GPIRG

This function computes the performance index for a generalized real eigensystem Az = λBz.

Function Return Value

GPIRG — Performance index. (Output)

Required Arguments

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

A — Real matrix of order N. (Input)

B — Real matrix of order N. (Input)

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

BETAV — Real 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: GPIRG (NEVAL, A, B, ALPHA, BETAV, EVEC, GPIRG [,…])

Specific: The specific interface names are S_GPIRG and D_GPIRG.

FORTRAN 77 Interface

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

Double: The double precision function name is DGPIRG.

Description

Let M = NEVAL, xj = EVEC(*,J) , the j-th column of EVEC. Also, let ɛ be the machine precision given by AMACH(4), see the Reference chapter of this manual. The performance index, , 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 is highly machine dependent, the performance of GVCRG 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 G2IRG/DG2IRG. The reference is:

G2IRG (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 GPIRG, see routine GVCRG.