Revised(5) Report on the Algorithmic Language Scheme - Example

# 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))