GPICG
This function computes the performance index for a generalized complex eigensystem Az = λBz.
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).
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).
LDB — Leading dimension of B exactly as specified in the dimension statement in the calling program. (Input)
Default: LDB = SIZE (B,1).
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).
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 ɛ be the machine precision given by AMACH(4). The performance index, 
, is defined to be
, is defined to beThe norms used are a modified form of the 1-norm. The norm of the complex vector v is
While the exact value of 
is highly machine dependent, the performance of GVCCG is considered excellent if 
< 1, good if 1 ≤ 
≤ 100, and poor if 
> 100.
is highly machine dependent, the performance of GVCCG is considered excellent if < 1, good if 1 ≤ ≤ 100, and poor if > 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 | Description |
|---|---|---|
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.
