NAG Library Function Document

nag_zggsvp (f08vsc)

to upper triangular form. This factorization is usually used as a preprocessing step for computing the generalized singular value decomposition (GSVD).

2 Specification

#include <nag.h>

#include <nagf08.h>

void

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

3 Description

nag_zggsvp (f08vsc) computes unitary matrices

U

V

and

Q

such that

\begin{array}{l} U^{H} 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^{H} 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

k

matrix

A_{12}

and

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.

(k + l)

is the effective numerical rank of the

(m + p)

n

matrix

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

This decomposition is usually used as the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see function nag_zggsvd (f08vnc).

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 3.2.1.3 in the Essential Introduction 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 unitary 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 unitary 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 unitary 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: 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: 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 $order = Nag_RowMajor$ .

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 (or trapezoidal) matrix described in Section 3.

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

10: 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 $order = Nag_RowMajor$ .

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 described in Section 3.

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

12: tola – doubleInput

13: tolb – doubleInput

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[ $\dim$ ] – ComplexOutput

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 unitary 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[ $\dim$ ] – ComplexOutput

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 unitary 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[ $\dim$ ] – ComplexOutput

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 unitary 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: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_ENUM_INT_2: On entry, $jobq = ⟨value⟩$ , $pdq = ⟨value⟩$ , $n = ⟨value⟩$ .
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 $pdv \geq 1$ .
On entry, $pdq = ⟨value⟩$ , $jobq = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: if $jobq = Nag_ComputeQ$ , $pdq \geq \max (1, n)$ ;
otherwise $pdq \geq 1$ .
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.

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

nag_zggsvp (f08vsc) is not threaded by NAG in any implementation.

nag_zggsvp (f08vsc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The real analogue of this function is nag_dggsvp (f08vec).

10 Example

This example finds the generalized factorization

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

of the matrix pair

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

, 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{matrix} 1 & 0 & - 1 & 0 \\ 0 & 1 & 0 & - 1 \end{matrix}) .

NAG Library Function Documentnag_zggsvp (f08vsc)

+− 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_zggsvp (f08vsc)