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<html>
<head>
<title>
TEST_ODE - Initial Value Test Problems
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TEST_ODE <br> Test Problems for <br> Initial Value Solvers
</h1>
<hr>
<p>
<b>TEST_ODE</b>
is a MATLAB library which
defines test initial value problems for ordinary
differential equations.
</p>
<p>
These problems have the common form:
<blockquote>
determine (some values of) the function <b>Y(T)</b>
</blockquote>
given
<blockquote>
<b>Y'(T) = F(T,Y)</b>
</blockquote>
with initial value
<blockquote>
<b>Y(T<sub>0</sub>) = Y<sub>0</sub></b>
</blockquote>
</p>
<p>
A uniform interface makes it easy to solve all the problems
automatically, or to concentrate on a single one.
</p>
<p>
The problems include:
<ol>
<li>
Enright and Pryce #A1,<br>
neqn = 1,<br>
y' = -y;
</li>
<li>
Enright and Pryce #A2,<br>
neqn = 1,<br>
y' = -y^3/2;
</li>
<li>
Enright and Pryce #A3,<br>
neqn = 1,<br>
y' = cos(t)*y;
</li>
<li>
Enright and Pryce #A4,<br>
neqn = 1,<br>
y' = y*(20-y)/80;
</li>
<li>
Enright and Pryce #A5,<br>
neqn = 1,<br>
y' = (y-t)/(y+t);
</li>
<li>
Enright and Pryce #B1,<br>
neqn = 2,<br>
y1' = 2*y1*(1-y2)<br>
y2' = - y2*(1-y1);
</li>
<li>
Enright and Pryce #B2,<br>
neqn = 3,<br>
y1' = -y1+y2<br>
y2' = y1-2*y2+y3<br>
y3' = y2-y3;
</li>
<li>
Enright and Pryce #B3,<br>
neqn = 3,<br>
y1' = -y1<br>
y2' = y1-y2^2<br>
y3' = y2^2;
</li>
<li>
Enright and Pryce #B4,<br>
neqn = 3,<br>
y1' = (-y2-y1*y3) / sqrt(y1^2+y2^2)<br>
y2' = (y1-y2*y3) / sqrt(y1^2+y2^2)<br>
y3' = y1 / sqrt(y1^2+y2^2);
</li>
<li>
Enright and Pryce #B5,<br>
neqn = 3,<br>
y1' = y2*y3<br>
y2' = -y1*y3<br>
y3' = -0.51*y1*y2;
</li>
<li>
Enright and Pryce #C1,<br>
neqn = 10;
</li>
<li>
Enright and Pryce #C2,<br>
neqn = 10;
</li>
<li>
Enright and Pryce #C3,<br>
neqn = 10;
</li>
<li>
Enright and Pryce #C4,<br>
neqn = 51;
</li>
<li>
Enright and Pryce #C5,<br>
neqn = 30;
</li>
<li>
Enright and Pryce #D1,<br>
neqn = 4;
</li>
<li>
Enright and Pryce #D2,<br>
neqn = 4;
</li>
<li>
Enright and Pryce #D3,<br>
neqn = 4;
</li>
<li>
Enright and Pryce #D4,<br>
neqn = 4;
</li>
<li>
Enright and Pryce #D5,<br>
neqn = 4;
</li>
<li>
Enright and Pryce #E1,<br>
neqn = 2;
</li>
<li>
Enright and Pryce #E2,<br>
neqn = 2;
</li>
<li>
Enright and Pryce #E3,<br>
neqn = 2;
</li>
<li>
Enright and Pryce #E4,<br>
neqn = 2;
</li>
<li>
Enright and Pryce #E5,<br>
neqn = 2;
</li>
<li>
Enright and Pryce #F1,<br>
neqn = 2;
</li>
<li>
Enright and Pryce #F2,<br>
neqn = 1;
</li>
<li>
Enright and Pryce #F3,<br>
neqn = 2;
</li>
<li>
Enright and Pryce #F4,<br>
neqn = 1;
</li>
<li>
Enright and Pryce #F5,<br>
neqn = 1;
</li>
<li>
Lotka-Volterra Predator-Prey Equations,<br>
neqn = 2;
</li>
<li>
The Lorenz System,<br>
neqn = 3;
</li>
<li>
The Van der Pol equation,<br>
neqn = 2;
</li>
<li>
The Linearized Damped Pendulum,<br>
neqn = 2;
</li>
<li>
The Nonlinear Damped Pendulum,<br>
neqn = 2;
</li>
<li>
Duffing's Equation,<br>
neqn = 2,
</li>
<li>
Duffing's Equation with Damping and Forcing,<br>
neqn = 2;
</li>
<li>
Shampine's Ball of Flame,<br>
neqn = 1,<br>
y' = y^2-y^3;
</li>
<li>
Polking's First Order System,<br>
neqn = 1,<br>
y' = y^2-a*t+b;
</li>
<li>
the Knee problem,<br>
neqn = 1,<br>
y' = y*(y-t)/eps;
</li>
</ol>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>TEST_ODE</b> is available in
<a href = "../../f_src/test_ode/test_ode.html">a FORTRAN90 version</a> and
<a href = "../../m_src/test_ode/test_ode.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/dfield8/dfield8.html">
DFIELD8</a>,
a MATLAB program which
interactively displays the direction field of a differential equation,
by John Polking.
