Appendix D

Example

This section describes an example consisting of the (runge-kutta) library, which provides an integrate-system procedure that integrates the system

of differential equations with the method of Runge-Kutta.

As the (runge-kutta) library makes use of the (rnrs base (6)) library, its skeleton is as follows:

#!r6rs
(library (runge-kutta)
  (export integrate-system
          head tail)
  (import (rnrs base (6)))
  <library body>)

The procedure definitions described below go in the place of <library body>.

The parameter system-derivative is a function that takes a system state (a vector of values for the state variables y₁, ..., y_n) and produces a system derivative (the values y₁^⁄, ..., 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
                      (lambda ()
                        (map-streams next states)))))
        states))))

The runge-kutta-4 procedure takes a function, f, that produces a system derivative from a system state. The runge-kutta-4 procedure 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)))))

The map-streams procedure 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))
          (lambda () (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 procedure that delivers the rest of the stream.

(define head car)
(define tail
  (lambda (stream) ((cdr stream))))

The following program illustrates the use of integrate-system in integrating the system

which models a damped oscillator.

#!r6rs
(import (rnrs base (6))
        (rnrs io simple (6))
        (runge-kutta))

(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))

(letrec ((loop (lambda (s)
                 (newline)
                 (write (head s))
                 (loop (tail s)))))
  (loop the-states))

This prints output like the following:

#(1 0)
#(0.99895054 9.994835e-6)
#(0.99780226 1.9978681e-5)
#(0.9965554 2.9950552e-5)
#(0.9952102 3.990946e-5)
#(0.99376684 4.985443e-5)
#(0.99222565 5.9784474e-5)
#(0.9905868 6.969862e-5)
#(0.9888506 7.9595884e-5)
#(0.9870173 8.94753e-5)