TEST_ODE
Test Problems for
Initial Value Solvers

TEST_ODE is a FORTRAN90 library which defines a set of test initial value problems for ordinary differential equations.

These problems have the common form:

determine (some values of) the function Y(T)

Y'(T) = F(T,Y)

Y(T₀) = Y₀

A uniform interface makes it easy to solve all the problems automatically, or to concentrate on a single one.

The problems include:

1. Enright and Pryce #A1,
   neqn = 1,
   y' = -y;
2. Enright and Pryce #A2,
   neqn = 1,
   y' = -y^3/2;
3. Enright and Pryce #A3,
   neqn = 1,
   y' = cos(t)*y;
4. Enright and Pryce #A4,
   neqn = 1,
   y' = y*(20-y)/80;
5. Enright and Pryce #A5,
   neqn = 1,
   y' = (y-t)/(y+t);
6. Enright and Pryce #B1,
   neqn = 2,
   y1' = 2*y1*(1-y2)
   y2' = - y2*(1-y1);
7. Enright and Pryce #B2,
   neqn = 3,
   y1' = -y1+y2
   y2' = y1-2*y2+y3
   y3' = y2-y3;
8. Enright and Pryce #B3,
   neqn = 3,
   y1' = -y1
   y2' = y1-y2^2
   y3' = y2^2;
9. Enright and Pryce #B4,
   neqn = 3,
   y1' = (-y2-y1*y3) / sqrt(y1^2+y2^2)
   y2' = (y1-y2*y3) / sqrt(y1^2+y2^2)
   y3' = y1 / sqrt(y1^2+y2^2);
10. Enright and Pryce #B5,
    neqn = 3,
    y1' = y2*y3
    y2' = -y1*y3
    y3' = -0.51*y1*y2;
11. Enright and Pryce #C1,
    neqn = 10;
12. Enright and Pryce #C2,
    neqn = 10;
13. Enright and Pryce #C3,
    neqn = 10;
14. Enright and Pryce #C4,
    neqn = 51;
15. Enright and Pryce #C5,
    neqn = 30;
16. Enright and Pryce #D1,
    neqn = 4;
17. Enright and Pryce #D2,
    neqn = 4;
18. Enright and Pryce #D3,
    neqn = 4;
19. Enright and Pryce #D4,
    neqn = 4;
20. Enright and Pryce #D5,
    neqn = 4;
21. Enright and Pryce #E1,
    neqn = 2;
22. Enright and Pryce #E2,
    neqn = 2;
23. Enright and Pryce #E3,
    neqn = 2;
24. Enright and Pryce #E4,
    neqn = 2;
25. Enright and Pryce #E5,
    neqn = 2;
26. Enright and Pryce #F1,
    neqn = 2;
27. Enright and Pryce #F2,
    neqn = 1;
28. Enright and Pryce #F3,
    neqn = 2;
29. Enright and Pryce #F4,
    neqn = 1;
30. Enright and Pryce #F5,
    neqn = 1;
31. Lotka-Volterra Predator-Prey Equations,
    neqn = 2;
32. The Lorenz System,
    neqn = 3;
33. The Van der Pol equation,
    neqn = 2;
34. The Linearized Damped Pendulum,
    neqn = 2;
35. The Nonlinear Damped Pendulum,
    neqn = 2;
36. Duffing's Equation,
    neqn = 2,
37. Duffing's Equation with Damping and Forcing,
    neqn = 2;
38. Shampine's Ball of Flame,
    neqn = 1,
    y' = y^2-y^3;
39. Polking's First Order System,
    neqn = 1,
    y' = y^2-a*t+b;

Related Data and Programs:

LSODI is a FORTRAN77 library which implements the Livermore Solver for Ordinary Differential Equations, Implicit case.

NMS is a FORTRAN90 library which includes the DDRIV package of ODE solvers.

ODE is a FORTRAN90 version of the Shampine and Allen ODE solver.

