R5RsScheme Revised(5) Report on the Algorithmic Language Scheme -- Example
Integrate-system integrates the system

y'k = 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 y'1, ..., y'n). 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 (delay (map-streams next states))))) states))))
Runge-Kutta-4 takes a function, f, that produces a system derivative from a system state. Runge-Kutta-4 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))))) (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)))))
Map-streams 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)) (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 head car) (define tail (lambda (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 (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))

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Last update Sun Dec 31 08:32:06 2017.