Computes the largest or smallest eigenvalues of a real symmetric matrix.

Required Arguments

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

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

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

EVAL — Real vector of length NEVAL containing the eigenvalues of A 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 EVASF (NEVAL, A, SMALL, EVAL [,…])

Specific:         The specific interface names are S_EVASF and D_EVASF.

FORTRAN 77 Interface

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

Double:          The double precision name is DEVASF.

Description

Routine EVASF computes the largest or smallest eigenvalues of a real symmetric matrix. Orthogonal similarity transformations are used to reduce the matrix to an equivalent symmetric tridiagonal matrix. Then, an implicit rational QR algorithm is used to compute the eigenvalues of this tridiagonal matrix.

The reduction routine is based on the EISPACK routine TRED2. See Smith et al. (1976). The rational QR algorithm is called the PWK algorithm. It is given in Parlett (1980, page 169).

Comments

1.         Workspace may be explicitly provided, if desired, by use of E4ASF/DE4ASF. The reference is:

CALL E4ASF (N, NEVAL, A, LDA, SMALL, EVAL, WORK, IWK)

WORK — Work array of length 4N.

IWK — Integer work array of length N.

2.         Informational error

Type   Code

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

Example

In this example, the three largest eigenvalues of the computed Hilbert matrix aij = 1/(i + j -1) of order N = 10 are computed and printed.

 

      USE EVASF_INT

      USE WRRRN_INT

 

      IMPLICIT   NONE

!                                 Declare variables

      INTEGER    LDA, N, NEVAL

      PARAMETER  (N=10, NEVAL=3, LDA=N)

!

      INTEGER    I, J

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

      LOGICAL    SMALL

      INTRINSIC  REAL

!                                 Set up Hilbert matrix

      DO 20  J=1, N

         DO 10  I=1, N

            A(I,J) = 1.0/REAL(I+J-1)

   10    CONTINUE

   20 CONTINUE

!                                 Find the 3 largest eigenvalues

      SMALL = .FALSE.

       CALL EVASF (NEVAL, A, SMALL, EVAL)

!                                 Print results

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

 

      END

Output

 

          EVAL

      1       2       3

  1.752   0.343   0.036

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