NAG C Library Function Document

nag_zhseqr (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>

#include <nagf08.h>

void	nag_zhseqr (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)

3

Description

nag_zhseqr (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 nag_zgehrd (f08nsc) has been called to reduce

A

to Hessenberg form, nag_zunghr (f08ntc) must be called to form

Q

explicitly;

Q

is then passed to nag_zhseqr (f08psc), which must be called with

compz = Nag_UpdateZ

The function can also take advantage of a previous call to nag_zgebal (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 nag_zhseqr (f08psc) directly. Also, nag_zgebak (f08nwc) must be called after this function to permute the Schur vectors of the balanced matrix to those of the original matrix. If nag_zgebal (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, nag_zgebal (f08nvc) must not be called with

job = Nag_Scale

Nag_DoBoth

, because the balancing transformation is not unitary.

nag_zhseqr (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_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $job$ – Nag_JobTypeInput

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_ComputeZTypeInput

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: $n$ – IntegerInput

On entry:

n

, the order of the matrix

H

Constraint:

n \geq 0

5: $ilo$ – IntegerInput

6: $ihi$ – IntegerInput

On entry: if the matrix

A

has been balanced by nag_zgebal (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: $h [\dim]$ – ComplexInput/Output

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

\max (1, pdh \times n)

Where

H (i, j)

appears in this document, it refers to the array element

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

On entry: the

n

n

upper Hessenberg matrix

H

, as returned by nag_zgehrd (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: $pdh$ – IntegerInput

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: $w [\dim]$ – ComplexOutput

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: $z [\dim]$ – ComplexInput/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: $pdz$ – IntegerInput

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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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 nag_ztrsna (f08qyc).

8

Parallelism and Performance

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

nag_zhseqr (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 nag_dhseqr (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 Section 10 in nag_zunghr (f08ntc), which illustrates the use of this function to compute the Schur factorization of a general matrix.

NAG C Library Function Document

nag_zhseqr (f08psc)

▸▿ 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

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