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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)))))
(*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 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)
(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 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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