ODE(1) GNU Plotting Utilities ODE(1)
**NAME**

ode - numerical solution of ordinary differential equations
**SYNOPSIS**

ode [ options ] [ file ]
**DESCRIPTION**

ode is a tool that solves, by numerical integration, the initial value problem for a specified system of first-order ordinary differential
equations. Three distinct numerical integration schemes are available: Runge-Kutta-Fehlberg (the default), Adams-Moulton, and Euler. The
Adams-Moulton and Runge-Kutta schemes are available with adaptive step size.
The operation of ode is specified by a program, written in its input language. The program is simply a list of expressions for the deriva-
tives of the variables to be integrated, together with some control statements. Some examples are given in the EXAMPLES section.
ode reads the program from the specified file, or from standard input if no file name is given. If reading from standard input, ode will
stop reading and exit when it sees a single period on a line by itself.
At each time step, the values of variables specified in the program are written to standard output. So a table of values will be produced,
with each column showing the evolution of a variable. If there are only two columns, the output can be piped to graph(1) or a similar
plotting program.
**OPTIONS**

Input Options
*-f* file
*--input-file* file
Read input from file before reading from standard input. This option makes it possible to work interactively, after reading a pro-
gram fragment that defines the system of differential equations.
Output Options
*-p* prec
*--precision* prec
When printing numerical results, use prec significant digits (the default is 6). If this option is given, the print format will be
scientific notation.
*-t
* *--title
* Print a title line at the head of the output, naming the variables in each column. If this option is given, the print format will
be scientific notation.
Integration Scheme Options
The following options specify the numerical integration scheme. Only one of the three basic options *-R*, *-A*, *-E* may be specified. The
default is *-R* (Runge-Kutta-Fehlberg).
*-R* [stepsize]
*--runge-kutta* [stepsize]
Use a fifth-order Runge-Kutta-Fehlberg algorithm, with an adaptive stepsize unless a constant stepsize is specified. When a con-
stant stepsize is specified and no error analysis is requested, then a classical fourth-order Runge-Kutta scheme is used.
*-A* [stepsize]
*--adams-moulton* [stepsize]
Use a fourth-order Adams-Moulton predictor-corrector scheme, with an adaptive stepsize unless a constant stepsize, stepsize, is
specified. The Runge-Kutta-Fehlberg algorithm is used to get past `bad' points (if any).
*-E* [stepsize]
*--euler* [stepsize]
Use a `quick and dirty' Euler scheme, with a constant stepsize. The default value of stepsize is 0.1. Not recommended for serious
applications.
The error bound options *-r* and *-e* (see below) may not be used if *-E* is specified.
*-h* hmin [hmax]
*--step-size-bound* hmin [hmax]
Use a lower bound hmin on the stepsize. The numerical scheme will not let the stepsize go below hmin. The default is to allow the
stepsize to shrink to the machine limit, i.e., the minimum nonzero double-precision floating point number.
The optional argument hmax, if included, specifies a maximum value for the stepsize. It is useful in preventing the numerical rou-
tine from skipping quickly over an interesting region.
Error Bound Options
*-r* rmax [rmin]
*--relative-error-bound* rmax [rmin]
The *-r* option sets an upper bound on the relative single-step error. If the *-r* option is used, the relative single-step error in
any dependent variable will never exceed rmax (the default for which is 10^-9). If this should occur, the solution will be aban-
doned and an error message will be printed. If the stepsize is not constant, the stepsize will be decreased `adaptively', so that
the upper bound on the single-step error is not violated. Thus, choosing a smaller upper bound on the single-step error will cause
smaller stepsizes to be chosen. A lower bound rmin may optionally be specified, to suggest when the stepsize should be increased
(the default for rmin is rmax/1000).
*-e* emax [emin]
*--absolute-error-bound* emax [emin]
Similar to *-r*, but bounds the absolute rather than the relative single-step error.
*-s
* *--suppress-error-bound
* Suppress the ceiling on single-step error, allowing ode to continue even if this ceiling is exceeded. This may result in large
numerical errors.
Informational Options
*--help* Print a list of command-line options, and exit.
*--version
* Print the version number of ode and the plotting utilities package, and exit.
**DIAGNOSTICS**

Mostly self-explanatory. The biggest exception is `syntax error', meaning there is a grammatical error. Language error messages are of
the form
ode: nnn: message...
where `nnn' is the number of the input line containing the error. If the *-f* option is used, the phrase "(file)" follows the `nnn' for
errors encountered inside the file. Subsequently, when ode begins reading the standard input, line numbers start over from 1.
No effort is made to recover successfully from syntactic errors in the input. However, there is a meager effort to resynchronize so more
than one error can be found in one scan.
Run-time errors elicit a message describing the problem, and the solution is abandoned.
**EXAMPLES**

The program
y' = y
y = 1
print t, y
step 0, 1
solves an initial value problem whose solution is y=e^t. When ode runs this program, it will write two columns of numbers to standard out-
put. Each line will show the value of the independent variable t, and the variable y, as t is stepped from 0 to 1.
A more sophisticated example would be
sine' = cosine
cosine' = *-sine
* sine = 0
cosine = 1
print t, sine
step 0, 2*PI
This program solves an initial value problem for a system of two differential equations. The initial value problem turns out to define the
sine and cosine functions. The program steps the system over a full period.
**AUTHORS**

ode was written by Nicholas B. Tufillaro (nbt@reed.edu), and slightly enhanced by Robert S. Maier (rsm@math.arizona.edu) to merge it into
the GNU plotting utilities.
**SEE ALSO**

"The GNU Plotting Utilities Manual".
**BUGS**

Email bug reports to bug-gnu-utils@gnu.org.
**FSF**

Dec 1998 ODE(1)