POLKING is a pair of MATLAB routines, DFIELD and PPLANE, for solving and displaying the solution of differential equations and their phase planes.

RKF45 is a FORTRAN90 Runge-Kutta-Fehlberg ODE solver.

TEST_ODE is available in a FORTRAN90 version and a MATLAB version.

TOMS648_NSD is a FORTRAN77 library which defines the non-stiff Enright and Pryce test ODE problems, using double precision arithmetic.

TOMS648_NSS is a FORTRAN77 library which defines the non-stiff Enright and Pryce test ODE problems, using single precision arithmetic.

References:

The following references include some reliable, sturdy, comprehensive test sets. (Sometimes I'm just doing this for fun or for classroom exercises. These people, on the other hand, are serious.)

1. David Arnold, John Polking,
   Ordinary Differential Equations using Matlab,
   Second Edition,
   Prentice Hall, 1999,
   ISBN: 0130113816.
2. Wayne Enright, John Pryce,
   Algorithm 648: NSDTST and STDTST,
   ACM Transactions on Mathematical Software,
   Volume 13, Number 1, March 1987, pages 28-34.
3. Thomas Hull, Wayne Enright, BM Fellen, Arthur Sedgwick,
   Comparing numerical methods for ordinary differential equations,
   SIAM Journal on Numerical Analysis,
   Volume 9, 1972, pages 603-637.
4. Cleve Moler,
   Cleve's Corner: Stiff Differential Equations,
   MATLAB News and Notes,
   May 2003, pages 12-13.

1. http://pitagora.dm.uniba.it/~testset/, Test Set for IVP Solvers.
2. http://www.unige.ch/math/~hairer/testset/testset.html Stiff ODE test set of Hairer and Wanner.

Source Code:

• test_ode.f90, the source code;
• test_ode.csh, commands to compile the source code.

Examples and Tests:

• test_ode_prb.f90, a sample calling program;
• test_ode_prb.csh, commands to compile, link and run the calling program;
• test_ode_prb_output.txt, output from the calling program;

List of Routines:

• CH_CAP capitalizes a single character.
• P00_AUTONOMOUS reports whether a given problem is autonomous.
• P00_EQUIL returns equilibrium solutions of any problem.
• P00_EULER_STEP takes a single Euler step from (T0,Y0) to (T1,Y1).
• P00_FUN evaluates the derivative function for any problem.
• P00_JAC evaluates the jacobian for any problem.
• P00_NEQN returns the number of equations for any problem.
• P00_RK_STEP takes a single Runge-Kutta step from (T0,Y0) to (T1,Y1).
• P00_SCALE returns scale factors for any problem.
• P00_START returns the starting point for any problem.
• P00_STOP returns the stopping point for any problem.
• P00_TEST_NUM returns the number of problems available.
• P00_TITLE returns the title for any problem.
• P01_AUTONOMOUS reports whether problem 1 is autonomous.
• P01_EQUIL returns equilibrium solutions of problem 1.
• P01_FUN evaluates the function for problem 1.
• P01_JAC evaluates the jacobian for problem 1.
• P01_NEQN returns the number of equations for problem 1.
• P01_SCALE returns scale factors for problem 1.
• P01_START returns the starting point for problem 1.
• P01_STOP returns the stopping point for problem 1.
• P01_TITLE returns the title of problem 1.
• P02_AUTONOMOUS reports whether problem 2 is autonomous.
• P02_EQUIL returns equilibrium solutions of problem 2.
• P02_FUN evaluates the function for problem 2.
• P02_JAC evaluates the jacobian for problem 2.
• P02_NEQN returns the number of equations for problem 2.
• P02_SCALE returns scale factors for problem 2.
• P02_START returns the starting point for problem 2.
• P02_STOP returns the stopping point for problem 2.
• P02_TITLE returns the title of problem 2.
• P03_AUTONOMOUS reports whether problem 3 is autonomous.
• P03_EQUIL returns equilibrium solutions of problem 3.
• P03_FUN evaluates the function for problem 3.
• P03_JAC evaluates the jacobian for problem 3.
• P03_NEQN returns the number of equations for problem 3.
• P03_SCALE returns scale factors for problem 3.
• P03_START returns the starting point for problem 3.
• P03_STOP returns the stopping point for problem 3.
• P03_TITLE returns the title of problem 3.
• P04_AUTONOMOUS reports whether problem 4 is autonomous.
• P04_EQUIL returns equilibrium solutions of problem 4.
• P04_FUN evaluates the function for problem 4.
• P04_JAC evaluates the jacobian for problem 4.
• P04_NEQN returns the number of equations for problem 4.
• P04_SCALE returns scale factors for problem 4.
• P04_START returns the starting point for problem 4.
• P04_STOP returns the stopping point for problem 4.
• P04_TITLE returns the title of problem 4.
• P05_AUTONOMOUS reports whether problem 5 is autonomous.
• P05_EQUIL returns equilibrium solutions of problem 5.
• P05_FUN evaluates the function for problem 5.
• P05_JAC evaluates the jacobian for problem 5.
• P05_NEQN returns the number of equations for problem 5.
• P05_SCALE returns scale factors for problem 5.
• P05_START returns the starting point for problem 5.
• P05_STOP returns the stopping point for problem 5.
• P05_TITLE returns the title of problem 5.
• P06_AUTONOMOUS reports whether problem 6 is autonomous.
• P06_EQUIL returns equilibrium solutions of problem 6.
• P06_FUN evaluates the function for problem 6.
• P06_JAC evaluates the jacobian for problem 6.
• P06_NEQN returns the number of equations for problem 6.
• P06_SCALE returns scale factors for problem 6.
• P06_START returns the starting point for problem 6.
• P06_STOP returns the stopping point for problem 6.
• P06_TITLE returns the title of problem 6.
• P07_AUTONOMOUS reports whether problem 7 is autonomous.
• P07_EQUIL returns equilibrium solutions of problem 7.
• P07_FUN evaluates the function for problem 7.
• P07_JAC evaluates the jacobian for problem 7.
• P07_NEQN returns the number of equations for problem 7.
• P07_SCALE returns scale factors for problem 7.
• P07_START returns the starting point for problem 7.
• P07_STOP returns the stopping point for problem 7.
• P07_TITLE returns the title of problem 7.
• P08_AUTONOMOUS reports whether problem 8 is autonomous.
