Computes the largest or smallest eigenvalues of a real symmetric matrix in band symmetric storage mode.

Required Arguments

NEVAL — Number of eigenvalues to be computed.   (Input)

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

NCODA — Number of codiagonals in A.   (Input)

SMALL — Logical variable.   (Input)
If .TRUE., the smallest NEVAL eigenvalues are computed. If .FALSE., the largest NEVAL eigenvalues are computed.

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

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).

FORTRAN 90 Interface

Generic:          CALL EVASB (NEVAL, A, NCODA, SMALL, EVAL [,…])

Specific:         The specific interface names are S_EVASB and D_EVASB.

FORTRAN 77 Interface

Single:            CALL EVASB (N, NEVAL, A, LDA, NCODA, SMALL, EVAL)

Double:          The double precision name is DEVASB.

Description

Routine EVASB computes the largest or smallest eigenvalues of a real band symmetric matrix. Orthogonal similarity transformations are used to reduce the matrix to an equivalent symmetric tridiagonal matrix. The rational QR algorithm with Newton corrections is used to compute the extreme eigenvalues of this tridiagonal matrix.

The reduction routine is based on the EISPACK routine BANDR; see Garbow et al. (1978). The QR routine is based on the EISPACK routine RATQR; see Smith et al. (1976).

Comments

1.         Workspace may be explicitly provided, if desired, by use of E3ASB/DE3ASB. The reference is:

CALL E3ASB (N, NEVAL, A, LDA, NCODA, SMALL, EVAL, ACOPY, WK)

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 + 1) elements of A will be destroyed.

WK — Work array of length 3N.

2.         Informational error

Type   Code

3           1                  The iteration for an eigenvalue failed to converge. The best estimate will be returned.

Example

The following example is given in Gregory and Karney (1969, page 63). The smallest four eigenvalues of the matrix

are computed and printed.

 

      USE EVASB_INT

      USE WRRRN_INT

      USE SSET_INT

 

      IMPLICIT   NONE

!                                 Declare variables

      INTEGER    LDA, N, NCODA, NEVAL

      PARAMETER  (N=11, NCODA=3, NEVAL=4, LDA=NCODA+1)

!

      REAL       A(LDA,N), EVAL(NEVAL)

      LOGICAL    SMALL

!                                 Set up matrix in band symmetric

!                                 storage mode

      CALL SSET (N, 6.0, A(4:,1), LDA)

      CALL SSET (N-1, 3.0, A(3:,2), LDA)

      CALL SSET (N-2, 1.0, A(2:,3), LDA)

      CALL SSET (N-3, 1.0, A(1:,4), LDA)

      CALL SSET (NCODA, 0.0, A(1:,1), 1)

      CALL SSET (NCODA-1, 0.0, A(1:,2), 1)

      CALL SSET (NCODA-2, 0.0, A(1:,3), 1)

      A(4,1) = 5.0

      A(4,N) = 5.0

      A(3,2) = 2.0

      A(3,N) = 2.0

!                                 Find the 4 smallest eigenvalues

      SMALL = .TRUE.

      CALL EVASB (NEVAL, A, NCODA, SMALL, EVAL)

!                                 Print results

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

      END

Output

 

              EVAL

      1       2       3       4

  4.000   3.172   1.804   0.522

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