Calculates the indices of eigenvalues of a Sturm-Liouville problem of the form for

with boundary conditions (at regular points)

in a specified subinterval of the real line, [, ].

Required Arguments

ALPHA — Value of the left end point of the search interval.   (Input)

BETAR — Value of the right end point of the search interval.   (Input)

CONS — Array of size eight containing

in locations CONS(1) CONS(8), respectively.   (Input)

COEFFN — User-supplied SUBROUTINE to evaluate the coefficient functions. The usage is
CALL COEFFN (X, PX, QX, RX)
X — Independent variable.   (Input)
PX — The value of p(x) at X.   (Output)
QX — The value of q(x) at X.   (Output)
RX — The value of r(x) at X.   (Output)
COEFFN must be declared EXTERNAL in the calling program.

ENDFIN — Logical array of size two. ENDFIN = .true. if and only if the endpoint a is finite. ENDFIN(2) = .true. if and only if endpoint b is finite.   (Input)

IFIRST — The index of the first eigenvalue greater than .   (Output)

NTOTAL — Total number of eigenvalues in the interval [, ].   (Output)

FORTRAN 90 Interface

Generic:          CALL SLCNT (ALPHA, BETAR, CONS, COEFFN, ENDFIN, IFIRST, NTOTAL)

Specific:         The specific interface names are S_SLCNT and D_SLCNT.

FORTRAN 77 Interface

Single:            CALL SLCNT (ALPHA, BETAR, CONS, COEFFN, ENDFIN, IFIRST, NTOTAL)

Double:          The double precision name is DSLCNT.

Description

This subroutine computes the indices of eigenvalues, if any, in a subinterval of the real line for Sturm-Liouville problems in the form

with boundary conditions (at regular points)

It is intended to be used in conjunction with SLEIG. SLCNT is based on the routine INTERV from the package SLEDGE.

Example

Consider the harmonic oscillator (Titchmarsh) defined by

                p(x) = 1

                q(x) = x2

                r(x) = 1

                [a, b] = [, ]

                u(a) = 0

                u(b) = 0

The eigenvalues of this problem are known to be

                k = 2k + 1, k = 0, 1,

Therefore in the interval [10, 16] we expect SLCNT to note three eigenvalues, with the first of these having index five.

 

      USE SLCNT_INT

      USE UMACH_INT

 

      IMPLICIT   NONE

!                                  SPECIFICATIONS FOR LOCAL VARIABLES

      INTEGER    IFIRST, NOUT, NTOTAL

      REAL       ALPHA, BETAR, CONS(8)

      LOGICAL    ENDFIN(2)

!                                  SPECIFICATIONS FOR SUBROUTINES

!                                  SPECIFICATIONS FOR FUNCTIONS

      EXTERNAL   COEFFN

!

      CALL UMACH (2, NOUT)

!                                  set u(a) = 0, u(b) = 0

      CONS(1) = 1.0E0

      CONS(2) = 0.0E0

      CONS(3) = 0.0E0

      CONS(4) = 0.0E0

      CONS(5) = 1.0E0

      CONS(6) = 0.0E0

      CONS(7) = 0.0E0

      CONS(8) = 0.0E0

!

      ENDFIN(1) = .FALSE.

      ENDFIN(2) = .FALSE.

!

      ALPHA = 10.0

      BETAR  = 16.0

!

      CALL SLCNT (ALPHA, BETAR, CONS, COEFFN, ENDFIN, IFIRST, NTOTAL)

!

      WRITE (NOUT,99998) ALPHA, BETAR, IFIRST

      WRITE (NOUT,99999) NTOTAL

!

99998 FORMAT (/, 'Index of first eigenvalue in [', F5.2, ',', F5.2, &

            '] IS ', I2)

99999 FORMAT ('Total number of eigenvalues in this interval: ', I2)

!

      END

!

      SUBROUTINE COEFFN (X, PX, QX, RX)

!                                  SPECIFICATIONS FOR ARGUMENTS

      REAL       X, PX, QX, RX

!

      PX = 1.0E0

      QX = X*X

      RX = 1.0E0

      RETURN

      END

Output

 

Index of first eigenvalue in [10.00,16.00] is 5
Total number of eigenvalues in this interval: 3

 

 

 

 

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