EVCHF

Computes all of the eigenvalues and eigenvectors of a complex Hermitian matrix.

Required Arguments

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

Only the upper triangle is used.

Only the upper triangle is used.

EVAL — Real vector of length N containing the eigenvalues of A in decreasing order of magnitude. (Output)

EVEC — Complex matrix of order N. (Output)

The J-th eigenvector, corresponding to EVAL(J), is stored in the J-th column. Each vector is normalized to have Euclidean length equal to the value one.

The J-th eigenvector, corresponding to EVAL(J), is stored in the J-th column. Each vector is normalized to have Euclidean length equal to the value one.

Optional Arguments

N — Order of the matrix A. (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).

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: CALL EVCHF (A, EVAL, EVEC [,…])

Specific: The specific interface names are S_EVCHF and D_EVCHF.

FORTRAN 77 Interface

Single: CALL EVCHF (N, A, LDA, EVAL, EVEC, LDEVEC)

Double: The double precision name is DEVCHF.

Description

Routine EVCHF computes the eigenvalues and eigenvectors of a complex Hermitian matrix. Unitary similarity transformations are used to reduce the matrix to an equivalent real symmetric tridiagonal matrix. The implicit QL algorithm is used to compute the eigenvalues and eigenvectors of this tridiagonal matrix. These eigenvectors and the transformations used to reduce the matrix to tridiagonal form are combined to obtain the eigenvectors for the user’s problem. The underlying code is based on either EISPACK or LAPACK code depending upon which supporting libraries are used during linking. For a detailed explanation, see “Using ScaLAPACK, LAPACK, LINPACK, and EISPACK” in the Introduction section of this manual.

Comments

1. Workspace may be explicitly provided, if desired, by use of E5CHF/DE5CHF. The reference is:

CALL E5CHF (N, A, LDA, EVAL, EVEC, LDEVEC, ACOPY, RWK, CWK, IWK)

The additional arguments are as follows:

ACOPY — Complex work array of length N2. A and ACOPY may be the same, in which case A will be destroyed.

RWK — Work array of length N2 + N.

CWK — Complex work array of length 2N.

IWK — Integer work array of length N.

2. Informational error

Type | Code | Description |
---|---|---|

3 | 1 | The matrix is not Hermitian. It has a diagonal entry with a small imaginary part. |

4 | 1 | The iteration for an eigenvalue failed to converge. |

4 | 2 | The matrix is not Hermitian. It has a diagonal entry with an imaginary part. |

3. The success of this routine can be checked using EPIHF.

4. Integer Options with Chapter 11 Options Manager

1 This option uses eight values to solve memory bank conflict (access inefficiency) problems. In routine E5CHF, the internal or working leading dimensions of ACOPY and ECOPY are both increased by IVAL(3) when N is a multiple of IVAL(4). The values IVAL(3) and IVAL(4) are temporarily replaced by IVAL(1) and IVAL(2), respectively, in routine EVCHF. Additional memory allocation and option value restoration are automatically done in EVCHF. There is no requirement that users change existing applications that use EVCHF or E5CHF. Default values for the option are IVAL(*) = 1, 16, 0, 1, 1, 16, 0, 1. Items 5–8 in IVAL(*) are for the generalized eigenvalue problem and are not used in EVCHF.

Example

In this example, a DATA statement is used to set A to a complex Hermitian matrix. The eigenvalues and eigenvectors of this matrix are computed and printed. The performance index is also computed and printed. This serves as a check on the computations, for more details, see routine EPIHF.

USE IMSL_libraries

IMPLICIT NONE

! Declare variables

INTEGER LDA, LDEVEC, N

PARAMETER (N=3, LDA=N, LDEVEC=N)

!

INTEGER NOUT

REAL EVAL(N), PI

COMPLEX A(LDA,N), EVEC(LDEVEC,N)

! Set values of A

!

! A = ((1, 0) ( 1,-7i) ( 0,- i))

! ((1,7i) ( 5, 0) (10,-3i))

! ((0, i) ( 10, 3i) (-2, 0))

!

DATA A/(1.0,0.0), (1.0,7.0), (0.0,1.0), (1.0,-7.0), (5.0,0.0), &

(10.0, 3.0), (0.0,-1.0), (10.0,-3.0), (-2.0,0.0)/

!

! Find eigenvalues and vectors of A

CALL EVCHF (A, EVAL, EVEC)

! Compute performance index

PI = EPIHF(N,A,EVAL,EVEC)

! Print results

CALL UMACH (2, NOUT)

CALL WRRRN ('EVAL', EVAL, 1, N, 1)

CALL WRCRN ('EVEC', EVEC)

WRITE (NOUT,'(/,A,F6.3)') ' Performance index = ', PI

END

Output

EVAL

1 2 3

15.38 -10.63 -0.75

EVEC

1 2 3

1 ( 0.0631,-0.4075) (-0.0598,-0.3117) ( 0.8539, 0.0000)

2 ( 0.7703, 0.0000) (-0.5939, 0.1841) (-0.0313,-0.1380)

3 ( 0.4668, 0.1366) ( 0.7160, 0.0000) ( 0.0808,-0.4942)

Performance index = 0.093