</p>
<p>
<a href = "../../m_src/flame_ode/flame_ode.html">
FLAME_ODE</a>,
a MATLAB library which
considers an ordinary differential equation (ODE) which models
the growth of a ball of flame in a combustion process.
</p>
<p>
<a href = "../../m_src/pplane8/pplane8.html">
PPLANE8</a>,
a MATLAB program which
interactively displays the phase plane of a differential equation,
by John Polking.
</p>
<p>
<a href = "../../m_src/rkf45/rkf45.html">
RKF45</a>,
a MATLAB library which
implements the Runge-Kutta-Fehlberg ODE solver.
</p>
<h3 align = "center">
References:
</h3>
<p>
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.)
<ol>
<li>
David Arnold, John Polking,<br>
Ordinary Differential Equations using Matlab,<br>
Prentice Hall, 1999.
</li>
<li>
Wayne Enright, John Pryce,<br>
Algorithm 648,<br>
NSDTST and STDTST,<br>
ACM Transactions on Mathematical Software,<br>
Volume 13, Number 1, 1987, pages 28-34.
</li>
<li>
T Hull, Wayne Enright, B Fellen, A Sedgwick,<br>
Comparing numerical methods for ordinary differential equations,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 9, 1972, pages 603-637.
<li>
</li>
Cleve Moler,<br>
Cleve's Corner: Stiff Differential Equations,<br>
MATLAB News and Notes,<br>
May 2003, pages 12-13.
</li>
<li>
<a href = "http://pitagora.dm.uniba.it/~testset/">
http://pitagora.dm.uniba.it/~testset/</a>,
Test Set for IVP Solvers.
</li>
<li>
<a href =
"http://www.unige.ch/math/~hairer/testset/testset.html">
http://www.unige.ch/math/~hairer/testset/testset.html</a>
Stiff ODE test set of Hairer and Wanner.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "ch_cap.m">ch_cap.m</a>,
capitalizes a character.
</li>
<li>
<a href = "euler_test.m">euler_test.m</a>,
uses Euler's method on a test problem.
</li>
<li>
<a href = "p00_autonomous.m">p00_autonomous.m</a>,
returns 1 if a given problem is autonomous.
</li>
<li>
<a href = "p00_equil.m">p00_equil.m</a>,
returns the equilibrium solutions for any problem.
</li>
<li>
<a href = "p00_euler_step.m">p00_euler_step.m</a>,
takes a single Euler step from (T0,Y0) to (T1,Y1).
</li>
<li>
<a href = "p00_fun.m">p00_fun.m</a>,
returns the right hand side of any problem.
</li>
<li>
<a href = "p00_jac.m">p00_jac.m</a>,
returns the jacobian for any test problem.
</li>
<li>
<a href = "p00_neqn.m">p00_neqn.m</a>,
returns the number of equations for any problem.
</li>
<li>
<a href = "p00_param_default.m">p00_param_default.m</a>,
defaults the parameters for any problem.
</li>
<li>
<a href = "p00_rk_step.m">p00_rk_step.m</a>,
takes a single Runge-Kutta step from (T0,Y0) to (T1,Y1).
</li>
<li>
<a href = "p00_scale.m">p00_scale.m</a>,
returns the scale vector for any test problem.
</li>
<li>
<a href = "p00_start.m">p00_start.m</a>,
returns the initial data for any test problem.
</li>
<li>
<a href = "p00_stop.m">p00_stop.m</a>,
returns the end data for any test problem.
</li>
<li>
<a href = "p00_test_num.m">p00_test_num.m</a>,
returns the number of problems available.
