Computes all of the eigenvalues of a complex matrix.

Required Arguments

A — Complex matrix of order N.   (Input)

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

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).

FORTRAN 90 Interface

Generic:          CALL EVLCG (A, EVAL [,…])

Specific:         The specific interface names are S_EVLCG and D_EVLCG.

FORTRAN 77 Interface

Single:            CALL EVLCG (N, A, LDA, EVAL)

Double:          The double precision name is EVLCG.

Description

Routine EVLCG computes the eigenvalues of a complex matrix. The matrix is first balanced. Unitary similarity transformations are used to reduce this balanced matrix to a complex upper Hessenberg matrix. The shifted QR algorithm is used to compute the eigenvalues of this Hessenberg matrix.

The underlying code is based on either EISPACK or LAPACK code depending upon which supporting libraries are used during linking. For a detailed explanation, see “Using ScaLAPACK, LAPACK, LINPACK, and EISPACK” in the Introduction section of this manual.

Comments

1.         Workspace may be explicitly provided, if desired, by use of E3LCG/DE3LCG. The reference is:

CALL E3LCG (N, A, LDA, EVAL, ACOPY, RWK, CWK, IWK)

The additional arguments are as follows:

ACOPY — Complex work array of length N2. A and ACOPY may be the same, in which case the first N2 elements of A will be destroyed.

RWK — Work array of length N.

CWK — Complex work array of length 2N.

IWK — Integer work array of length N.

2.         Informational error

TypeCode

4           1                  The iteration for an eigenvalue failed to converge.

3.         Integer Options with Chapter 11 Options Manager

1          This option uses eight values to solve memory bank conflict (access inefficiency) problems. In routine E3LCG, the internal or working, leading dimension of ACOPY is increased by IVAL(3) when N is a multiple of IVAL(4). The values IVAL(3) and IVAL (4) are temporarily replaced by IVAL(1) and IVAL(2), respectively, in routine EVLCG . Additional memory allocation and option value restoration are automatically done in EVLCG. There is no requirement that users change existing applications that use EVLCG or E3LCG. Default values for the option are
IVAL(*) = 1, 16, 0, 1, 1, 16, 0, 1. Items 5-8 in IVAL(*) are for the generalized eigenvalue problem and are not used in EVLCG.

Example

In this example, a DATA statement is used to set A to a matrix given by Gregory and Karney  (1969, page 115). The program computes the eigenvalues of this matrix.

 

      USE EVLCG_INT

      USE WRCRN_INT

!                                 Declare variables

      INTEGER    LDA, N

      PARAMETER  (N=3, LDA=N)

!

      COMPLEX    A(LDA,N), EVAL(N)

!                                 Set values of A

!

!                                 A = ( 1+2i    3+4i   21+22i)

!                                     (43+44i  13+14i  15+16i)

!                                     ( 5+6i    7+8i   25+26i)

!

      DATA A/(1.0,2.0), (43.0,44.0), (5.0,6.0), (3.0,4.0), &

          (13.0,14.0), (7.0,8.0), (21.0,22.0), (15.0,16.0), &

          (25.0,26.0)/

!

!                                 Find eigenvalues of A

      CALL EVLCG (A, EVAL)

!                                 Print results

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

      END

Output

 

                          EVAL
              1                2                3

( 39.78, 43.00)  (  6.70, -7.88)  ( -7.48,  6.88)

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