ODE(1)                GNU Plotting Utilities               ODE(1)



NAME
       ode  -  numerical  solution of ordinary differential equa-
       tions

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 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, nam-
              ing the variables in each column.  If  this  option
              is given, the print format will be scientific nota-
              tion.

   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
              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

              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.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.



FSF                          Dec 1998                      ODE(1)