</li>
<li>
<a href = "p00_title.m">p00_title.m</a>,
returns the title of any problem.
</li>
<li>
<a href = "p01_autonomous.m">p01_autonomous.m</a>,
returns 1 if problem p01 is autonomous.
</li>
<li>
<a href = "p01_equil.m">p01_equil.m</a>,
returns the equilibrium solutions for problem p01.
</li>
<li>
<a href = "p01_fun.m">p01_fun.m</a>,
returns the right hand side of problem p01.
</li>
<li>
<a href = "p01_jac.m">p01_jac.m</a>,
returns the jacobian for problem p01.
</li>
<li>
<a href = "p01_neqn.m">p01_neqn.m</a>,
returns the number of equations for problem p01.
</li>
<li>
<a href = "p01_scale.m">p01_scale.m</a>,
returns the scale vector for problem p01.
</li>
<li>
<a href = "p01_start.m">p01_start.m</a>,
returns the initial data for problem p01.
</li>
<li>
<a href = "p01_stop.m">p01_stop.m</a>,
returns the end data for problem p01.
</li>
<li>
<a href = "p01_title.m">p01_title.m</a>,
returns the title of problem p01.
</li>
<li>
<a href = "p02_autonomous.m">p02_autonomous.m</a>,
returns 1 if problem p02 is autonomous.
</li>
<li>
<a href = "p02_equil.m">p02_equil.m</a>,
returns the equilibrium solutions for problem p02.
</li>
<li>
<a href = "p02_fun.m">p02_fun.m</a>,
returns the right hand side of problem p02.
</li>
<li>
<a href = "p02_jac.m">p02_jac.m</a>,
returns the jacobian for problem p02.
</li>
<li>
<a href = "p02_neqn.m">p02_neqn.m</a>,
returns the number of equations for problem p02.
</li>
<li>
<a href = "p02_scale.m">p02_scale.m</a>,
returns the scale vector for problem p02.
</li>
<li>
<a href = "p02_start.m">p02_start.m</a>,
returns the initial data for problem p02.
</li>
<li>
<a href = "p02_stop.m">p02_stop.m</a>,
returns the end data for problem p02.
</li>
<li>
<a href = "p02_title.m">p02_title.m</a>,
returns the title of problem p02.
</li>
<li>
<a href = "p03_autonomous.m">p03_autonomous.m</a>,
returns 1 if problem p03 is autonomous.
</li>
<li>
<a href = "p03_equil.m">p03_equil.m</a>,
returns the equilibrium solutions for problem p03.
</li>
<li>
<a href = "p03_fun.m">p03_fun.m</a>,
returns the right hand side of problem p03.
</li>
<li>
<a href = "p03_jac.m">p03_jac.m</a>,
returns the jacobian for problem p03.
</li>
<li>
<a href = "p03_neqn.m">p03_neqn.m</a>,
returns the number of equations for problem p03.
</li>
<li>
<a href = "p03_scale.m">p03_scale.m</a>,
returns the scale vector for problem p03.
</li>
<li>
<a href = "p03_start.m">p03_start.m</a>,
returns the initial data for problem p03.
</li>
<li>
<a href = "p03_stop.m">p03_stop.m</a>,
returns the end data for problem p03.
</li>
<li>
<a href = "p03_title.m">p03_title.m</a>,
returns the title of problem p03.
</li>
<li>
<a href = "p04_autonomous.m">p04_autonomous.m</a>,
returns 1 if problem p04 is autonomous.
</li>
<li>
<a href = "p04_equil.m">p04_equil.m</a>,
returns the equilibrium solutions for problem p04.
</li>
<li>
<a href = "p04_fun.m">p04_fun.m</a>,
returns the right hand side of problem p04.
</li>
<li>
<a href = "p04_jac.m">p04_jac.m</a>,
returns the jacobian for problem p04.
</li>
<li>
<a href = "p04_neqn.m">p04_neqn.m</a>,
returns the number of equations for problem p04.
</li>
<li>
<a href = "p04_scale.m">p04_scale.m</a>,
returns the scale vector for problem p04.
</li>
<li>
<a href = "p04_start.m">p04_start.m</a>,
returns the initial data for problem p04.
</li>
<li>
<a href = "p04_stop.m">p04_stop.m</a>,
returns the end data for problem p04.
</li>
<li>
<a href = "p04_title.m">p04_title.m</a>,
returns the title of problem p04.
</li>
<li>
<a href = "p05_autonomous.m">p05_autonomous.m</a>,
returns 1 if problem p05 is autonomous.
