NAG Library Routine Document

f08pef (dhseqr) computes all the eigenvalues and, optionally, the Schur factorization of a real Hessenberg matrix or a real general matrix which has been reduced to Hessenberg form.

2

Specification

Fortran Interface

Subroutine f08pef (

job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz, work, lwork, info)

Integer, Intent (In)	::	n, ilo, ihi, ldh, ldz, lwork
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	h(ldh,), wr(), wi(), z(ldz,)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))
Character (1), Intent (In)	::	job, compz

C Header Interface

#include <nagmk26.h>

void	f08pef_ (const char job, const char compz, const Integer n, const Integer ilo, const Integer ihi, double h[], const Integer ldh, double wr[], double wi[], double z[], const Integer ldz, double work[], const Integer lwork, Integer *info, const Charlen length_job, const Charlen length_compz)

The routine may be called by its LAPACK name dhseqr.

3

Description

f08pef (dhseqr) computes all the eigenvalues and, optionally, the Schur factorization of a real upper Hessenberg matrix

H

H = Z T Z^{T},

where

T

is an upper quasi-triangular matrix (the Schur form of

H

), and

Z

is the orthogonal matrix whose columns are the Schur vectors

z_{i}

. See Section 9 for details of the structure of

T

The routine may also be used to compute the Schur factorization of a real general matrix

A

which has been reduced to upper Hessenberg form

H

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

In this case, after f08nef (dgehrd) has been called to reduce

A

to Hessenberg form, f08nff (dorghr) must be called to form

Q

explicitly;

Q

is then passed to f08pef (dhseqr), which must be called with

compz ='V'

The routine can also take advantage of a previous call to f08nhf (dgebal) 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 f08pef (dhseqr) directly. Also, f08njf (dgebak) must be called after this routine to permute the Schur vectors of the balanced matrix to those of the original matrix. If f08nhf (dgebal) 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, f08nhf (dgebal) must not be called with

job ='S'

'B'

, because the balancing transformation is not orthogonal.

f08pef (dhseqr) 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 factor

\pm 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: $job$ – Character(1)Input

On entry: indicates whether eigenvalues only or the Schur form

T

is required.

$job ='E'$: Eigenvalues only are required.
$job ='S'$: The Schur form $T$ is required.

Constraint:

job ='E'

'S'

2: $compz$ – Character(1)Input

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

$compz ='N'$: No Schur vectors are computed (and the array z is not referenced).
$compz ='V'$: The Schur vectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
$compz ='I'$: The Schur vectors of $H$ are computed (and the array z is initialized by the routine).

Constraint:

compz ='N'

'V'

'I'

3: $n$ – IntegerInput

On entry:

n

, the order of the matrix

H

Constraint:

n \geq 0

4: $ilo$ – IntegerInput

5: $ihi$ – IntegerInput

On entry: if the matrix

A

has been balanced by f08nhf (dgebal), ilo and ihi must contain the values returned by that routine. Otherwise, ilo must be set to

1

and ihi to n.

Constraint:

ilo \geq 1

and

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

6: $h (ldh *)$ – Real (Kind=nag_wp) arrayInput/Output

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

\max (1, n)

On entry: the

n

n

upper Hessenberg matrix

H

, as returned by f08nef (dgehrd).

On exit: if

job ='E'

, the array contains no useful information.

job ='S'

, h is overwritten by the upper quasi-triangular matrix

T

from the Schur decomposition (the Schur form) unless

info > 0

7: $ldh$ – IntegerInput

On entry: the first dimension of the array h as declared in the (sub)program from which f08pef (dhseqr) is called.

Constraint:

ldh \geq \max (1, n)

8: $wr (*)$ – Real (Kind=nag_wp) arrayOutput

9: $wi (*)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the arrays wr and wi must be at least

\max (1, n)

On exit: the real and imaginary parts, respectively, of the computed eigenvalues, unless

info > 0

(in which case see Section 6). Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having positive imaginary part first. The eigenvalues are stored in the same order as on the diagonal of the Schur form

T

(if computed); see Section 9 for details.

10: $z (ldz *)$ – Real (Kind=nag_wp) arrayInput/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 the orthogonal matrix

Q

from the reduction to Hessenberg form.

compz ='I'

, z need not be set.

On exit: if

compz ='V'

'I'

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

info > 0

compz ='N'

, z is not referenced.

11: $ldz$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f08pef (dhseqr) is called.

Constraints:

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

12: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

13: $lwork$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08pef (dhseqr) is called, unless

lwork = - 1

, in which case a workspace query is assumed and the routine only calculates the minimum dimension of work.

Constraint:

lwork \geq \max (1, n)

lwork = - 1

14: $info$ – IntegerOutput

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.

$info > 0$

The algorithm has failed to find all the eigenvalues after a total of

30 (ihi - ilo + 1)

iterations.

info = i

, elements

1, 2, \dots, ilo - 1

and

i + 1, i + 2, \dots, n

of wr and wi contain the real and imaginary parts of contain the eigenvalues which have been found.

job ='E'

, then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix

\hat{H}

, formed from

h (ilo : info, ilo : info)

, i.e., the ilo through info rows and columns of the final output matrix

H

job ='S'

, then on exit

(*) H_{i} U = U \tilde{H}

for some matrix

U

, where

H_{i}

is the input upper Hessenberg matrix and

\tilde{H}

is an upper Hessenberg matrix formed from

h (info + 1 : ihi, info + 1 : ihi)

compz ='V'

, then on exit

Z_{out} = Z_{in} U

where

U

is defined in

(*)

(regardless of the value of job).

compz ='I'

, then on exit

Z_{out} = U

where

U

is defined in

(*)

(regardless of the value of job).

info > 0

and

compz ='N'

, z is not accessed.

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 f08qlf (dtrsna).

8

Parallelism and Performance

f08pef (dhseqr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08pef (dhseqr) 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

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

$7 n^{3}$ if only eigenvalues are computed;
$10 n^{3}$ if the Schur form is computed;
$20 n^{3}$ if the full Schur factorization is computed.

The Schur form

T

has the following structure (referred to as canonical Schur form).

If all the computed eigenvalues are real,

T

is upper triangular, and the diagonal elements of

T

are the eigenvalues;

wr (i) = t_{i i}

, for

i = 1, 2, \dots, n

, and

wi (i) = 0.0

If some of the computed eigenvalues form complex conjugate pairs, then

T

has

2

2

diagonal blocks. Each diagonal block has the form

(\begin{matrix} t_{i i} & t_{i, i + 1} \\ t_{i + 1, i} & t_{i + 1, i + 1} \end{matrix}) = (\begin{matrix} α & β \\ γ & α \end{matrix})

where

β γ < 0

. The corresponding eigenvalues are

α \pm \sqrt{β γ}

;

wr (i) = wr (i + 1) = α

;

wi (i) = + \sqrt{|β γ|}

;

wi (i + 1) = - wi (i)

The complex analogue of this routine is f08psf (zhseqr).

10

Example

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

H

, where

H = (\begin{array}{r} 0.3500 & - 0.1160 & - 0.3886 & - 0.2942 \\ - 0.5140 & 0.1225 & 0.1004 & 0.1126 \\ 0.0000 & 0.6443 & - 0.1357 & - 0.0977 \\ 0.0000 & 0.0000 & 0.4262 & 0.1632 \end{array}) .

See also Section 10 in f08nff (dorghr), which illustrates the use of this routine to compute the Schur factorization of a general matrix.

NAG Library Routine Document

f08pef (dhseqr)

▸▿ 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