NAG Library Routine Document

F02GCF

INTEGER	N, LDA, MEST, M, LDV, LWORK, IWORK(N), IFAIL
REAL (KIND=nag_wp)	WL, WU, RWORK(2*N)
COMPLEX (KIND=nag_wp)	A(LDA,*), W(N), V(LDV,MEST), WORK(LWORK)
LOGICAL	BWORK(N)
CHARACTER(1)	CRIT

3 Description

F02GCF computes selected eigenvalues and the corresponding right eigenvectors of a complex general matrix

A

A x_{i} = λ_{i} x_{i} .

Eigenvalues

λ_{i}

may be selected either by modulus, satisfying

w_{l} \leq |λ_{i}| \leq w_{u},

or by real part, satisfying

w_{l} \leq Re (λ_{i}) \leq w_{u} .

4 References

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

5 Parameters

1: $CRIT$ – CHARACTER(1)Input

On entry: indicates the criterion for selecting eigenvalues.

$CRIT ='M'$: Eigenvalues are selected according to their moduli: $w_{l} \leq |λ_{i}| \leq w_{u}$ .
$CRIT ='R'$: Eigenvalues are selected according to their real parts: $w_{l} \leq Re (λ_{i}) \leq w_{u}$ .

Constraint:

CRIT ='M'

'R'

2: $N$ – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

3: $A (LDA *)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

On entry: the

n

n

general matrix

A

On exit: contains the Hessenberg form of the balanced input matrix

A^{'}

(see Section 9).

4: $LDA$ – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F02GCF is called.

Constraint:

LDA \geq \max (1, N)

5: $WL$ – REAL (KIND=nag_wp)Input

6: $WU$ – REAL (KIND=nag_wp)Input

On entry:

w_{l}

and

w_{u}

, the lower and upper bounds on the criterion for the selected eigenvalues (see CRIT).

Constraint:

WU > WL

7: $MEST$ – INTEGERInput

On entry: the second dimension of the array V as declared in the (sub)program from which F02GCF is called. MEST must be an upper bound on

m

, the number of eigenvalues and eigenvectors selected. No eigenvectors are computed if

MEST < m

Constraint:

MEST \geq \max (1, m)

8: $M$ – INTEGEROutput

On exit:

m

, the number of eigenvalues actually selected.

9: $W (N)$ – COMPLEX (KIND=nag_wp) arrayOutput

On exit: the first M elements of W hold the selected eigenvalues; elements

M + 1

to N contain the other eigenvalues.

10: $V (LDV, MEST)$ – COMPLEX (KIND=nag_wp) arrayOutput

On exit: contains the selected eigenvectors, with the

i

th column holding the eigenvector associated with the eigenvalue

λ_{i}

(stored in

W (i)

11: $LDV$ – INTEGERInput

On entry: the first dimension of the array V as declared in the (sub)program from which F02GCF is called.

Constraint:

LDV \geq \max (1, N)

12: $WORK (LWORK)$ – COMPLEX (KIND=nag_wp) arrayWorkspace

13: $LWORK$ – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F02GCF is called.

Constraint:

LWORK \geq \max (1, N \times (N + 2))

14: $RWORK (2 \times N)$ – REAL (KIND=nag_wp) arrayWorkspace

15: $IWORK (N)$ – INTEGER arrayWorkspace

16: $BWORK (N)$ – LOGICAL arrayWorkspace

17: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$CRIT \neq'M'$ or $'R'$ ,
or	$N < 0$ ,
or	$LDA < \max (1, N)$ ,
or	$WU \leq WL$ ,
or	$MEST < 1$ ,
or	$LDV < \max (1, N)$ ,
or	$LWORK < \max (1, N \times (N + 2))$ .

$IFAIL = 2$: The $Q R$ algorithm failed to compute all the eigenvalues. No eigenvectors have been computed.

$IFAIL = 3$: There are more than MEST eigenvalues in the specified range. The actual number of eigenvalues in the range is returned in M. No eigenvectors have been computed. Rerun with the second dimension of $V = MEST \geq M$ .

$IFAIL = 4$: Inverse iteration failed to compute all the specified eigenvectors. If an eigenvector failed to converge, the corresponding column of V is set to zero.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

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 {‖A^{'}‖}_{2}}{s_{i}},

where

c (n)

is a modestly increasing function of

n

ε

is the machine precision, and

s_{i}

is the reciprocal condition number of

λ_{i}

;

A^{'}

is the balanced form of the original matrix

A

(see Section 9), and

‖A^{'}‖ \leq ‖A‖

x_{i}

is the corresponding exact eigenvector, and

{\tilde{x}}_{i}

is the corresponding computed eigenvector, then the angle

θ ({\tilde{x}}_{i}, x_{i})

between them is bounded as follows:

θ ({\tilde{x}}_{i}, x_{i}) \leq \frac{c (n) ε {‖A^{'}‖}_{2}}{{sep}_{i}}

where

{sep}_{i}

is the reciprocal condition number of

x_{i}

The condition numbers

s_{i}

and

{sep}_{i}

may be computed from the Hessenberg form of the balanced matrix

A^{'}

which is returned in the array A. This requires calling F08PSF (ZHSEQR) with

JOB ='S'

to compute the Schur form of

A^{'}

, followed by F08QYF (ZTRSNA).

8 Parallelism and Performance

F02GCF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F02GCF 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

F02GCF calls routines from LAPACK in Chapter F08. It first balances the matrix, using a diagonal similarity transformation to reduce its norm; and then reduces the balanced matrix

A^{'}

to upper Hessenberg form

H

, using a unitary similarity transformation:

A^{'} = Q H Q^{H}

. The routine uses the Hessenberg

Q R

algorithm to compute all the eigenvalues of

H

, which are the same as the eigenvalues of

A

. It computes the eigenvectors of

H

which correspond to the selected eigenvalues, using inverse iteration. It premultiplies the eigenvectors by

Q

to form the eigenvectors of

A^{'}

; and finally transforms the eigenvectors to those of the original matrix

A

Each eigenvector

x

is normalized so that

{‖x‖}_{2} = 1

, and the element of largest absolute value is real and positive.

The inverse iteration routine may make a small perturbation to the real parts of close eigenvalues, and this may shift their moduli just outside the specified bounds. If you are relying on eigenvalues being within the bounds, you should test them on return from F02GCF.

The time taken by the routine is approximately proportional to

n^{3}

The routine can be used to compute all eigenvalues and eigenvectors, by setting WL large and negative, and WU large and positive.

10 Example

This example computes those eigenvalues of the matrix

A

which lie in the range

[- 5.5, + 5.5]

, and their corresponding eigenvectors, where

A = (\begin{array}{r} - 3.97 - 5.04 i & - 4.11 + 3.70 i & - 0.34 + 1.01 i & 1.29 - 0.86 i \\ 0.34 - 1.50 i & 1.52 - 0.43 i & 1.88 - 5.38 i & 3.36 + 0.65 i \\ 3.31 - 3.85 i & 2.50 + 3.45 i & 0.88 - 1.08 i & 0.64 - 1.48 i \\ - 1.10 + 0.82 i & 1.81 - 1.59 i & 3.25 + 1.33 i & 1.57 - 3.44 i \end{array}) .

NAG Library Routine DocumentF02GCF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Library Routine Document

F02GCF