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 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 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))
 |