ODE(1)							      GNU Plotting Utilities							    ODE(1)

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 deriva-
       tives 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 pro-
	      gram 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 con-
	      stant 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 rou-
	      tine 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 aban-
	      doned 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 out-
       put.  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.

SEE ALSO

       "The GNU Plotting Utilities Manual".

BUGS

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

FSF
								     Dec 1998								    ODE(1)