R

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: $tola$ – doubleInput

15: $tolb$ – doubleInput

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 nag_zggsvp (f08vsc) or nag_zggsvp3 (f08vuc), say

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

where

ε

is the machine precision.

16: $alpha [n]$ – doubleOutput

On exit: see the description of beta.

17: $beta [n]$ – doubleOutput

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

A

and

B

;

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

Furthermore, if

k + l < n

alpha [i] = beta [i] = 0

, for

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

18: $u [\dim]$ – ComplexInput/Output

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

$\max (1, pdu \times m)$ when $jobu = Nag_AllU$ or $Nag_InitU$ ;
$1$ 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 entry: if

jobu = Nag_AllU

, u must contain an

m

m

matrix

U_{1}

(usually the unitary matrix returned by nag_zggsvp (f08vsc) or nag_zggsvp3 (f08vuc)).

On exit: if

jobu = Nag_AllU

, u contains the product

U_{1} U

jobu = Nag_InitU

, u contains the unitary matrix

U

jobu = Nag_NotU

, u is not referenced.

19: $pdu$ – IntegerInput

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

Constraints:

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

20: $v [\dim]$ – ComplexInput/Output

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

$\max (1, pdv \times p)$ when $jobv = Nag_ComputeV$ or $Nag_InitV$ ;
$1$ 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 entry: if

jobv = Nag_ComputeV

, v must contain an

p

p

matrix

V_{1}

(usually the unitary matrix returned by nag_zggsvp (f08vsc) or nag_zggsvp3 (f08vuc)).

On exit: if

jobv = Nag_InitV

, v contains the unitary matrix

V

jobv = Nag_ComputeV

, v contains the product

V_{1} V

jobv = Nag_NotV

, v is not referenced.

21: $pdv$ – IntegerInput

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

Constraints:

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

22: $q [\dim]$ – ComplexInput/Output

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

$\max (1, pdq \times n)$ when $jobq = Nag_ComputeQ$ or $Nag_InitQ$ ;
$1$ 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 entry: if

jobq = Nag_ComputeQ

, q must contain an

n

n

matrix

Q_{1}

(usually the unitary matrix returned by nag_zggsvp (f08vsc) or nag_zggsvp3 (f08vuc)).

On exit: if

jobq = Nag_InitQ

, q contains the unitary matrix

Q

jobq = Nag_ComputeQ

, q contains the product

Q_{1} Q

jobq = Nag_NotQ

, q is not referenced.

23: $pdq$ – IntegerInput

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

Constraints:

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

24: $ncycle$ – Integer *Output

On exit: the number of cycles required for convergence.

25: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 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 procedure does not converge after $40$ cycles.
NE_ENUM_INT_2: On entry, $jobq = 〈value〉$ , $pdq = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $jobq = Nag_ComputeQ$ or $Nag_InitQ$ , $pdq \geq \max (1, n)$ ;
otherwise $pdq \geq 1$ .

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

On entry, $jobv = 〈value〉$ , $pdv = 〈value〉$ and $p = 〈value〉$ .
Constraint: if $jobv = Nag_ComputeV$ or $Nag_InitV$ , $pdv \geq \max (1, p)$ ;
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.
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_ztgsja (f08ysc) 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

The real analogue of this function is nag_dtgsja (f08yec).

10

Example

This example finds the generalized singular value decomposition

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

of the matrix pair

(A, B)

, where

A = (\begin{array}{r} 0.96 - 0.81 i & - 0.03 + 0.96 i & - 0.91 + 2.06 i & - 0.05 + 0.41 i \\ - 0.98 + 1.98 i & - 1.20 + 0.19 i & - 0.66 + 0.42 i & - 0.81 + 0.56 i \\ 0.62 - 0.46 i & 1.01 + 0.02 i & 0.63 - 0.17 i & - 1.11 + 0.60 i \\ 0.37 + 0.38 i & 0.19 - 0.54 i & - 0.98 - 0.36 i & 0.22 - 0.20 i \\ 0.83 + 0.51 i & 0.20 + 0.01 i & - 0.17 - 0.46 i & 1.47 + 1.59 i \\ 1.08 - 0.28 i & 0.20 - 0.12 i & - 0.07 + 1.23 i & 0.26 + 0.26 i \end{array})

and

B = (\begin{array}{r} 1 & 0 & - 1 & 0 \\ 0 & 1 & 0 & - 1 \end{array}) .

NAG Library Function Document

nag_ztgsja (f08ysc)

▸▿ 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