Example

The procedure integrate-system integrates the system

 yk⁄ = fk(y1, y2, …, yn), k = 1, …, n

of differential equations with the method of Runge-Kutta.

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 system-derivative
initial-state
h)
(let ((next (runge-kutta-4 system-derivative h)))
(letrec ((states
(cons initial-state
(delay (map-streams next
states)))))
states)))
The procedure runge-kutta-4 takes a function, f, that produces a system derivative from a system state. It produces a function that takes a system state and produces a new system state.

(define (runge-kutta-4 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 f)
(lambda vectors
(generate-vector
(vector-length (car vectors))
(lambda (i)
(apply f
(map (lambda (v) (vector-ref  v i))
vectors))))))

(define (generate-vector 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 (scale-vector 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 f s)
(delay (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 promise to deliver the rest of the stream.

(define (tail stream)
(force (cdr stream)))

The following 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.

(define (damped-oscillator 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))