Evaluates a sequence of even, periodic, integer order, real Mathieu functions.

Required Arguments

X — Argument for which the sequence of Mathieu functions is to be evaluated.   (Input)

Q — Parameter.   (Input)
The parameter Q must be positive.

N — Number of elements in the sequence.   (Input)

CE — Vector of length N containing the values of the function through the series.   (Output)
CE(I) contains the value of the Mathieu function of order I − 1 at X for I = 1 to N.

FORTRAN 90 Interface

Generic:                              CALL MATCE (X, Q, N, CE)

Specific:                             The specific interface names are S_MATCE and D_MATCE.

FORTRAN 77 Interface

Single:                                CALL MATCE (X, Q, N, CE)

Double:                              The double precision name is DMATCE.

Description

The eigenvalues of Mathieu's equation are computed using MATEE. The function values are then computed using a sum of Bessel functions, see Gradshteyn and Ryzhik (1965), equation 8.661.

Comments

1.         Workspace may be explicitly provided, if desired, by use of M2TCE/DM2TCE. The reference is

CALL M2TCE (X, Q, N, CE, NORDER, NEEDEV, EVAL0, EVAL1, COEF, WORK, BSJ)

The additional arguments are as follows:

NORDER — Order of the matrix used to compute the eigenvalues.   (Input)
It must be greater than N. Routine MATSE computes NORDER by the following call to M3TEE.

CALL M3TEE(Q, N, NORDER)

NEEDEV — Logical variable, if .TRUE., the eigenvalues must be computed.   (Input)

EVAL0 — Real work vector of length NORDER containing the eigenvalues computed by MATEE with ISYM = 0 and IPER = 0.   (Input/Output)
If NEEDEV is .TRUE., then EVAL0 is computed by M2TCE; otherwise, it must be set as an input value.

EVAL1 — Real work vector of length NORDER containing the eigenvalues computed by MATEE with ISYM = 0 and IPER = 1.   (Input/Output)
If NEEDEV is .TRUE., then EVAL1 is computed by M2TCE; otherwise, it must be set as an input value.

COEF — Real work vector of length NORDER + 4.

WORK — Real work vector of length NORDER + 4.

BSJ — Real work vector of length 2 * NORDER − 2.

2.         Informational error

Type Code

4         1                  The iteration for the eigenvalues did not converge.

Example 1

In this example, cen(x = π/4, q = 1), n = 0, …, 9 is computed and printed.

 

      USE CONST_INT

      USE MATCE_INT

      USE UMACH_INT

 

      IMPLICIT    NONE

!                                 Declare variables

      INTEGER    N

      PARAMETER  (N=10)

 

!

      INTEGER    K, NOUT

      REAL       CE(N), Q, X

!                                 Compute

      Q = 1.0

      X = CONST('PI')

      X = 0.25* X

      CALL MATCE (X, Q, N, CE)

!                                 Print the results

      CALL UMACH (2, NOUT)

      DO 10  K=1, N

         WRITE (NOUT,99999) K-1, X, Q, CE(K)

   10 CONTINUE

99999 FORMAT (' ce sub', I2, ' (', F6.3, ',', F6.3, ') = ', F6.3)

      END

Output

 

 ce sub 0 ( 0.785, 1.000) =  0.654

 ce sub 1 ( 0.785, 1.000) =  0.794

 ce sub 2 ( 0.785, 1.000) =  0.299

 ce sub 3 ( 0.785, 1.000) = -0.555

 ce sub 4 ( 0.785, 1.000) = -0.989

 ce sub 5 ( 0.785, 1.000) = -0.776

 ce sub 6 ( 0.785, 1.000) = -0.086

 ce sub 7 ( 0.785, 1.000) =  0.654

 ce sub 8 ( 0.785, 1.000) =  0.998

 ce sub 9 ( 0.785, 1.000) =  0.746

Additional Examples

Example 2

In this example, we compute cen(x, q) for various values of n and x and a fixed value of q. To avoid having to recompute the eigenvalues, which depend on q but not on x, we compute the eigenvalues once and pass in their value to M2TCE. The eigenvalues are computed using MATEE. The routine M3TEE is used to compute NORDER based on Q and N. The arrays BSJ, COEF and WORK are used as temporary storage in M2TCE.

 

      USE IMSL_LIBRARIES

 

      IMPLICIT    NONE

 

!                                 Declare variables

      INTEGER    MAXORD, N, NX

      PARAMETER  (MAXORD=100, N=4, NX=5)

!

      INTEGER    ISYM, K, NORDER, NOUT

      REAL       BSJ(2*MAXORD-2), CE(N), COEF(MAXORD+4)

      REAL       EVAL0(MAXORD), EVAL1(MAXORD), PI, Q, WORK(MAXORD+4), X

!                                 Compute NORDER

      Q = 1.0

      CALL M3TEE (Q, N, NORDER)

!

      CALL UMACH (2, NOUT)

      WRITE (NOUT, 99997) NORDER

!                                 Compute eigenvalues

      ISYM = 0

      CALL MATEE (Q, ISYM, 0, EVAL0)

      CALL MATEE (Q, ISYM, 1, EVAL1)

!

      PI = CONST('PI')

!                                 Compute function values

      WRITE (NOUT, 99998)

      DO 10  K=0, NX

         X = (K*PI)/NX

         CALL M2TCE(X, Q, N, CE, NORDER, .FALSE., EVAL0, EVAL1, &

          COEF, WORK, BSJ)

         WRITE (NOUT,99999) X, CE(1), CE(2), CE(3), CE(4)

   10 CONTINUE

!

99997 FORMAT (' NORDER = ', I3)

99998 FORMAT (/, 28X, 'Order', /, 20X, '0', 7X, '1', 7X, &

          '2', 7X, '3')

99999 FORMAT (' ce(', F6.3, ') = ', 4F8.3)

      END

Output

 

NORDER =  23

 

                            Order

                    0       1       2       3

 ce( 0.000) =    0.385   0.857   1.086   1.067

 ce( 0.628) =    0.564   0.838   0.574  -0.131

 ce( 1.257) =    0.926   0.425  -0.575  -0.820

 ce( 1.885) =    0.926  -0.425  -0.575   0.820

 ce( 2.513) =    0.564  -0.838   0.574   0.131

 ce( 3.142) =    0.385  -0.857   1.086  -1.067

Figure 12- 1  Plot of cen(x, q = 1)

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