ODE(1) GNU Plotting Utilities ODE(1)NAMEode - numerical solution of ordinary differential equa
tions
SYNOPSISode [ options ] [ file ]
DESCRIPTIONode 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 derivatives 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 evo
lution of a variable. If there are only two columns, the
output can be piped to graph(1) or a similar plotting pro
gram.
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 program fragment
that defines the system of differential equations.
Output Options
-p prec
--precision prec
When printing numerical results, use prec signifi
cant digits (the default is 6). If this option is
given, the print format will be scientific nota
tion.
-t
--title
Print a title line at the head of the output,
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naming the variables in each column. If this
option is given, the print format will be scien
tific 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 step
size is specified. When a constant 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-correc
tor scheme, with an adaptive stepsize unless a con
stant 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 con
stant 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 numer
ical 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 routine 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
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relative single-step error in any dependent vari
able will never exceed rmax (the default for which
is 10^-9). If this should occur, the solution will
be abandoned 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)" fol
lows 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
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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 output. 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.
AUTHORSode was written by Nicholas B. Tufillaro (nbt@reed.edu),
and slightly enhanced by Robert S. Maier (rsm@math.ari
zona.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.
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