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, l = EVAL, xj = EVEC(*, J), the j-th column of EVEC. Also, let ε be the machine precision, given by AMACH(4), see the Reference chapter of this manual. 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 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

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.

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