TOMS352
Characteristic Values and Associated Solutions of Matthieu's Differential Equation

TOMS352 is a FORTRAN77 library which implements ACM TOMS algorithm 352, for determining characteristic values and associated solutions of Matthieu's differential equation.

The text of many ACM TOMS algorithms is available online through ACM: http://www.acm.org/pubs/calgo or NETLIB: http://www.netlib.org/toms/index.html.

Usage:

The routine MFCVAL computes characteristic values of Mathieu's differential equation:

call mfcval ( n, r, qq, cv, j )

input, integer N

the number of characteristic values desired.

input, integer R

given as N-1 or N according as the characteristic values are to be associated with the even or odd solutions, respectively.

input, double precision QQ

the nonnegative parameter.

output, double precision CV(6,N)

the computed array of characteristic values and bounds.

output, integer J

the number of characteristic values successfully computed. If J is not equal to N then an error occurred while trying to compute value J+1.

The routine MATH computes solutions and their derivatives of either Mathieu's differential equation or Mathieu's modified equation, associated with the characteristic values. the Mathieu equation:

call math ( xx, qq, r, cv, sol, fnc, norm, f, k, m )

Languages:

TOMS352 is available in a FORTRAN77 version.

Related Data and Programs:

TEST_VALUES, a FORTRAN77 library which contains routines which return sample values of various functions, including the Bessel functions used by this package.

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Gertrude Blanch,
   Numerical evaluation of continued fractions,
   SIAM Review,
   Volume 6, Number 4, 1964, pages 383-421.
3. Gertrude Blanch,
   Numerical aspects of Mathieu eigenvalues,
   Rendiconti Circolo Matematico di Palermo,
   Volume 2, Number 15, 1966, pages 51-97.
4. Gertrude Blanch, Donald Clemm,
   Tables Relating to the Radial Mathieu Functions, Volume 1: Functions of the First Kind,
   US Government Printing Office, 1962.
5. Gertrude Blanch, Donald Clemm,
   Tables Relating to the Radial Mathieu Functions, Volume 2: Functions of the Second Kind,
   US Government Printing Office, 1962.
6. Donald Clemm,
   Algorithm 352: Characteristic Values and Associated Solutions of Matthieu's Differential Equation,
   Communications of the ACM,
   Volume 12, Number 7, pages 399-407, June 1969.
7. Edward Ince,
   Tables of the Elliptic Cylinder Functions,
   Proceedings of the Royal Society of Edinburgh,
   Volume 52, 1932, pages 355-423.
8. Julius Stratton, Philip Morse, Lan Jen Chu, Reina Hutner,
   Elliptic Cylinder and Spheroidal Wave Functions,
   Wiley, 1941.

Source Code:

• toms352.f, the source code.
• toms352.sh, commands to compile the source code.

Examples and Tests:

• toms352_prb.f, a sample calling program.
• toms352_prb.sh, commands to compile and run the sample program.
• toms352_prb_output.txt, the output file.

List of Routines:

• MFCVAL computes the first N characteristic values.
• MATH computes solutions of Mathieu's differential equation.
• BOUNDS gets crude upper and lower bounds for the characteristic values.
• MFITR8 improves estimated characteristic values.
• TMOFA determines the accuracy of the solution.
• COEF computes the neutral coefficients.
• SUM_SERIES does series evaluation.
• BESSEL evaluates Bessel functions of first or second kind.
• J0J1 computes J0(X) and J1(X) for X < 8.
• Y0Y1 computes Y0(X) and Y1(X) for X < 8.
• LUKE evaluates Bessel functions of order 0 or 1, of the first or second kind, for 8 <= X.
• JNS evaluates Bessel functions of the first kind, given J0(X) and J1(X).
• DS evaluates one term of the radial solution for B(Q).
• DC evaluates one term of the radial solution for A(Q).
• DDS evaluates one term of the derivative of the radial solution for B(Q).
• DDC evaluates one term of the derivative of the radial solution for A(Q).
• PS evaluates one term of the odd periodic solution.
• PC evaluates one term of the even periodic solution.
• DPS evaluates one term of the derivative of the odd periodic solution.
• DPC evaluates one term of the derivative of the even periodic solution.
• FJ produces Bessel functions of the first kind.
• FY produces Bessel functions of the second kind.
• DJ produces derivatives of Bessel functions of the first kind.
• DY produces derivatives of Bessel functions of the second kind.

You can go up one level to the FORTRAN77 source codes.