</li>
<li>
<a href = "p05_equil.m">p05_equil.m</a>,
returns the equilibrium solutions for problem p05.
</li>
<li>
<a href = "p05_fun.m">p05_fun.m</a>,
returns the right hand side of problem p05.
</li>
<li>
<a href = "p05_jac.m">p05_jac.m</a>,
returns the jacobian for problem p05.
</li>
<li>
<a href = "p05_neqn.m">p05_neqn.m</a>,
returns the number of equations for problem p05.
</li>
<li>
<a href = "p05_scale.m">p05_scale.m</a>,
returns the scale vector for problem p05.
</li>
<li>
<a href = "p05_start.m">p05_start.m</a>,
returns the initial data for problem p05.
</li>
<li>
<a href = "p05_stop.m">p05_stop.m</a>,
returns the end data for problem p05.
</li>
<li>
<a href = "p05_title.m">p05_title.m</a>,
returns the title of problem p05.
</li>
<li>
<a href = "p06_autonomous.m">p06_autonomous.m</a>,
returns 1 if problem p06 is autonomous.
</li>
<li>
<a href = "p06_equil.m">p06_equil.m</a>,
returns the equilibrium solutions for problem p06.
</li>
<li>
<a href = "p06_fun.m">p06_fun.m</a>,
returns the right hand side of problem p06.
</li>
<li>
<a href = "p06_jac.m">p06_jac.m</a>,
returns the jacobian for problem p06.
</li>
<li>
<a href = "p06_neqn.m">p06_neqn.m</a>,
returns the number of equations for problem p06.
</li>
<li>
<a href = "p06_scale.m">p06_scale.m</a>,
returns the scale vector for problem p06.
</li>
<li>
<a href = "p06_start.m">p06_start.m</a>,
returns the initial data for problem p06.
</li>
<li>
<a href = "p06_stop.m">p06_stop.m</a>,
returns the end data for problem p06.
</li>
<li>
<a href = "p06_title.m">p06_title.m</a>,
returns the title of problem p06.
</li>
<li>
<a href = "p07_autonomous.m">p07_autonomous.m</a>,
returns 1 if problem p07 is autonomous.
</li>
<li>
<a href = "p07_equil.m">p07_equil.m</a>,
returns the equilibrium solutions for problem p07.
</li>
<li>
<a href = "p07_fun.m">p07_fun.m</a>,
returns the right hand side of problem p07.
</li>
<li>
<a href = "p07_jac.m">p07_jac.m</a>,
returns the jacobian for problem p07.
</li>
<li>
<a href = "p07_neqn.m">p07_neqn.m</a>,
returns the number of equations for problem p07.
</li>
<li>
<a href = "p07_scale.m">p07_scale.m</a>,
returns the scale vector for problem p07.
</li>
<li>
<a href = "p07_start.m">p07_start.m</a>,
returns the initial data for problem p07.
</li>
<li>
<a href = "p07_stop.m">p07_stop.m</a>,
returns the end data for problem p07.
</li>
<li>
<a href = "p07_title.m">p07_title.m</a>,
returns the title of problem p07.
</li>
<li>
<a href = "p08_autonomous.m">p08_autonomous.m</a>,
returns 1 if problem p08 is autonomous.
</li>
<li>
<a href = "p08_equil.m">p08_equil.m</a>,
returns the equilibrium solutions for problem p08.
</li>
<li>
<a href = "p08_fun.m">p08_fun.m</a>,
returns the right hand side of problem p08.
</li>
<li>
<a href = "p08_jac.m">p08_jac.m</a>,
returns the jacobian for problem p08.
</li>
<li>
<a href = "p08_neqn.m">p08_neqn.m</a>,
returns the number of equations for problem p08.
</li>
<li>
<a href = "p08_scale.m">p08_scale.m</a>,
returns the scale vector for problem p08.
</li>
<li>
<a href = "p08_start.m">p08_start.m</a>,
returns the initial data for problem p08.
</li>
<li>
<a href = "p08_stop.m">p08_stop.m</a>,
returns the end data for problem p08.
</li>
<li>
<a href = "p08_title.m">p08_title.m</a>,
returns the title of problem p08.
</li>
<li>
<a href = "p09_autonomous.m">p09_autonomous.m</a>,
returns 1 if problem p09 is autonomous.
</li>
<li>
<a href = "p09_equil.m">p09_equil.m</a>,
returns the equilibrium solutions for problem p09.
</li>
<li>
<a href = "p09_fun.m">p09_fun.m</a>,
returns the right hand side of problem p09.
