to upper triangular form. This factorization is usually used as a preprocessing step for computing the generalized singular value decomposition (GSVD). For sufficiently large problems, a blocked algorithm is used to make best use of Level 3 BLAS.

2 Specification

#include <nag.h>

void

f08vgc (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVType jobv, Nag_ComputeQType jobq, Integer m, Integer p, Integer n, double a[], Integer pda, double b[], Integer pdb, double tola, double tolb, Integer *k, Integer *l, double u[], Integer pdu, double v[], Integer pdv, double q[], Integer pdq, NagError *fail)

The function may be called by the names: f08vgc, nag_lapackeig_dggsvp3 or nag_dggsvp3.

3 Description

f08vgc computes orthogonal matrices

U

V

and

Q

such that

\begin{array}{l} U^{T} A Q = {\begin{array}{r} \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{array} \\ V^{T} B Q = \begin{array}{rc} n - k - l & k & l \\ l & 0 & 0 & B_{13} \\ p - l & 0 & 0 & 0 \\ ( & ) \end{array} \end{array}

where the

k \times k

matrix

A_{12}

and

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.

(k + l)

is the effective numerical rank of the

(m + p) \times n

matrix

{(\begin{matrix} A^{T} & B^{T} \end{matrix})}^{T}

This decomposition is usually used as the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see function f08yec; the two steps are combined in f08vcc.

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 (2012) Matrix Computations (4th 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

, the orthogonal matrix

U

is computed.

jobu = Nag_NotU

U

is not computed.

Constraint:

jobu = Nag_AllU

Nag_NotU

3: $jobv$ – Nag_ComputeVType Input

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_ComputeQType Input

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: $m$ – 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: $a [\dim]$ – 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 $order = Nag_RowMajor$ .

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 \times n

matrix

A

On exit: contains the triangular (or trapezoidal) matrix described in Section 3.

9: $pda$ – 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 $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

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

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 \times n

matrix

B

On exit: contains the triangular matrix described in Section 3.

11: $pdb$ – 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 $order = Nag_RowMajor$ , $pdb \geq \max (1, n)$ .

12: $tola$ – double Input

13: $tolb$ – double Input

On entry: tola and tolb are the thresholds to determine the effective numerical rank of matrix

B

and a subblock of

A

. Generally, they are set to

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

where

ε

is the machine precision.

The size of tola and tolb may affect the size of backward errors of the decomposition.

14: $k$ – Integer * Output

15: $l$ – 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^{T} & b^{T} \end{matrix})}^{T}

16: $u [\dim]$ – double Output

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 orthogonal matrix

U

jobu = Nag_NotU

, u is not referenced.

17: $pdu$ – Integer Input

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

Constraints:

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

18: $v [\dim]$ – double Output

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 orthogonal matrix

V

jobv = Nag_NotV

, v is not referenced.

19: $pdv$ – Integer Input

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

Constraints:

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

20: $q [\dim]$ – double Output

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 orthogonal matrix

Q

jobq = Nag_NotQ

, q is not referenced.

21: $pdq$ – Integer Input

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

Constraints:

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

22: $fail$ – 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_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.
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 factorization is nearly the exact factorization 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.

8 Parallelism and Performance

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

f08vgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08vgc 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 f08vec which used an unblocked algorithm and, therefore, did not make best use of Level 3 BLAS functions.

The complex analogue of this function is f08vuc.

10 Example

This example finds the generalized factorization

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

of the matrix pair

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

, 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}) .

f08vg: FL CL CPP AD PY MB

NAG CL Interfacef08vgc (dggsvp3)

▸▿ 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
f08vgc (dggsvp3)