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.

Optional Arguments

N — Order of the matrix A.   (Input)
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).

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

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

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