Integer, Intent (In)	::	n, ilo, ihi, lda, ldb, ldq, ldz
Integer, Intent (Out)	::	info
Complex (Kind=nag_wp), Intent (Inout)	::	a(lda,), b(ldb,), q(ldq,), z(ldz,)
Character (1), Intent (In)	::	compq, compz

C Header Interface

#include <nag.h>

void

f08wsf_ (const char *compq, const char *compz, const Integer *n, const Integer *ilo, const Integer *ihi, Complex a[], const Integer *lda, Complex b[], const Integer *ldb, Complex q[], const Integer *ldq, Complex z[], const Integer *ldz, Integer *info, const Charlen length_compq, const Charlen length_compz)

The routine may be called by the names f08wsf, nagf_lapackeig_zgghrd or its LAPACK name zgghrd.

3 Description

f08wsf is usually the third step in the solution of the complex generalized eigenvalue problem

A x = λ B x .

The (optional) first step balances the two matrices using f08wvf. In the second step, matrix

B

is reduced to upper triangular form using the

Q R

factorization routine f08asf and this unitary transformation

Q

is applied to matrix

A

by calling f08auf.

f08wsf reduces a pair of complex matrices

(A, B)

, where

B

is triangular, to the generalized upper Hessenberg form using unitary transformations. This two-sided transformation is of the form

\begin{matrix} Q^{H} A Z = H \\ Q^{H} B Z = T \end{matrix}

where

H

is an upper Hessenberg matrix,

T

is an upper triangular matrix and

Q

and

Z

are unitary matrices determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices

Q_{1}

and

Z_{1}

, so that

\begin{matrix} Q_{1} A Z_{1}^{H} = (Q_{1} Q) H {(Z_{1} Z)}^{H}, \\ Q_{1} B Z_{1}^{H} = (Q_{1} Q) T {(Z_{1} Z)}^{H} . \end{matrix}

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256

5 Arguments

1: $compq$ – Character(1) Input

On entry: specifies the form of the computed unitary matrix

Q

$compq ='N'$: Do not compute $Q$ .
$compq ='I'$: The unitary matrix $Q$ is returned.
$compq ='V'$: q must contain a unitary matrix $Q_{1}$ , and the product $Q_{1} Q$ is returned.

Constraint:

compq ='N'

'I'

'V'

2: $compz$ – Character(1) Input

On entry: specifies the form of the computed unitary matrix

Z

$compz ='N'$: Do not compute $Z$ .
$compz ='V'$: z must contain a unitary matrix $Z_{1}$ , and the product $Z_{1} Z$ is returned.
$compz ='I'$: The unitary matrix $Z$ is returned.

Constraint:

compz ='N'

'V'

'I'

3: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

4: $ilo$ – Integer Input

5: $ihi$ – Integer Input

On entry:

i_{lo}

and

i_{hi}

as determined by a previous call to f08wvf. Otherwise, they should be set to

1

and

n

, respectively.

Constraints:

if $n > 0$ , $1 \leq ilo \leq ihi \leq n$ ;
if $n = 0$ , $ilo = 1$ and $ihi = 0$ .

6: $a (lda, *)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array a must be at least

\max (1, n)

On entry: the matrix

A

of the matrix pair

(A, B)

. Usually, this is the matrix

A

returned by f08auf.

On exit: a is overwritten by the upper Hessenberg matrix

H

7: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08wsf is called.

Constraint:

lda \geq \max (1, n)

8: $b (ldb, *)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array b must be at least

\max (1, n)

On entry: the upper triangular matrix

B

of the matrix pair

(A, B)

. Usually, this is the matrix

B

returned by the

Q R

factorization routine f08asf.

On exit: b is overwritten by the upper triangular matrix

T

9: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f08wsf is called.

Constraint:

ldb \geq \max (1, n)

10: $q (ldq, *)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array q must be at least

\max (1, n)

compq ='I'

'V'

and at least

1

compq ='N'

On entry: if

compq ='V'

, q must contain a unitary matrix

Q_{1}

compq ='N'

, q is not referenced.

On exit: if

compq ='I'

, q contains the unitary matrix

Q

Iif

compq ='V'

, q is overwritten by

Q_{1} Q

11: $ldq$ – Integer Input

On entry: the first dimension of the array q as declared in the (sub)program from which f08wsf is called.

Constraints:

if $compq ='I'$ or $'V'$ , $ldq \geq \max (1, n)$ ;
if $compq ='N'$ , $ldq \geq 1$ .

12: $z (ldz, *)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array z must be at least

\max (1, n)

compz ='V'

'I'

and at least

1

compz ='N'

On entry: if

compz ='V'

, z must contain a unitary matrix

Z_{1}

compz ='N'

, z is not referenced.

On exit: if

compz ='I'

, z contains the unitary matrix

Z

compz ='V'

, z is overwritten by

Z_{1} Z

13: $ldz$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08wsf is called.

Constraints:

if $compz ='V'$ or $'I'$ , $ldz \geq \max (1, n)$ ;
if $compz ='N'$ , $ldz \geq 1$ .

14: $info$ – Integer Output

On exit:

info = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

The reduction to the generalized Hessenberg form is implemented using unitary transformations which are backward stable.

8 Parallelism and Performance

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

f08wsf is not threaded in any implementation.

9 Further Comments

This routine is usually followed by f08xsf which implements the

Q Z

algorithm for computing generalized eigenvalues of a reduced pair of matrices.

The real analogue of this routine is f08wef.

10 Example

See Section 10 in f08xsf and f08yxf.

NAG Library Manual, Mark 28.7

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

F08 (Lapackeig) Chapter Contents

F08 (Lapackeig) Chapter Introduction

f08ws: FL CL CPP AD PY MB

NAG FL Interfacef08wsf (zgghrd)

▸▿ 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

NAG FL Interface
f08wsf (zgghrd)