This function computes the performance index for a generalized complex eigensystem Az = lBz.

Function Return Value

GPICG — Performance index.   (Output)

Required Arguments

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

A — Complex matrix of order N.   (Input)

B — Complex matrix of order N.   (Input)

ALPHA — Complex vector of length NEVAL containing the numerators of eigenvalues.   (Input)

BETAV — Complex vector of length NEVAL containing the denominators of eigenvalues.   (Input)

EVEC — Complex N by NEVAL array containing the eigenvectors.   (Input)

Optional Arguments

N — Order of the matrices A and B.   (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).

LDB — Leading dimension of B exactly as specified in the dimension statement in the calling program.   (Input)
Default: LDB = size (B,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:          GPICG (NEVAL, A, B, ALPHA, BETAV, EVEC [,…])

Specific:         The specific interface names are S_GPICG and D_GPICG.

FORTRAN 77 Interface

Single:            GPICG (N, NEVAL, A, LDA, B, LDB, ALPHA, BETAV, EVEC, LDEVEC)

Double:          The double precision name is DGPICG.

Description

Let M = NEVAL, 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 Garbow et al. (1977, pages 77− 79).

Comments

1.         Workspace may be explicitly provided, if desired, by use of G2ICG/DG2ICG. The reference is:

G2ICG (N, NEVAL, A, LDA, B, LDB, ALPHA, BETAV, EVEC, LDEVEC, WK)

The additional argument is:

WK — Complex work array of length 2N.

2.         Informational errors

Type   Code

3           1                  Performance index is greater than 100.

3           2                  An eigenvector is zero.

3           3                  The matrix A is zero.

3           4                  The matrix B is zero.

3.         The J-th eigenvalue should be ALPHA(J)/BETAV (J), its eigenvector should be in the J-th column of EVEC.

Example

For an example of GPICG, see routine GVCCG.

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