NAG Library Routine Document

F02ECF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F02ECF computes selected eigenvalues and eigenvectors of a real general matrix.

2 Specification

SUBROUTINE F02ECF (

CRIT, N, A, LDA, WL, WU, MEST, M, WR, WI, VR, LDVR, VI, LDVI, WORK, LWORK, IWORK, BWORK, IFAIL)

INTEGER	N, LDA, MEST, M, LDVR, LDVI, LWORK, IWORK(N), IFAIL
REAL (KIND=nag_wp)	A(LDA,N), WL, WU, WR(N), WI(N), VR(LDVR,MEST), VI(LDVI,MEST), WORK(LWORK)
LOGICAL	BWORK(N)
CHARACTER(1)	CRIT

3 Description

F02ECF computes selected eigenvalues and the corresponding right eigenvectors of a real 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} .

Note that even though

A

is real,

λ_{i}

and

x_{i}

may be complex. If

x_{i}

is an eigenvector corresponding to a complex eigenvalue

λ_{i}

, then the complex conjugate vector

{\bar{x}}_{i}

is the eigenvector corresponding to the complex conjugate eigenvalue

{\bar{λ}}_{i}

. The eigenvalues in a complex conjugate pair

λ_{i}

and

{\bar{λ}}_{i}

are either both selected or both not selected.

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,N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the

n

n

general matrix

A

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

A^{'}

(see Section 8).

4: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F02ECF 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 arrays VR and VI as declared in the (sub)program from which F02ECF 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: WR(N) – REAL (KIND=nag_wp) arrayOutput

10: WI(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the first M elements of WR and WI hold the real and imaginary parts, respectively, of the selected eigenvalues; elements

M + 1

to N contain the other eigenvalues. Complex conjugate pairs of eigenvalues are stored in consecutive elements of the arrays, with the eigenvalue having positive imaginary part first. See also Section 8.

11: VR(LDVR,MEST) – REAL (KIND=nag_wp) arrayOutput

On exit: contains the real parts of the selected eigenvectors, with the

i

th column holding the real part of the eigenvector associated with the eigenvalue

λ_{i}

(stored in

WR (i)

and

WI (i)

12: LDVR – INTEGERInput

On entry: the first dimension of the array VR as declared in the (sub)program from which F02ECF is called.

Constraint:

LDVR \geq \max (1, N)

13: VI(LDVI,MEST) – REAL (KIND=nag_wp) arrayOutput

On exit: contains the imaginary parts of the selected eigenvectors, with the

i

th column holding the imaginary part of the eigenvector associated with the eigenvalue

λ_{i}

(stored in

WR (i)

and

WI (i)

14: LDVI – INTEGERInput

On entry: the first dimension of the array VI as declared in the (sub)program from which F02ECF is called.

Constraint:

LDVI \geq \max (1, N)

15: WORK(LWORK) – REAL (KIND=nag_wp) arrayWorkspace

16: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F02ECF is called.

Constraint:

LWORK \geq \max (1, N \times (N + 4))

17: IWORK(N) – INTEGER arrayWorkspace

18: BWORK(N) – LOGICAL arrayWorkspace

19: 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	$LDVR < \max (1, N)$ ,
or	$LDVI < \max (1, N)$ ,
or	$LWORK < \max (1, N \times (N + 4))$ .

$IFAIL = 2$: 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 VR and $VI = MEST \geq M$ .

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

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 8), 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 F08PEF (DHSEQR) with

JOB ='S'

to compute the Schur form of

A^{'}

, followed by F08QLF (DTRSNA).

8 Further Comments

F02ECF 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 an orthogonal similarity transformation:

A^{'} = Q H Q^{T}

. 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

(real or complex) 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 F02ECF.

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.

9 Example

This example computes those eigenvalues of the matrix

A

whose moduli lie in the range

[0.2, 0.5]

, and their corresponding eigenvectors, where

A = (\begin{array}{r} 0.35 & 0.45 & - 0.14 & - 0.17 \\ 0.09 & 0.07 & - 0.54 & 0.35 \\ - 0.44 & - 0.33 & - 0.03 & 0.17 \\ 0.25 & - 0.32 & - 0.13 & 0.11 \end{array}) .

NAG Library Routine DocumentF02ECF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F02ECF