• P08_EQUIL returns equilibrium solutions of problem 8.
• P08_FUN evaluates the function for problem 8.
• P08_JAC evaluates the jacobian for problem 8.
• P08_NEQN returns the number of equations for problem 8.
• P08_SCALE returns scale factors for problem 8.
• P08_START returns the starting point for problem 8.
• P08_STOP returns the stopping point for problem 8.
• P08_TITLE returns the title of problem 8.
• P09_AUTONOMOUS reports whether problem 9 is autonomous.
• P09_EQUIL returns equilibrium solutions of problem 9.
• P09_FUN evaluates the function for problem 9.
• P09_JAC evaluates the jacobian for problem 9.
• P09_NEQN returns the number of equations for problem 9.
• P09_SCALE returns scale factors for problem 9.
• P09_START returns the starting point for problem 9.
• P09_STOP returns the stopping point for problem 9.
• P09_TITLE returns the title of problem 9.
• P10_AUTONOMOUS reports whether problem 10 is autonomous.
• P10_EQUIL returns equilibrium solutions of problem 10.
• P10_FUN evaluates the function for problem 10.
• P10_JAC evaluates the jacobian for problem 10.
• P10_NEQN returns the number of equations for problem 10.
• P10_SCALE returns scale factors for problem 10.
• P10_START returns the starting point for problem 10.
• P10_STOP returns the stopping point for problem 10.
• P10_TITLE returns the title of problem 10.
• P11_AUTONOMOUS reports whether problem 11 is autonomous.
• P11_EQUIL returns equilibrium solutions of problem 11.
• P11_FUN evaluates the function for problem 11.
• P11_JAC evaluates the jacobian for problem 11.
• P11_NEQN returns the number of equations for problem 11.
• P11_SCALE returns scale factors for problem 11.
• P11_START returns the starting point for problem 11.
• P11_STOP returns the stopping point for problem 11.
• P11_TITLE returns the title of problem 11.
• P12_AUTONOMOUS reports whether problem 12 is autonomous.
• P12_EQUIL returns equilibrium solutions of problem 12.
• P12_FUN evaluates the function for problem 12.
• P12_JAC evaluates the jacobian for problem 12.
• P12_NEQN returns the number of equations for problem 12.
• P12_SCALE returns scale factors for problem 12.
• P12_START returns the starting point for problem 12.
• P12_STOP returns the stopping point for problem 12.
• P12_TITLE returns the title of problem 12.
• P13_AUTONOMOUS reports whether problem 13 is autonomous.
• P13_EQUIL returns equilibrium solutions of problem 13.
• P13_FUN evaluates the function for problem 13.
• P13_JAC evaluates the jacobian for problem 13.
• P13_NEQN returns the number of equations for problem 13.
• P13_SCALE returns scale factors for problem 13.
• P13_START returns the starting point for problem 13.
• P13_STOP returns the stopping point for problem 13.
• P13_TITLE returns the title of problem 13.
• P14_AUTONOMOUS reports whether problem 14 is autonomous.
• P14_EQUIL returns equilibrium solutions of problem 14.
• P14_FUN evaluates the function for problem 14.
• P14_JAC evaluates the jacobian for problem 14.
• P14_NEQN returns the number of equations for problem 14.
• P14_SCALE returns scale factors for problem 14.
• P14_START returns the starting point for problem 14.
• P14_STOP returns the stopping point for problem 14.
• P14_TITLE returns the title of problem 14.
• P15_AUTONOMOUS reports whether problem 15 is autonomous.
• P15_EQUIL returns equilibrium solutions of problem 15.
• P15_FUN evaluates the function for problem 15.
• P15_JAC evaluates the jacobian for problem 15.
• P15_NEQN returns the number of equations for problem 15.
• P15_PARAM handles the parameters for problem 15.
• P15_SCALE returns scale factors for problem 15.
• P15_START returns the starting point for problem 15.
• P15_STOP returns the stopping point for problem 15.
• P15_TITLE returns the title of problem 15.
• P16_AUTONOMOUS reports whether problem 16 is autonomous.
• P16_EQUIL returns equilibrium solutions of problem 16.
• P16_FUN evaluates the function for problem 16.
• P16_JAC evaluates the jacobian for problem 16.
