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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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