</li>
<li>
<a href = "p09_jac.m">p09_jac.m</a>,
returns the jacobian for problem p09.
</li>
<li>
<a href = "p09_neqn.m">p09_neqn.m</a>,
returns the number of equations for problem p09.
</li>
<li>
<a href = "p09_scale.m">p09_scale.m</a>,
returns the scale vector for problem p09.
</li>
<li>
<a href = "p09_start.m">p09_start.m</a>,
returns the initial data for problem p09.
</li>
<li>
<a href = "p09_stop.m">p09_stop.m</a>,
returns the end data for problem p09.
</li>
<li>
<a href = "p09_title.m">p09_title.m</a>,
returns the title of problem p09.
</li>
<li>
<a href = "p10_autonomous.m">p10_autonomous.m</a>,
returns 1 if problem p10 is autonomous.
</li>
<li>
<a href = "p10_equil.m">p10_equil.m</a>,
returns the equilibrium solutions for problem p10.
</li>
<li>
<a href = "p10_fun.m">p10_fun.m</a>,
returns the right hand side of problem p10.
</li>
<li>
<a href = "p10_jac.m">p10_jac.m</a>,
returns the jacobian for problem p10.
</li>
<li>
<a href = "p10_neqn.m">p10_neqn.m</a>,
returns the number of equations for problem p10.
</li>
<li>
<a href = "p10_scale.m">p10_scale.m</a>,
returns the scale vector for problem p10.
</li>
<li>
<a href = "p10_start.m">p10_start.m</a>,
returns the initial data for problem p10.
</li>
<li>
<a href = "p10_stop.m">p10_stop.m</a>,
returns the end data for problem p10.
</li>
<li>
<a href = "p10_title.m">p10_title.m</a>,
returns the title of problem p10.
</li>
<li>
<a href = "p11_autonomous.m">p11_autonomous.m</a>,
returns 1 if problem p11 is autonomous.
</li>
<li>
<a href = "p11_equil.m">p11_equil.m</a>,
returns the equilibrium solutions for problem p11.
</li>
<li>
<a href = "p11_fun.m">p11_fun.m</a>,
returns the right hand side of problem p11.
</li>
<li>
<a href = "p11_jac.m">p11_jac.m</a>,
returns the jacobian for problem p11.
</li>
<li>
<a href = "p11_neqn.m">p11_neqn.m</a>,
returns the number of equations for problem p11.
</li>
<li>
<a href = "p11_scale.m">p11_scale.m</a>,
returns the scale vector for problem p11.
</li>
<li>
<a href = "p11_start.m">p11_start.m</a>,
returns the initial data for problem p11.
</li>
<li>
<a href = "p11_stop.m">p11_stop.m</a>,
returns the end data for problem p11.
</li>
<li>
<a href = "p11_title.m">p11_title.m</a>,
returns the title of problem p11.
</li>
<li>
<a href = "p12_autonomous.m">p12_autonomous.m</a>,
returns 1 if problem p12 is autonomous.
</li>
<li>
<a href = "p12_equil.m">p12_equil.m</a>,
returns the equilibrium solutions for problem p12.
</li>
<li>
<a href = "p12_fun.m">p12_fun.m</a>,
returns the right hand side of problem p12.
</li>
<li>
<a href = "p12_jac.m">p12_jac.m</a>,
returns the jacobian for problem p12.
</li>
<li>
<a href = "p12_neqn.m">p12_neqn.m</a>,
returns the number of equations for problem p12.
</li>
<li>
<a href = "p12_scale.m">p12_scale.m</a>,
returns the scale vector for problem p12.
</li>
<li>
<a href = "p12_start.m">p12_start.m</a>,
returns the initial data for problem p12.
</li>
<li>
<a href = "p12_stop.m">p12_stop.m</a>,
returns the end data for problem p12.
</li>
<li>
<a href = "p12_title.m">p12_title.m</a>,
returns the title of problem p12.
</li>
<li>
<a href = "p13_autonomous.m">p13_autonomous.m</a>,
returns 1 if problem p13 is autonomous.
</li>
<li>
<a href = "p13_equil.m">p13_equil.m</a>,
returns the equilibrium solutions for problem p13.
</li>
<li>
<a href = "p13_fun.m">p13_fun.m</a>,
returns the right hand side of problem p13.
</li>
<li>
<a href = "p13_jac.m">p13_jac.m</a>,
returns the jacobian for problem p13.
