EVCCH

Computes all of the eigenvalues and eigenvectors of a complex upper Hessenberg matrix.

Required Arguments

A — Complex upper Hessenberg matrix of order N. (Input)

EVAL — Complex vector of length N containing the eigenvalues of A in decreasing order of magnitude. (Output)

EVEC — Complex matrix of order N. (Output)

The J-th eigenvector, corresponding to EVAL(J), is stored in the J-th column. Each vector is normalized to have Euclidean length equal to the value one.

The J-th eigenvector, corresponding to EVAL(J), is stored in the J-th column. Each vector is normalized to have Euclidean length equal to the value one.

Optional Arguments

N — Order of the matrix A. (Input)

Default: N = SIZE (A,2).

Default: N = SIZE (A,2).

LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)

Default: LDA = SIZE (A,1).

Default: LDA = SIZE (A,1).

LDEVEC — Leading dimension of EVEC exactly as specified in the dimension statement in the calling program. (Input)

Default: LDEVEC = SIZE (EVEC,1).

Default: LDEVEC = SIZE (EVEC,1).

FORTRAN 90 Interface

Generic: CALL EVCCH (A, EVAL, EVEC [,…])

Specific: The specific interface names are S_EVCCH and D_EVCCH.

FORTRAN 77 Interface

Single: CALL EVCCH (N, A, LDA, EVAL, EVEC, LDEVEC)

Double: The double precision name is DEVCCH.

Description

Routine EVCCH computes the eigenvalues and eigenvectors of a complex upper Hessenberg matrix using the QR algorithm. This routine is based on the EISPACK routine COMQR2; see Smith et al. (1976).

Comments

1. Workspace may be explicitly provided, if desired, by use of E4CCH/DE4CCH. The reference is:

CALL E4CCH (N, A, LDA, EVAL, EVEC, LDEVEC, ACOPY, CWORK, RWK, IWK)

The additional arguments are as follows:

ACOPY — Complex N by N work array. A and ACOPY may be the same, in which case A is destroyed.

CWORK — Complex work array of length 2N.

RWK — Real work array of length N.

IWK — Integer work array of length N.

2 Informational error

Type | Code | Description |
---|---|---|

4 | 1 | The iteration for the eigenvalues failed to converge. |

3. The results of EVCCH can be checked using EPICG. This requires that the matrix A explicitly contains the zeros in A(I, J) for (I – 1) > J which are assumed by EVCCH.

Example

In this example, a DATA statement is used to set the matrix A. The program computes the eigenvalues and eigenvectors of this matrix. The performance index is also computed and printed. This serves as a check on the computations; for more details, see IMSL routine EPICG. The zeros in the lower part of the matrix are not referenced by EVCCH, but they are required by EPICG.

USE EVCCH_INT

USE EPICG_INT

USE UMACH_INT

USE WRCRN_INT

IMPLICIT NONE

! Declare variables

INTEGER LDA, LDEVEC, N

PARAMETER (N=4, LDA=N, LDEVEC=N)

!

INTEGER NOUT

REAL PI

COMPLEX A(LDA,N), EVAL(N), EVEC(LDEVEC,N)

! Set values of A

!

! A = (5+9i 5+5i -6-6i -7-7i)

! (3+3i 6+10i -5-5i -6-6i)

! ( 0 3+3i -1+3i -5-5i)

! ( 0 0 -3-3i 4i)

!

DATA A/(5.0,9.0), (3.0,3.0), (0.0,0.0), (0.0,0.0), (5.0,5.0), &

(6.0,10.0), (3.0,3.0), (0.0,0.0), (-6.0,-6.0), (-5.0,-5.0), &

(-1.0,3.0), (-3.0,-3.0), (-7.0,-7.0), (-6.0,-6.0), &

(-5.0,-5.0), (0.0,4.0)/

!

! Find eigenvalues and vectors of A

CALL EVCCH (A, EVAL, EVEC)

! Compute performance index

PI = EPICG(N,A,EVAL,EVEC)

! Print results

CALL UMACH (2, NOUT)

CALL WRCRN ('EVAL', EVAL, 1, N, 1)

CALL WRCRN ('EVEC', EVEC)

WRITE (NOUT,'(/,A,F6.3)') ' Performance index = ', PI

END

Output

EVAL

1 2 3 4

( 8.22, 12.22) ( 3.40, 7.40) ( 1.60, 5.60) ( -3.22, 0.78)

EVEC

1 2 3 4

1 ( 0.7167, 0.0000) (-0.0704, 0.0000) (-0.3678, 0.0000) ( 0.5429, 0.0000)

2 ( 0.6402,-0.0000) (-0.0046,-0.0000) ( 0.6767, 0.0000) ( 0.4298,-0.0000)

3 ( 0.2598, 0.0000) ( 0.7477, 0.0000) (-0.3005, 0.0000) ( 0.5277,-0.0000)

4 (-0.0948,-0.0000) (-0.6603,-0.0000) ( 0.5625, 0.0000) ( 0.4920,-0.0000)

Performance index = 0.020