This function computes the performance index for a complex eigensystem.

Function Return Value

EPICG — Performance index.   (Output)

Required Arguments

NEVAL — Number of eigenvalue/eigenvector pairs on which the performance index computation is based.   (Input)

A — Complex matrix of order N.   (Input)

EVAL —  Complex vector of length N containing the eigenvalues of A.   (Input)

EVEC — Complex matrix of order N containing the eigenvectors of A.   (Input)
The J-th eigenvalue/eigenvector pair should be in EVAL(J) and in the J-th column of EVEC.

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:          EPICG (NEVAL, A, EVAL, EVEC [,…])

Specific:         The specific interface names are S_EPICG and D_EPICG.

FORTRAN 77 Interface

Single:            EPICG (N, NEVAL, A, LDA, EVAL, EVEC, LDEVEC)

Double:          The double precision function name is DEPICG.

Description

Let M = NEVAL, l = EVAL, xj = EVEC(*, J), the j-th column of EVEC. Also, let ε be the machine precision given by AMACH(4). The performance index, τ, is defined to be

The norms used are a modified form of the 1-norm. The norm of the complex vector v is

While the exact value of τ is highly machine dependent, the performance of EVCSF is considered excellent if τ < 1, good if 1 ≤ τ ≤ 100, and poor if τ > 100. The performance index was first developed by the EISPACK project at Argonne National Laboratory; see Smith et al. (1976, pages 124− 125).

Comments

1.         Workspace may be explicitly provided, if desired, by use of E2ICG/DE2ICG. The reference is:

E2ICG (N, NEVAL, A, LDA, EVAL, EVEC, LDEVEC, WK)

The additional argument is:

WK — Complex work array of length N.

2.         Informational errors

Type   Code

3           1                  Performance index is greater than 100.

3           2                  An eigenvector is zero.

3           3                  The matrix is zero.

Example

For an example of EPICG, see IMSL routine EVCCG.

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