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\chapter{Example }
\label{exampleappendix}
\nobreak
This section describes an example consisting of the
\library{runge-kutta} library, which provides an {\cf integrate-system}
procedure that integrates the system
$$y_k^\prime = f_k(y_1, y_2, \ldots, y_n), \; k = 1, \ldots, n$$
of differential equations with the method of Runge-Kutta.
As the \library{runge-kutta} library makes use of the \rsixlibrary{base}
library, its skeleton is as follows:
\begin{scheme}
\#!r6rs
(library (runge-kutta)
(export integrate-system
head tail)
(import (rnrs base))
\hyper{library body})
\end{scheme}
The procedure definitions described below go in the place of \hyper{library body}.
The parameter {\tt system-derivative} is a function that takes a system
state (a vector of values for the state variables $y_1, \ldots, y_n$)
and produces a system derivative (the values $y_1^\prime, \ldots,
y_n^\prime$). The parameter {\tt initial-state} provides an initial
system state, and {\tt h} is an initial guess for the length of the
integration step.
The value returned by {\cf integrate-system} is an infinite stream of
system states.
\begin{schemenoindent}
(define integrate-system
(lambda (system-derivative initial-state h)
(let ((next (runge-kutta-4 system-derivative h)))
(letrec ((states
(cons initial-state
(lambda ()
(map-streams next states)))))
states))))%
\end{schemenoindent}
The {\cf runge-kutta-4} procedure takes a function, {\tt f}, that produces a
system derivative from a system state. The {\cf runge-kutta-4} procedure
produces a function that takes a system state and
produces a new system state.
\begin{schemenoindent}
(define runge-kutta-4
(lambda (f h)
(let ((*h (scale-vector h))
(*2 (scale-vector 2))
(*1/2 (scale-vector (/ 1 2)))
(*1/6 (scale-vector (/ 1 6))))
(lambda (y)
;; y {\rm{}is a system state}
(let* ((k0 (*h (f y)))
(k1 (*h (f (add-vectors y (*1/2 k0)))))
(k2 (*h (f (add-vectors y (*1/2 k1)))))
(k3 (*h (f (add-vectors y k2)))))
(add-vectors y
(*1/6 (add-vectors k0
(*2 k1)
(*2 k2)
k3))))))))
%|--------------------------------------------------|
(define elementwise
(lambda (f)
(lambda vectors
(generate-vector
(vector-length (car vectors))
(lambda (i)
(apply f
(map (lambda (v) (vector-ref v i))
vectors)))))))
%|--------------------------------------------------|
(define generate-vector
(lambda (size proc)
(let ((ans (make-vector size)))
(letrec ((loop
(lambda (i)
(cond ((= i size) ans)
(else
(vector-set! ans i (proc i))
(loop (+ i 1)))))))
(loop 0)))))
(define add-vectors (elementwise +))
(define scale-vector
(lambda (s)
(elementwise (lambda (x) (* x s)))))%
\end{schemenoindent}
The {\cf map-streams} procedure is analogous to {\cf map}: it applies its first
argument (a procedure) to all the elements of its second argument (a
stream).
\begin{schemenoindent}
(define map-streams
(lambda (f s)
(cons (f (head s))
(lambda () (map-streams f (tail s))))))%
\end{schemenoindent}
Infinite streams are implemented as pairs whose car holds the first
element of the stream and whose cdr holds a procedure that delivers the rest
of the stream.
\begin{schemenoindent}
(define head car)
(define tail
(lambda (stream) ((cdr stream))))%
\end{schemenoindent}
\bigskip
The following program illustrates the use of {\cf integrate-\hp{}system} in
integrating the system
$$ C {dv_C \over dt} = -i_L - {v_C \over R}$$\nobreak
$$ L {di_L \over dt} = v_C$$
which models a damped oscillator.
\begin{schemenoindent}
\#!r6rs
(import (rnrs base)
(rnrs io simple)
(runge-kutta))
(define damped-oscillator
(lambda (R L C)
(lambda (state)
(let ((Vc (vector-ref state 0))
(Il (vector-ref state 1)))
(vector (- 0 (+ (/ Vc (* R C)) (/ Il C)))
(/ Vc L))))))
(define the-states
(integrate-system
(damped-oscillator 10000 1000 .001)
'\#(1 0)
.01))
(letrec ((loop (lambda (s)
(newline)
(write (head s))
(loop (tail s)))))
(loop the-states))%
\end{schemenoindent}
This prints output like the following:
\begin{scheme}
\#(1 0)
\#(0.99895054 9.994835e-6)
\#(0.99780226 1.9978681e-5)
\#(0.9965554 2.9950552e-5)
\#(0.9952102 3.990946e-5)
\#(0.99376684 4.985443e-5)
\#(0.99222565 5.9784474e-5)
\#(0.9905868 6.969862e-5)
\#(0.9888506 7.9595884e-5)
\#(0.9870173 8.94753e-5)
\end{scheme}
%%% Local Variables:
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