• P16_NEQN returns the number of equations for problem 16.
• P16_PARAM handles the parameters for problem 16.
• P16_SCALE returns scale factors for problem 16.
• P16_START returns the starting point for problem 16.
• P16_STOP returns the stopping point for problem 16.
• P16_TITLE returns the title of problem 16.
• P17_AUTONOMOUS reports whether problem 17 is autonomous.
• P17_EQUIL returns equilibrium solutions of problem 17.
• P17_FUN evaluates the function for problem 17.
• P17_JAC evaluates the jacobian for problem 17.
• P17_NEQN returns the number of equations for problem 17.
• P17_PARAM handles the parameters for problem 17.
• P17_SCALE returns scale factors for problem 17.
• P17_START returns the starting point for problem 17.
• P17_STOP returns the stopping point for problem 17.
• P17_TITLE returns the title of problem 17.
• P18_AUTONOMOUS reports whether problem 18 is autonomous.
• P18_EQUIL returns equilibrium solutions of problem 18.
• P18_FUN evaluates the function for problem 18.
• P18_JAC evaluates the jacobian for problem 18.
• P18_NEQN returns the number of equations for problem 18.
• P18_PARAM handles the parameters for problem 18.
• P18_SCALE returns scale factors for problem 18.
• P18_START returns the starting point for problem 18.
• P18_STOP returns the stopping point for problem 18.
• P18_TITLE returns the title of problem 18.
• P19_AUTONOMOUS reports whether problem 19 is autonomous.
• P19_EQUIL returns equilibrium solutions of problem 19.
• P19_FUN evaluates the function for problem 19.
• P19_JAC evaluates the jacobian for problem 19.
• P19_NEQN returns the number of equations for problem 19.
• P19_PARAM handles the parameters for problem 19.
• P19_SCALE returns scale factors for problem 19.
• P19_START returns the starting point for problem 19.
• P19_STOP returns the stopping point for problem 19.
• P19_TITLE returns the title of problem 19.
• P20_AUTONOMOUS reports whether problem 20 is autonomous.
• P20_EQUIL returns equilibrium solutions of problem 20.
• P20_FUN evaluates the function for problem 20.
• P20_JAC evaluates the jacobian for problem 20.
• P20_NEQN returns the number of equations for problem 20.
• P20_PARAM handles the parameters for problem 20.
• P20_SCALE returns scale factors for problem 20.
• P20_START returns the starting point for problem 20.
• P20_STOP returns the stopping point for problem 20.
• P20_TITLE returns the title of problem 20.
• P21_AUTONOMOUS reports whether problem 21 is autonomous.
• P21_EQUIL returns equilibrium solutions of problem 21.
• P21_FUN evaluates the function for problem 21.
• P21_JAC evaluates the jacobian for problem 21.
• P21_NEQN returns the number of equations for problem 21.
• P21_SCALE returns scale factors for problem 21.
• P21_START returns the starting point for problem 21.
• P21_STOP returns the stopping point for problem 21.
• P21_TITLE returns the title of problem 21.
• P22_AUTONOMOUS reports whether problem 22 is autonomous.
• P22_EQUIL returns equilibrium solutions of problem 22.
• P22_FUN evaluates the function for problem 22.
• P22_JAC evaluates the jacobian for problem 22.
• P22_NEQN returns the number of equations for problem 22.
• P22_SCALE returns scale factors for problem 22.
• P22_START returns the starting point for problem 22.
• P22_STOP returns the stopping point for problem 22.
• P22_TITLE returns the title of problem 22.
• P23_AUTONOMOUS reports whether problem 23 is autonomous.
• P23_EQUIL returns equilibrium solutions of problem 23.
• P23_FUN evaluates the function for problem 23.
• P23_JAC evaluates the jacobian for problem 23.
• P23_NEQN returns the number of equations for problem 23.
• P23_SCALE returns scale factors for problem 23.
• P23_START returns the starting point for problem 23.
• P23_STOP returns the stopping point for problem 23.
• P23_TITLE returns the title of problem 23.
• P24_AUTONOMOUS reports whether problem 24 is autonomous.
• P24_EQUIL returns equilibrium solutions of problem 24.
• P24_FUN evaluates the function for problem 24.
• P24_JAC evaluates the jacobian for problem 24.
