# Appendix DExample

This section describes an example consisting of the (runge-kutta) library, which provides an integrate-system procedure that integrates the system

 yk⁄ = fk(y1, y2, ..., yn), k = 1, ..., n

of differential equations with the method of Runge-Kutta.

As the (runge-kutta) library makes use of the (rnrs base (6)) library, its skeleton is as follows:

#!r6rs
(library (runge-kutta)
(export integrate-system
(import (rnrs base))
<library body>)

The procedure definitions described below go in the place of <library body>.

The parameter system-derivative is a function that takes a system state (a vector of values for the state variables y1, ..., yn) and produces a system derivative (the values y1, ..., yn). The parameter initial-state provides an initial system state, and h is an initial guess for the length of the integration step.

The value returned by integrate-system is an infinite stream of system states.

(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))))

The runge-kutta-4 procedure takes a function, f, that produces a system derivative from a system state. The runge-kutta-4 procedure produces a function that takes a system state and produces a new system state.

(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 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)))))
(*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)))))

The map-streams procedure is analogous to map: it applies its first argument (a procedure) to all the elements of its second argument (a stream).

(define map-streams
(lambda (f s)
(cons (f (head s))
(lambda () (map-streams f (tail s))))))

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.

(define tail
(lambda (stream) ((cdr stream))))

The following program illustrates the use of integrate-system in integrating the system

 C dvC / dt = - iL - vC / R

 L diL / dt = vC

which models a damped oscillator.

#!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)
(loop (tail s)))))
(loop the-states))

This prints output like the following:

#(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)