f08yec (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVType jobv, Nag_ComputeQType jobq, Integer m, Integer p, Integer n, Integer k, Integer l, double a[], Integer pda, double b[], Integer pdb, double tola, double tolb, double alpha[], double beta[], double u[], Integer pdu, double v[], Integer pdv, double q[], Integer pdq, Integer *ncycle, NagError *fail)

The function may be called by the names: f08yec, nag_lapackeig_dtgsja or nag_dtgsja.

3 Description

f08yec computes the GSVD of the matrices

A

and

B

which are assumed to have the form as returned by f08vgc

\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 \times k

matrix

A_{12}

and the

l \times l

matrix

B_{13}

are nonsingular upper triangular,

A_{23}

l \times l

upper triangular if

m - k - l \geq 0

and is

(m - k) \times l

upper trapezoidal otherwise.

f08yec 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 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: $order$ – Nag_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $jobu$ – Nag_ComputeUType Input

On entry: if

jobu = Nag_AllU

, u must contain an orthogonal 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 orthogonal matrix

U

is returned.

jobu = Nag_NotU

U

is not computed.

Constraint:

jobu = Nag_AllU

Nag_InitU

Nag_NotU

3: $jobv$ – Nag_ComputeVType Input

On entry: if

jobv = Nag_ComputeV

, v must contain an orthogonal 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 orthogonal matrix

V

is returned.

jobv = Nag_NotV

V

is not computed.

Constraint:

jobv = Nag_ComputeV

Nag_InitV

Nag_NotV

4: $jobq$ – Nag_ComputeQType Input

On entry: if

jobq = Nag_ComputeQ

, q must contain an orthogonal 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 orthogonal matrix

Q

is returned.

jobq = Nag_NotQ

Q

is not computed.

Constraint:

jobq = Nag_ComputeQ

Nag_InitQ

Nag_NotQ

5: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

6: $p$ – Integer Input

On entry:

p

, the number of rows of the matrix

B

Constraint:

p \geq 0

7: $n$ – Integer Input

On entry:

n

, the number of columns of the matrices

A

and

B

Constraint:

n \geq 0

8: $k$ – Integer Input

9: $l$ – Integer Input

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 f08yec.

10: $a [\dim]$ – double Input/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 \times n

matrix

A

On exit: if

m - k - l \geq 0

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

contains the

(k + l) \times (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) \times (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: $pda$ – Integer Input

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: $b [\dim]$ – double Input/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 \times 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: $pdb$ – Integer Input

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: $tola$ – double Input

15: $tolb$ – double 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 f08vgc, say

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

where

ε

is the machine precision.

16: $alpha [n]$ – double Output

On exit: see the description of beta.

17: $beta [n]$ – double Output

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: $u [\dim]$ – double Input/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 \times m

matrix

U_{1}

(usually the orthogonal matrix returned by f08vgc).

On exit: if

jobu = Nag_AllU

, u contains the product

U_{1} U

jobu = Nag_InitU

, u contains the orthogonal matrix

U

jobu = Nag_NotU

, u is not referenced.

19: $pdu$ – Integer Input

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: $v [\dim]$ – double Input/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 \times p

matrix

V_{1}

(usually the orthogonal matrix returned by f08vgc).

On exit: if

jobv = Nag_InitV

, v contains the orthogonal matrix

V

jobv = Nag_ComputeV

, v contains the product

V_{1} V

jobv = Nag_NotV

, v is not referenced.

21: $pdv$ – Integer Input

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: $q [\dim]$ – double Input/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 \times n

matrix

Q_{1}

(usually the orthogonal matrix returned by f08vgc).

On exit: if

jobq = Nag_InitQ

, q contains the orthogonal matrix

Q

jobq = Nag_ComputeQ

, q contains the product

Q_{1} Q

jobq = Nag_NotQ

, q is not referenced.

23: $pdq$ – Integer Input

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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface 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

Background information to multithreading can be found in the Multithreading documentation.

f08yec 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 complex analogue of this function is f08ysc.

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}) .

f08ye: FL CL CPP AD PY MB

NAG CL Interfacef08yec (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

NAG CL Interface
f08yec (dtgsja)