• P24_NEQN returns the number of equations for problem 24.
• P24_SCALE returns scale factors for problem 24.
• P24_START returns the starting point for problem 24.
• P24_STOP returns the stopping point for problem 24.
• P24_TITLE returns the title of problem 24.
• P25_AUTONOMOUS reports whether problem 25 is autonomous.
• P25_EQUIL returns equilibrium solutions of problem 25.
• P25_FUN evaluates the function for problem 25.
• P25_JAC evaluates the jacobian for problem 25.
• P25_NEQN returns the number of equations for problem 25.
• P25_SCALE returns scale factors for problem 25.
• P25_START returns the starting point for problem 25.
• P25_STOP returns the stopping point for problem 25.
• P25_TITLE returns the title of problem 25.
• P26_AUTONOMOUS reports whether problem 26 is autonomous.
• P26_EQUIL returns equilibrium solutions of problem 26.
• P26_FUN evaluates the function for problem 26.
• P26_JAC evaluates the jacobian for problem 26.
• P26_NEQN returns the number of equations for problem 26.
• P26_PARAM handles the parameters for problem 26.
• P26_SCALE returns scale factors for problem 26.
• P26_START returns the starting point for problem 26.
• P26_STOP returns the stopping point for problem 26.
• P26_TITLE returns the title of problem 26.
• P27_AUTONOMOUS reports whether problem 27 is autonomous.
• P27_EQUIL returns equilibrium solutions of problem 27.
• P27_FUN evaluates the function for problem 27.
• P27_JAC evaluates the jacobian for problem 27.
• P27_NEQN returns the number of equations for problem 27.
• P27_SCALE returns scale factors for problem 27.
• P27_START returns the starting point for problem 27.
• P27_STOP returns the stopping point for problem 27.
• P27_TITLE returns the title of problem 27.
• P28_AUTONOMOUS reports whether problem 14 is autonomous.
• P28_EQUIL returns equilibrium solutions of problem 28.
• P28_FUN evaluates the function for problem 28.
• P28_JAC evaluates the jacobian for problem 28.
• P28_NEQN returns the number of equations for problem 28.
• P28_SCALE returns scale factors for problem 28.
• P28_START returns the starting point for problem 28.
• P28_STOP returns the stopping point for problem 28.
• P28_TITLE returns the title of problem 28.
• P29_AUTONOMOUS reports whether problem 29 is autonomous.
• P29_EQUIL returns equilibrium solutions of problem 29.
• P29_FUN evaluates the function for problem 29.
• P29_JAC evaluates the jacobian for problem 29.
• P29_NEQN returns the number of equations for problem 29.
• P29_SCALE returns scale factors for problem 29.
• P29_START returns the starting point for problem 29.
• P29_STOP returns the stopping point for problem 29.
• P29_TITLE returns the title of problem 29.
• P30_AUTONOMOUS reports whether problem 30 is autonomous.
• P30_EQUIL returns equilibrium solutions of problem 30.
• P30_FUN evaluates the function for problem 30.
• P30_JAC evaluates the jacobian for problem 30.
• P30_NEQN returns the number of equations for problem 30.
• P30_SCALE returns scale factors for problem 30.
• P30_START returns the starting point for problem 30.
• P30_STOP returns the stopping point for problem 30.
• P30_TITLE returns the title of problem 30.
• P31_AUTONOMOUS reports whether problem 31 is autonomous.
• P31_EQUIL returns equilibrium solutions of problem 31.
• P31_FUN evaluates the function for problem 31.
• P31_JAC evaluates the jacobian for problem 31.
• P31_NEQN returns the number of equations for problem 31.
• P31_PARAM handles the parameters for problem 31.
• P31_SCALE returns scale factors for problem 31.
• P31_START returns the starting point for problem 31.
• P31_STOP returns the stopping point for problem 31.
• P31_TITLE returns the title of problem 31.
• P32_AUTONOMOUS reports whether problem 32 is autonomous.
• P32_EQUIL returns equilibrium solutions of problem 32.
• P32_FUN evaluates the function for problem 32.
• P32_JAC evaluates the jacobian for problem 32.
• P32_NEQN returns the number of equations for problem 32.
• P32_PARAM handles the parameters for problem 32.
• P32_SCALE returns scale factors for problem 32.
