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# Example

`Integrate-system' integrates the system

y_k^^ = f_k(y_1, y_2, ..., y_n), 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 y_1, ..., y_n) 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 dv_C / dt = -i_L - v_C / R

L di_L / dt = v_C

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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