EPIHF

This function computes the performance index for a complex Hermitian eigensystem.

Function Return Value

EPIHF — Performance index. (Output)

Required Arguments

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

A — Complex Hermitian matrix of order N. (Input)

EVAL — Vector of length NEVAL containing eigenvalues of A. (Input)

EVEC — Complex N by NEVAL array containing eigenvectors of A. (Input)
The eigenvector corresponding to the eigenvalue EVAL(J) must be 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: EPIHF (NEVAL, A, EVAL, EVEC [,…])

Specific: The specific interface names are S_EPIHF and D_EPIHF.

FORTRAN 77 Interface

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

Double: The double precision function name is DEPIHF.

Description

Let M = NEVAL, λ = EVAL, 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 EVCSF 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 Smith et al. (1976, pages 124 ‑ 125).

Comments

1. Workspace may be explicitly provided, if desired, by use of E2IHF/DE2IHF. The reference is:

E2IHF(N, NEVAL, A, LDA, EVAL, EVEC, LDEVEC, WK)

The additional argument is

WK — Complex work array of length N.

2. Informational errors

 

Type	Code	Description
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 EPIHF, see IMSL routine EVCHF.