Integer, Intent (In)	::	n, ilo, ihi, lda, ldb, ldq, ldz, lwork
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Out)	::	rwork(n)
Complex (Kind=nag_wp), Intent (Inout)	::	a(lda,), b(ldb,), q(ldq,), z(ldz,)
Complex (Kind=nag_wp), Intent (Out)	::	alpha(n), beta(n), work(max(1,lwork))
Character (1), Intent (In)	::	job, compq, compz

C Header Interface

#include <nag.h>

void

f08xsf_ (const char *job, 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 alpha[], Complex beta[], Complex q[], const Integer *ldq, Complex z[], const Integer *ldz, Complex work[], const Integer *lwork, double rwork[], Integer *info, const Charlen length_job, const Charlen length_compq, const Charlen length_compz)

The routine may be called by the names f08xsf, nagf_lapackeig_zhgeqz or its LAPACK name zhgeqz.

3 Description

f08xsf implements a single-shift version of the

Q Z

method for finding the generalized eigenvalues of the complex matrix pair

(A, B)

which is in the generalized upper Hessenberg form. If the matrix pair

(A, B)

is not in the generalized upper Hessenberg form, then the routine f08wtf should be called before invoking f08xsf.

This problem is mathematically equivalent to solving the matrix equation

\det (A - λ B) = 0 .

Note that, to avoid underflow, overflow and other arithmetic problems, the generalized eigenvalues

λ_{j}

are never computed explicitly by this routine but defined as ratios between two computed values,

α_{j}

and

β_{j}

λ_{j} = α_{j} / β_{j} .

The arguments

α_{j}

, in general, are finite complex values and

β_{j}

are finite real non-negative values.

If desired, the matrix pair

(A, B)

may be reduced to generalized Schur form. That is, the transformed matrices

A

and

B

are upper triangular and the diagonal values of

A

and

B

provide

α

and

β

The argument job specifies two options. If

job ='S'

then the matrix pair

(A, B)

is simultaneously reduced to Schur form by applying one unitary transformation (usually called

Q

) on the left and another (usually called

Z

) on the right. That is,

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

job ='E'

, then at each iteration the same transformations are computed but they are only applied to those parts of

A

and

B

which are needed to compute

α

and

β

. This option could be used if generalized eigenvalues are required but not generalized eigenvectors.

job ='S'

and

compq ='V'

'I'

, and

compz ='V'

'I'

, then the unitary transformations used to reduce the pair

(A, B)

are accumulated into the input arrays q and z. If generalized eigenvectors are required then job must be set to

job ='S'

and if left (right) generalized eigenvectors are to be computed then compq (compz) must be set to

compq ='V'

'I'

rather than

compq ='N'

compq ='I'

, then eigenvectors are accumulated on the identity matrix and on exit the array q contains the left eigenvector matrix

Q

. However, if

compq ='V'

then the transformations are accumulated in the user-supplied matrix

Q_{0}

in array q on entry and thus on exit q contains the matrix product

Q Q_{0}

. A similar convention is used for compz.

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

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

Stewart G W and Sun J-G (1990) Matrix Perturbation Theory Academic Press, London

5 Arguments

1: $job$ – Character(1) Input

On entry: specifies the operations to be performed on

(A, B)

$job ='E'$: The matrix pair $(A, B)$ on exit might not be in the generalized Schur form.
$job ='S'$: The matrix pair $(A, B)$ on exit will be in the generalized Schur form.

Constraint:

job ='E'

'S'

2: $compq$ – Character(1) Input

On entry: specifies the operations to be performed on

Q

$compq ='N'$: The array q is unchanged.
$compq ='V'$: The left transformation $Q$ is accumulated on the array q.
$compq ='I'$: The array q is initialized to the identity matrix before the left transformation $Q$ is accumulated in q.

Constraint:

compq ='N'

'V'

'I'

3: $compz$ – Character(1) Input

On entry: specifies the operations to be performed on

Z

$compz ='N'$: The array z is unchanged.
$compz ='V'$: The right transformation $Z$ is accumulated on the array z.
$compz ='I'$: The array z is initialized to the identity matrix before the right transformation $Z$ is accumulated in z.

Constraint:

compz ='N'

'V'

'I'

4: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

B

Q

and

Z

Constraint:

n \geq 0

5: $ilo$ – Integer Input

6: $ihi$ – Integer Input

On entry: the indices

i_{lo}

and

i_{hi}

, respectively which define the upper triangular parts of

A

. The submatrices

A (1 : i_{lo} - 1, 1 : i_{lo} - 1)

and

A (i_{hi} + 1 : n, i_{hi} + 1 : n)

are then upper triangular. These arguments are provided by f08wvf if the matrix pair was previously balanced; otherwise,

ilo = 1

and

ihi = n

Constraints:

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

7: $a (lda, *)$ – 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

n \times n

upper Hessenberg matrix

A

. The elements below the first subdiagonal must be set to zero.

On exit: if

job ='S'

, the matrix pair

(A, B)

will be simultaneously reduced to generalized Schur form.

job ='E'

, the

1 \times 1

and

2 \times 2

diagonal blocks of the matrix pair

(A, B)

will give generalized eigenvalues but the remaining elements will be irrelevant.

