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

yk = fk(y1, y2, ..., yn), k = 1, ..., n

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:


(library (runge-kutta)

  (export integrate-system

          head tail)

  (import (rnrs base))

  <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, ..., yn) and produces a system derivative (the values y1, ..., yn). 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)))))


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)


(define elementwise

  (lambda (f)

    (lambda vectors


        (vector-length (car vectors))

        (lambda (i)

          (apply f

                 (map (lambda (v) (vector-ref  v i))


(define generate-vector

  (lambda (size proc)

    (let ((ans (make-vector size)))

      (letrec ((loop

                (lambda (i)

                  (cond ((= i size) ans)


                         (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

= −iL

= vC

which models a damped oscillator.


(import (rnrs base)

        (rnrs io simple)


(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


     (damped-oscillator 10000 1000 .001)

     '#(1 0)


(letrec ((loop (lambda (s)


                 (write (head s))

                 (loop (tail s)))))

  (loop the-states))

(close-output-port (current-output-port))

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)