f08psc computes all the eigenvalues and, optionally, the Schur factorization of a complex Hessenberg matrix or a complex general matrix which has been reduced to Hessenberg form.

2 Specification

#include <nag.h>

void	f08psc (Nag_OrderType order, Nag_JobType job, Nag_ComputeZType compz, Integer n, Integer ilo, Integer ihi, Complex h[], Integer pdh, Complex w[], Complex z[], Integer pdz, NagError *fail)

The function may be called by the names: f08psc, nag_lapackeig_zhseqr or nag_zhseqr.

3 Description

f08psc computes all the eigenvalues and, optionally, the Schur factorization of a complex upper Hessenberg matrix

H

H = Z T Z^{H},

where

T

is an upper triangular matrix (the Schur form of

H

), and

Z

is the unitary matrix whose columns are the Schur vectors

z_{i}

. The diagonal elements of

T

are the eigenvalues of

H

The function may also be used to compute the Schur factorization of a complex general matrix

A

which has been reduced to upper Hessenberg form

H

\begin{array}{l} A & = & Q H Q^{H}, where ​ Q ​ is unitary, \\ = & (Q Z) T {(Q Z)}^{H} . \end{array}

In this case, after f08nsc has been called to reduce

A

to Hessenberg form, f08ntc must be called to form

Q

explicitly;

Q

is then passed to f08psc, which must be called with

compz = Nag_UpdateZ

The function can also take advantage of a previous call to f08nvc which may have balanced the original matrix before reducing it to Hessenberg form, so that the Hessenberg matrix

H

has the structure:

(\begin{matrix} H_{11} & H_{12} & H_{13} \\ H_{22} & H_{23} \\ H_{33} \end{matrix})

where

H_{11}

and

H_{33}

are upper triangular. If so, only the central diagonal block

H_{22}

(in rows and columns

i_{lo}

i_{hi}

) needs to be further reduced to Schur form (the blocks

H_{12}

and

H_{23}

are also affected). Therefore, the values of

i_{lo}

and

i_{hi}

can be supplied to f08psc directly. Also, f08nwc must be called after this function to permute the Schur vectors of the balanced matrix to those of the original matrix. If f08nvc has not been called however, then

i_{lo}

must be set to

1

and

i_{hi}

n

. Note that if the Schur factorization of

A

is required, f08nvc must not be called with

job = Nag_Schur

Nag_DoBoth

, because the balancing transformation is not unitary.

f08psc uses a multishift form of the upper Hessenberg

Q R

algorithm, due to Bai and Demmel (1989). The Schur vectors are normalized so that

{‖ z_{i} ‖}_{2} = 1

, but are determined only to within a complex factor of absolute value

1

4 References

Bai Z and Demmel J W (1989) On a block implementation of Hessenberg multishift

Q R

iteration Internat. J. High Speed Comput. 1 97–112

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: $job$ – Nag_JobType Input

On entry: indicates whether eigenvalues only or the Schur form

T

is required.

$job = Nag_EigVals$: Eigenvalues only are required.
$job = Nag_Schur$: The Schur form $T$ is required.

Constraint:

job = Nag_EigVals

Nag_Schur

3: $compz$ – Nag_ComputeZType Input

On entry: indicates whether the Schur vectors are to be computed.

$compz = Nag_NotZ$: No Schur vectors are computed (and the array z is not referenced).
$compz = Nag_UpdateZ$: The Schur vectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
$compz = Nag_InitZ$: The Schur vectors of $H$ are computed (and the array z is initialized by the function).

Constraint:

compz = Nag_NotZ

Nag_UpdateZ

Nag_InitZ

4: $n$ – Integer Input

On entry:

n

, the order of the matrix

H

Constraint:

n \geq 0

5: $ilo$ – Integer Input

6: $ihi$ – Integer Input

On entry: if the matrix

A

has been balanced by f08nvc, ilo and ihi must contain the values returned by that function. Otherwise, ilo must be set to

1

and ihi to n.

Constraint:

ilo \geq 1

and

\min (ilo, n) \leq ihi \leq n

7: $h [\dim]$ – Complex Input/Output

Note: the dimension, dim, of the array h must be at least

\max (1, pdh \times n)

The

(i, j)

th element of the matrix

H

is stored in

$h [(j - 1) \times pdh + i - 1]$ when $order = Nag_ColMajor$ ;
$h [(i - 1) \times pdh + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n \times n

upper Hessenberg matrix

H

, as returned by f08nsc.

On exit: if

job = Nag_EigVals

, the array contains no useful information.

job = Nag_Schur

, h is overwritten by the upper triangular matrix

T

from the Schur decomposition (the Schur form) unless

fail . code =

NE_CONVERGENCE.

