Ubuntu Manpage: ode - numerical solution of ordinary differential equations

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 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
       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 program 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 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-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 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 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 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)" 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 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.

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.

BUGS

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