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ODE(1)                 GNU Plotting Utilities                 ODE(1)

       ode - numerical solution of ordinary differential equations

       ode [ options ] [ file ]

       ode is a tool that solves, by numerical integration, the ini‐
       tial 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 evolution of a
       variable.  If there are only two columns, the output  can  be
       piped to graph(1) or a similar plotting program.

   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.

              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-preci‐
              sion floating point number.

              The  optional  argument hmax, if included, specifies a
              maximum value for the stepsize.  It is useful in  pre‐
              venting  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 sin‐
              gle-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
              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

       -e emax [emin]
       --absolute-error-bound emax [emin]
              Similar to -r, but bounds the absolute rather than the
              relative single-step error.

              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.

              Print the version number of ode and the plotting util‐
              ities package, and exit.

       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.  Subse‐
       quently, 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

       Run-time errors elicit a message describing the problem,  and
       the solution is abandoned.

       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 num‐
       bers  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.

       ode was written by Nicholas B. Tufillaro (,  and
       slightly  enhanced  by Robert S. Maier (
       to merge it into the GNU plotting utilities.

       "The GNU Plotting Utilities Manual".

       Email bug reports to

FSF                           Dec 1998                        ODE(1)
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