8: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08xsf is called.

Constraint:

lda \geq \max (1, n)

9: $b (ldb, *)$ – 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

n \times n

upper triangular matrix

B

. The elements below the diagonal must be zero.

On exit: if

job ='S'

, the matrix pair

(A, B)

will be simultaneously reduced to generalized Schur form.

job ='E'

, the

1 \times 1

and

2 \times 2

diagonal blocks of the matrix pair

(A, B)

will give generalized eigenvalues but the remaining elements will be irrelevant.

10: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f08xsf is called.

Constraint:

ldb \geq \max (1, n)

11: $alpha (n)$ – Complex (Kind=nag_wp) array Output

On exit:

α_{j}

, for

j = 1, 2, \dots, n

12: $beta (n)$ – Complex (Kind=nag_wp) array Output

On exit:

β_{j}

, for

j = 1, 2, \dots, n

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

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

\max (1, n)

compq ='V'

'I'

and at least

1

compq ='N'

On entry: if

compq ='V'

, the matrix

Q_{0}

. The matrix

Q_{0}

is usually the matrix

Q

returned by f08wtf.

compq ='N'

, q is not referenced.

On exit: if

compq ='V'

, q contains the matrix product

Q Q_{0}

compq ='I'

, q contains the transformation matrix

Q

14: $ldq$ – Integer Input

On entry: the first dimension of the array q as declared in the (sub)program from which f08xsf is called.

Constraints:

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

15: $z (ldz, *)$ – 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'

, the matrix

Z_{0}

. The matrix

Z_{0}

is usually the matrix

Z

returned by f08wtf.

compz ='N'

, z is not referenced.

On exit: if

compz ='V'

, z contains the matrix product

Z Z_{0}

compz ='I'

, z contains the transformation matrix

Z

16: $ldz$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08xsf is called.

Constraints:

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

17: $work (\max (1, lwork))$ – Complex (Kind=nag_wp) array Workspace

On exit: if

info = 0

, the real part of

work (1)

contains the minimum value of lwork required for optimal performance.

18: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08xsf is called.

lwork = −1

, a workspace query is assumed; the routine only calculates the minimum size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

Constraint:

lwork \geq \max (1, n)

lwork = −1

19: $rwork (n)$ – Real (Kind=nag_wp) array Workspace

20: $info$ – 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.

$info = 1, \dots, n$: The $Q Z$ iteration did not converge and the matrix pair $(A, B)$ is not in the generalized Schur form. The computed $α_{i}$ and $β_{i}$ should be correct for $i = ⟨ value ⟩, \dots, ⟨ value ⟩$ .

$info > n$ and $info \leq 2 \times n$: The computation of shifts failed and the matrix pair $(A, B)$ is not in the generalized Schur form. The computed $α_{i}$ and $β_{i}$ should be correct for $i = ⟨ value ⟩, \dots, ⟨ value ⟩$ .

$info > 2 \times n$: An unexpected Library error has occurred.

7 Accuracy

Please consult Section 4.11 of the LAPACK Users' Guide (see Anderson et al. (1999)) and Chapter 6 of Stewart and Sun (1990), for more information.

8 Parallelism and Performance

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

f08xsf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

f08xsf is the fifth step in the solution of the complex generalized eigenvalue problem and is called after f08wtf.

The number of floating-point operations taken by this routine is proportional to

n^{3}

The real analogue of this routine is f08xef.

10 Example

This example computes the

α

and

β

arguments, which defines the generalized eigenvalues, of the matrix pair

(A, B)

given by

A = (\begin{array}{r} 1.0 + 3.0 i & 1.0 + 4.0 i & 1.0 + 5.0 i & 1.0 + 6.0 i \\ 2.0 + 2.0 i & 4.0 + 3.0 i & 8.0 + 4.0 i & 16.0 + 5.0 i \\ 3.0 + 1.0 i & 9.0 + 2.0 i & 27.0 + 3.0 i & 81.0 + 4.0 i \\ 4.0 + 0.0 i & 16.0 + 1.0 i & 64.0 + 2.0 i & 256.0 + 3.0 i \end{array})

and

B = (\begin{array}{r} 1.0 + 0.0 i & 2.0 + 1.0 i & 3.0 + 2.0 i & 4.0 + 3.0 i \\ 1.0 + 1.0 i & 4.0 + 2.0 i & 9.0 + 3.0 i & 16.0 + 4.0 i \\ 1.0 + 2.0 i & 8.0 + 3.0 i & 27.0 + 4.0 i & 64.0 + 5.0 i \\ 1.0 + 3.0 i & 16.0 + 4.0 i & 81.0 + 5.0 i & 256.0 + 6.0 i \end{array}) .

This requires calls to five routines: f08wvf to balance the matrix, f08asf to perform the

Q R

factorization of

B

, f08auf to apply

Q

A

, f08wtf to reduce the matrix pair to the generalized Hessenberg form and f08xsf to compute the eigenvalues using the

Q Z

algorithm.

f08xs: FL CL CPP AD PY MB

NAG FL Interfacef08xsf (zhgeqz)

▸▿ 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 FL Interface
f08xsf (zhgeqz)