Integrate-system integrates the system
y'k = fk(y1, y2, ..., yn), 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))))
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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)))))
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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))))))
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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))))
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The following illustrates the use of
integrate-system in
integrating the system
C dvC / dt = -iL - vC / R
L diL / dt = vC
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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