# 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))

(close-output-port (current-output-port))

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)