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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 expres-
     sions 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 stan-
     dard  input  if no file name is given. If reading from stan-
     dard 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 evolu-
     tion of a variable.  If there are only two columns, the out-
     put 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 interac-
          tively, 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 rela-
          tive 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  `adap-
          tively',  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 step-
          sizes 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 utili-
          ties 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.  Sub-
     sequently,  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 ([email protected]), and
     slightly  enhanced by Robert S. Maier ([email protected])
     to merge it into the GNU plotting utilities.

SEE ALSO
     "The GNU Plotting Utilities Manual".

BUGS
     Email bug reports to [email protected].

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