EVCSB

Computes all of the eigenvalues and eigenvectors of a real symmetric matrix in band symmetric storage mode.

Required Arguments

A — Band symmetric matrix of order N. (Input)

NCODA — Number of codiagonals in A. (Input)

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

EVEC — Matrix of order N containing the eigenvectors. (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 EVCSB (A, NCODA, EVAL, EVEC [,…])

Specific: The specific interface names are S_EVCSB and D_EVCSB.

FORTRAN 77 Interface

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

Double: The double precision name is DEVCSB.

Description

Routine EVCSB computes the eigenvalues and eigenvectors of a real band symmetric matrix. Orthogonal similarity transformations are used to reduce the matrix to an equivalent symmetric tridiagonal matrix. These transformations are accumulated. The implicit QL algorithm is used to compute the eigenvalues and eigenvectors of the resulting tridiagonal matrix.

The reduction routine is based on the EISPACK routine BANDR; see Garbow et al. (1977). The QL routine is based on the EISPACK routine IMTQL2; see Smith et al. (1976).

Comments

1. Workspace may be explicitly provided, if desired, by use of E4CSB/DE4CSB. The reference is:

CALL E4CSB (N, A, LDA, NCODA, EVAL, EVEC, LDEVEC, COPY, WK, IWK)

The additional arguments are as follows:

ACOPY — Work array of length N(NCODA + 1). A and ACOPY may be the same, in which case the first N * NCODA elements of A will be destroyed.

WK — Work array of length N.

IWK — Integer work array of length N.

2. Informational error

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

4 | 1 | The iteration for the eigenvalues failed to converge. |

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

Example

In this example, a DATA statement is used to set A to a band matrix given by Gregory and Karney (1969, page 75). The eigenvalues, λk, of this matrix are given by

The eigenvalues and eigenvectors of this real band symmetric 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 IMSL routine EPISB.

USE EVCSB_INT

USE EPISB_INT

USE UMACH_INT

USE WRRRN_INT

IMPLICIT NONE

! Declare variables

INTEGER LDA, LDEVEC, N, NCODA

PARAMETER (N=6, NCODA=2, LDA=NCODA+1, LDEVEC=N)

!

INTEGER NOUT

REAL A(LDA,N), EVAL(N), EVEC(LDEVEC,N), PI

! Define values of A:

! A = ( 5 -4 1 )

! ( -4 6 -4 1 )

! ( 1 -4 6 -4 1 )

! ( 1 -4 6 -4 1 )

! ( 1 -4 6 -4 )

! ( 1 -4 5 )

! Represented in band symmetric

! form this is:

! A = ( 0 0 1 1 1 1 )

! ( 0 -4 -4 -4 -4 -4 )

! ( 5 6 6 6 6 5 )

!

DATA A/0.0, 0.0, 5.0, 0.0, -4.0, 6.0, 1.0, -4.0, 6.0, 1.0, -4.0, &

6.0, 1.0, -4.0, 6.0, 1.0, -4.0, 5.0/

!

! Find eigenvalues and vectors

CALL EVCSB (A, NCODA, EVAL, EVEC)

! Compute performance index

PI = EPISB(N,A,NCODA,EVAL,EVEC)

! Print results

CALL UMACH (2, NOUT)

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

CALL WRRRN ('EVEC', EVEC)

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

END

Output

EVAL

1 2 3 4 5 6

14.45 10.54 5.98 2.42 0.57 0.04

EVEC

1 2 3 4 5 6

1 -0.2319 -0.4179 -0.5211 0.5211 -0.4179 0.2319

2 0.4179 0.5211 0.2319 0.2319 -0.5211 0.4179

3 -0.5211 -0.2319 0.4179 -0.4179 -0.2319 0.5211

4 0.5211 -0.2319 -0.4179 -0.4179 0.2319 0.5211

5 -0.4179 0.5211 -0.2319 0.2319 0.5211 0.4179

6 0.2319 -0.4179 0.5211 0.5211 0.4179 0.2319

Performance index = 0.029