Computes selected eigenvalues and eigenvectors of a real symmetric matrix.

Required Arguments

MXEVAL — Maximum number of eigenvalues to be computed.   (Input)

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

ELOW — Lower limit of the interval in which the eigenvalues are sought.   (Input)

EHIGH — Upper limit of the interval in which the eigenvalues are sought.   (Input)

NEVAL — Number of eigenvalues found.   (Output)

EVAL — Real vector of length MXEVAL containing the eigenvalues of A in the interval
(ELOW, EHIGH) in decreasing order of magnitude.   (Output)
Only the first NEVAL elements of EVAL are significant.

EVEC — Real matrix of dimension N by MXEVAL.   (Output)
The J-th eigenvector corresponding to EVAL(J), is stored in the J-th column. Only the first NEVAL columns of EVEC are significant. 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).

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:          CALL EVFSF (MXEVAL, A, ELOW, EHIGH, NEVAL, EVAL, EVEC [,…])

Specific:         The specific interface names are S_EVFSF and D_EVFSF.

FORTRAN 77 Interface

Single:            CALL EVFSF (N, MXEVAL, A, LDA, ELOW, EHIGH, NEVAL, EVAL, EVEC, LDEVEC)

Double:          The double precision name is DEVFSF.

Description

Routine EVFSF computes the eigenvalues in a given interval and the corresponding eigenvectors 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. Inverse iteration is used to compute the eigenvectors of the tridiagonal matrix. This is followed by orthogonalization of these vectors. The eigenvectors of the original matrix are computed by back transforming those of the tridiagonal matrix.

The reduction step is based on the EISPACK routine TRED1. The rational QR algorithm is called the PWK algorithm. It is given in Parlett (1980, page 169). The inverse iteration and orthogonalization processes are discussed in Hanson et al. (1990). The transformation back to the users's input matrix is based on the EISPACK routine TRBAK1. See Smith et al. (1976) for the EISPACK routines.

Comments

1.         Workspace may be explicitly provided, if desired, by use of E3FSF/DE3FSF. The reference is:

            CALL E3FSF (N, MXEVAL, A, LDA, ELOW, EHIGH, NEVAL, VAL, EVEC, LDEVEC, WK, IWK)

The additional arguments are as follows:

WK — Work array of length 9N.

IWK — Integer work array of length N.

2.         Informational errors

Type   Code

3           1                  The number of eigenvalues in the specified range exceeds MXEVAL. NEVAL contains the number of eigenvalues in the range. No eigenvalues will be computed.

3           2                  Inverse iteration did not converge. Eigenvector is not correct for the specified eigenvalue.

3           3                  The eigenvectors have lost orthogonality.

Example

In this example, A is set to the computed Hilbert matrix. The eigenvalues in the interval [0.001, 1] and their corresponding eigenvectors are computed and printed. This example uses MXEVAL = 3. The routine EVFSF computes the number of eigenvalues NEVAL in the given interval. The value of NEVAL is 2. The performance index is also computed and printed. For more details, see IMSL routine EPISF.

 

      USE EVFSF_INT

      USE EPISF_INT

      USE WRRRN_INT

      USE UMACH_INT

 

      IMPLICIT   NONE

!                                 Declare variables

      INTEGER    LDA, LDEVEC, MXEVAL, N, J, I

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

!

      INTEGER    NEVAL, NOUT

      REAL       A(LDA,N), EHIGH, ELOW, EVAL(MXEVAL), &

                EVEC(LDEVEC,MXEVAL), PI

!                                 Compute Hilbert matrix

      DO 20 J=1,N

         DO 10 I=1,N

            A(I,J) = 1.0/FLOAT(I+J-1)

   10    CONTINUE

   20 CONTINUE

!                                 Find eigenvalues and vectors

      ELOW  = 0.001

      EHIGH = 1.0

      CALL EVFSF (MXEVAL, A, ELOW, EHIGH, NEVAL, EVAL, EVEC, LDEVEC)

!                                 Compute performance index

      PI = EPISF(NEVAL,A,EVAL,EVEC)

!                                 Print results

      CALL UMACH (2, NOUT)

      WRITE (NOUT,'(/,A,I2)') ' NEVAL = ', NEVAL

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

      CALL WRRRN ('EVEC', EVEC, N, NEVAL, LDEVEC)

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

      END

Output

 

 NEVAL =  2

       EVAL

       1        2

  0.1223   0.0027

        EVEC

          1        2

 1  -0.5474  -0.1277

 2   0.5283   0.7137

 3   0.6490  -0.6887

 

 Performance index =  0.008

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