• P32_START returns the starting point for problem 32.
• P32_STOP returns the stopping point for problem 32.
• P32_TITLE returns the title of problem 32.
• P33_AUTONOMOUS reports whether problem 33 is autonomous.
• P33_EQUIL returns equilibrium solutions of problem 33.
• P33_FUN evaluates the function for problem 33.
• P33_JAC evaluates the jacobian for problem 33.
• P33_NEQN returns the number of equations for problem 33.
• P33_PARAM handles the parameters for problem 33.
• P33_SCALE returns scale factors for problem 33.
• P33_START returns the starting point for problem 33.
• P33_STOP returns the stopping point for problem 33.
• P33_TITLE returns the title of problem 33.
• P34_AUTONOMOUS reports whether problem 34 is autonomous.
• P34_EQUIL returns equilibrium solutions of problem 34.
• P34_FUN evaluates the function for problem 34.
• P34_JAC evaluates the jacobian for problem 34.
• P34_NEQN returns the number of equations for problem 34.
• P34_PARAM handles the parameters for problem 34.
• P34_SCALE returns scale factors for problem 34.
• P34_START returns the starting point for problem 34.
• P34_STOP returns the stopping point for problem 34.
• P34_TITLE returns the title of problem 34.
• P35_AUTONOMOUS reports whether problem 35 is autonomous.
• P35_EQUIL returns equilibrium solutions of problem 35.
• P35_FUN evaluates the function for problem 35.
• P35_JAC evaluates the jacobian for problem 35.
• P35_NEQN returns the number of equations for problem 35.
• P35_PARAM handles the parameters for problem 35.
• P35_SCALE returns scale factors for problem 35.
• P35_START returns the starting point for problem 35.
• P35_STOP returns the stopping point for problem 35.
• P35_TITLE returns the title of problem 35.
• P36_AUTONOMOUS reports whether problem 36 is autonomous.
• P36_EQUIL returns equilibrium solutions of problem 36.
• P36_FUN evaluates the function for problem 36.
• P36_JAC evaluates the jacobian for problem 36.
• P36_NEQN returns the number of equations for problem 36.
• P36_SCALE returns scale factors for problem 36.
• P36_START returns the starting point for problem 36.
• P36_STOP returns the stopping point for problem 36.
• P36_TITLE returns the title of problem 36.
• P37_AUTONOMOUS reports whether problem 37 is autonomous.
• P37_EQUIL returns equilibrium solutions of problem 37.
• P37_FUN evaluates the function for problem 37.
• P37_JAC evaluates the jacobian for problem 37.
• P37_NEQN returns the number of equations for problem 37.
• P37_PARAM handles the parameters for problem 37.
• P37_SCALE returns scale factors for problem 37.
• P37_START returns the starting point for problem 37.
• P37_STOP returns the stopping point for problem 37.
• P37_TITLE returns the title of problem 37.
• P38_AUTONOMOUS reports whether problem 38 is autonomous.
• P38_EQUIL returns equilibrium solutions of problem 38.
• P38_FUN evaluates the function for problem 38.
• P38_JAC evaluates the jacobian for problem 38.
• P38_NEQN returns the number of equations for problem 38.
• P38_PARAM handles the parameters for problem 38.
• P38_SCALE returns scale factors for problem 38.
• P38_START returns the starting point for problem 38.
• P38_STOP returns the stopping point for problem 38.
• P38_TITLE returns the title of problem 38.
• P39_AUTONOMOUS reports whether problem 39 is autonomous.
• P39_EQUIL returns equilibrium solutions of problem 39.
• P39_FUN evaluates the function for problem 39.
• P39_JAC evaluates the jacobian for problem 39.
• P39_NEQN returns the number of equations for problem 39.
• P39_PARAM handles the parameters for problem 39.
• P39_SCALE returns scale factors for problem 39.
• P39_START returns the starting point for problem 39.
• P39_STOP returns the stopping point for problem 39.
• P39_TITLE returns the title of problem 39.
• R8_CUBE_ROOT returns the cube root of a real number.
• S_EQI is a case insensitive comparison of two strings for equality.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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