8: $pdh$ – Integer Input

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

Constraint:

pdh \geq \max (1, n)

9: $w [\dim]$ – Complex Output

Note: the dimension, dim, of the array w must be at least

\max (1, n)

On exit: the computed eigenvalues, unless

fail . code =

NE_CONVERGENCE (in which case see Section 6). The eigenvalues are stored in the same order as on the diagonal of the Schur form

T

(if computed).

10: $z [\dim]$ – Complex Input/Output

Note: the dimension, dim, of the array z must be at least

$\max (1, pdz \times n)$ when $compz = Nag_UpdateZ$ or $Nag_InitZ$ ;
$1$ when $compz = Nag_NotZ$ .

The

(i, j)

th element of the matrix

Z

is stored in

$z [(j - 1) \times pdz + i - 1]$ when $order = Nag_ColMajor$ ;
$z [(i - 1) \times pdz + j - 1]$ when $order = Nag_RowMajor$ .

On entry: if

compz = Nag_UpdateZ

, z must contain the unitary matrix

Q

from the reduction to Hessenberg form.

compz = Nag_InitZ

, z need not be set.

On exit: if

compz = Nag_UpdateZ

Nag_InitZ

, z contains the unitary matrix of the required Schur vectors, unless

fail . code =

NE_CONVERGENCE.

compz = Nag_NotZ

, z is not referenced.

11: $pdz$ – Integer Input

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

Constraints:

if $compz = Nag_UpdateZ$ or $Nag_InitZ$ , $pdz \geq \max (1, n)$ ;
if $compz = Nag_NotZ$ , $pdz \geq 1$ .

12: $fail$ – 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 algorithm has failed to find all the eigenvalues after a total of $30 (ihi - ilo + 1)$ iterations.
NE_ENUM_INT_2: On entry, $compz = ⟨ value ⟩$ , $pdz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $compz = Nag_UpdateZ$ or $Nag_InitZ$ , $pdz \geq \max (1, n)$ ;
if $compz = Nag_NotZ$ , $pdz \geq 1$ .
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pdh = ⟨ value ⟩$ .
Constraint: $pdh > 0$ .

On entry, $pdz = ⟨ value ⟩$ .
Constraint: $pdz > 0$ .
NE_INT_2: On entry, $pdh = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdh \geq \max (1, n)$ .
NE_INT_3: On entry, $n = ⟨ value ⟩$ , $ilo = ⟨ value ⟩$ and $ihi = ⟨ value ⟩$ .
Constraint: $ilo \geq 1$ and $\min (ilo, n) \leq ihi \leq n$ .
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 Schur factorization is the exact factorization of a nearby matrix

(H + E)

, where

{‖ E ‖}_{2} = O (ε) {‖ H ‖}_{2},

and

ε

is the machine precision.

λ_{i}

is an exact eigenvalue, and

{\tilde{λ}}_{i}

is the corresponding computed value, then

| {\tilde{λ}}_{i} - λ_{i} | \leq \frac{c (n) ε {‖ H ‖}_{2}}{s_{i}},

where

c (n)

is a modestly increasing function of

n

, and

s_{i}

is the reciprocal condition number of

λ_{i}

. The condition numbers

s_{i}

may be computed by calling f08qyc.

8 Parallelism and Performance

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

f08psc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08psc 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 total number of real floating-point operations depends on how rapidly the algorithm converges, but is typically about:

$25 n^{3}$ if only eigenvalues are computed;
$35 n^{3}$ if the Schur form is computed;
$70 n^{3}$ if the full Schur factorization is computed.

The real analogue of this function is f08pec.

10 Example

This example computes all the eigenvalues and the Schur factorization of the upper Hessenberg matrix

H

, where

H = (\begin{array}{r} - 3.9700 - 5.0400 i & - 1.1318 - 2.5693 i & - 4.6027 - 0.1426 i & - 1.4249 + 1.7330 i \\ - 5.4797 + 0.0000 i & 1.8585 - 1.5502 i & 4.4145 - 0.7638 i & - 0.4805 - 1.1976 i \\ 0.0000 + 0.0000 i & 6.2673 + 0.0000 i & - 0.4504 - 0.0290 i & - 1.3467 + 1.6579 i \\ 0.0000 + 0.0000 i & 0.0000 + 0.0000 i & - 3.5000 + 0.0000 i & 2.5619 - 3.3708 i \end{array}) .

See also f08ntc, which illustrates the use of this function to compute the Schur factorization of a general matrix.

f08ps: FL CL CPP AD PY MB

NAG CL Interfacef08psc (zhseqr)

▸▿ 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
f08psc (zhseqr)