</li>
<li>
<a href = "p13_neqn.m">p13_neqn.m</a>,
returns the number of equations for problem p13.
</li>
<li>
<a href = "p13_scale.m">p13_scale.m</a>,
returns the scale vector for problem p13.
</li>
<li>
<a href = "p13_start.m">p13_start.m</a>,
returns the initial data for problem p13.
</li>
<li>
<a href = "p13_stop.m">p13_stop.m</a>,
returns the end data for problem p13.
</li>
<li>
<a href = "p13_title.m">p13_title.m</a>,
returns the title of problem p13.
</li>
<li>
<a href = "p14_autonomous.m">p14_autonomous.m</a>,
returns 1 if problem p14 is autonomous.
</li>
<li>
<a href = "p14_equil.m">p14_equil.m</a>,
returns the equilibrium solutions for problem p14.
</li>
<li>
<a href = "p14_fun.m">p14_fun.m</a>,
returns the right hand side of problem p14.
</li>
<li>
<a href = "p14_jac.m">p14_jac.m</a>,
returns the jacobian for problem p14.
</li>
<li>
<a href = "p14_neqn.m">p14_neqn.m</a>,
returns the number of equations for problem p14.
</li>
<li>
<a href = "p14_scale.m">p14_scale.m</a>,
returns the scale vector for problem p14.
</li>
<li>
<a href = "p14_start.m">p14_start.m</a>,
returns the initial data for problem p14.
</li>
<li>
<a href = "p14_stop.m">p14_stop.m</a>,
returns the end data for problem p14.
</li>
<li>
<a href = "p14_title.m">p14_title.m</a>,
returns the title of problem p14.
</li>
<li>
<a href = "p15_autonomous.m">p15_autonomous.m</a>,
returns 1 if problem p15 is autonomous.
</li>
<li>
<a href = "p15_equil.m">p15_equil.m</a>,
returns the equilibrium solutions for problem p15.
</li>
<li>
<a href = "p15_fun.m">p15_fun.m</a>,
returns the right hand side of problem p15.
</li>
<li>
<a href = "p15_jac.m">p15_jac.m</a>,
returns the jacobian for problem p15.
</li>
<li>
<a href = "p15_neqn.m">p15_neqn.m</a>,
returns the number of equations for problem p15.
</li>
<li>
<a href = "p15_param.m">p15_param.m</a>,
manages parameters for problem p15.
</li>
<li>
<a href = "p15_scale.m">p15_scale.m</a>,
returns the scale vector for problem p15.
</li>
<li>
<a href = "p15_start.m">p15_start.m</a>,
returns the initial data for problem p15.
</li>
<li>
<a href = "p15_stop.m">p15_stop.m</a>,
returns the end data for problem p15.
</li>
<li>
<a href = "p15_title.m">p15_title.m</a>,
returns the title of problem p15.
</li>
<li>
<a href = "p16_autonomous.m">p16_autonomous.m</a>,
returns 1 if problem p16 is autonomous.
</li>
<li>
<a href = "p16_equil.m">p16_equil.m</a>,
returns the equilibrium solutions for problem p16.
</li>
<li>
<a href = "p16_fun.m">p16_fun.m</a>,
returns the right hand side of problem p16.
</li>
<li>
<a href = "p16_jac.m">p16_jac.m</a>,
returns the jacobian for problem p16.
</li>
<li>
<a href = "p16_neqn.m">p16_neqn.m</a>,
returns the number of equations for problem p16.
</li>
<li>
<a href = "p16_param.m">p16_param.m</a>,
manages parameters for problem p16.
</li>
<li>
<a href = "p16_scale.m">p16_scale.m</a>,
returns the scale vector for problem p16.
</li>
<li>
<a href = "p16_start.m">p16_start.m</a>,
returns the initial data for problem p16.
</li>
<li>
<a href = "p16_stop.m">p16_stop.m</a>,
returns the end data for problem p16.
</li>
<li>
<a href = "p16_title.m">p16_title.m</a>,
returns the title of problem p16.
</li>
<li>
<a href = "p17_autonomous.m">p17_autonomous.m</a>,
returns 1 if problem p17 is autonomous.
</li>
<li>
<a href = "p17_equil.m">p17_equil.m</a>,
returns the equilibrium solutions for problem p17.
</li>
<li>
<a href = "p17_fun.m">p17_fun.m</a>,
returns the right hand side of problem p17.
</li>
<li>
<a href = "p17_jac.m">p17_jac.m</a>,
returns the jacobian for problem p17.
</li>