211 lines
12 KiB
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211 lines
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<head>
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<title>
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Revised^6 Report on the Algorithmic Language Scheme
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</title>
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<link rel="stylesheet" type="text/css" href="r6rs-Z-S.css" title=default>
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<meta name=robots content="index,follow">
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</head>
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<body>
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<div id=slidecontent>
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<div align=right class=navigation>[Go to <span><a href="r6rs.html">first</a>, <a href="r6rs-Z-H-17.html">previous</a></span><span>, <a href="r6rs-Z-H-19.html">next</a></span> page<span>; </span><span><a href="r6rs-Z-H-2.html#node_toc_start">contents</a></span><span><span>; </span><a href="r6rs-Z-H-21.html#node_index_start">index</a></span>]</div>
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<p></p>
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<a name="node_chap_D"></a>
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<h1 class=chapter>
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<div class=chapterheading><a href="r6rs-Z-H-2.html#node_toc_node_chap_D">Appendix D</a></div><br>
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<a href="r6rs-Z-H-2.html#node_toc_node_chap_D">Example </a></h1>
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<p></p>
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<p>
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This section describes an example consisting of the
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<tt>(runge-kutta)</tt> library, which provides an <tt>integrate-system</tt>
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procedure that integrates the system
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</p>
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<div class=mathdisplay align=left><table><tr><td><em>y</em><sub><em>k</em></sub><sup>⁄</sup> = <em>f</em><sub><em>k</em></sub>(<em>y</em><sub>1</sub>, <em>y</em><sub>2</sub>, <tt>...</tt>, <em>y</em><sub><em>n</em></sub>), <em>k</em> = 1, <tt>...</tt>, <em>n</em></td></tr></table></div>
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<p class=noindent>
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of differential equations with the method of Runge-Kutta.</p>
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<p>
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As the <tt>(runge-kutta)</tt> library makes use of the <tt>(rnrs base (6))</tt>
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library, its skeleton is as follows:</p>
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<p>
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</p>
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<tt>#!r6rs<br>
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(library (runge-kutta)<br>
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(export integrate-system<br>
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head tail)<br>
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(import (rnrs base))<br>
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<library body>)<br>
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<p></tt></p>
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<p>
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The procedure definitions described below go in the place of <library body>.</p>
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<p>
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The parameter <tt>system-derivative</tt> is a function that takes a system
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state (a vector of values for the state variables <em>y</em><sub>1</sub>, <tt>...</tt>, <em>y</em><sub><em>n</em></sub>)
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and produces a system derivative (the values <em>y</em><sub>1</sub><sup>⁄</sup>, <tt>...</tt>,
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<em>y</em><sub><em>n</em></sub><sup>⁄</sup>). The parameter <tt>initial-state</tt> provides an initial
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system state, and <tt>h</tt> is an initial guess for the length of the
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integration step.</p>
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<p>
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The value returned by <tt>integrate-system</tt> is an infinite stream of
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system states.</p>
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<p>
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</p>
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<tt>(define integrate-system<br>
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(lambda (system-derivative initial-state h)<br>
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(let ((next (runge-kutta-4 system-derivative h)))<br>
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(letrec ((states<br>
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(cons initial-state<br>
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(lambda ()<br>
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(map-streams next states)))))<br>
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states))))<p></tt></p>
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<p>
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The <tt>runge-kutta-4</tt> procedure takes a function, <tt>f</tt>, that produces a
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system derivative from a system state. The <tt>runge-kutta-4</tt> procedure
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produces a function that takes a system state and
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produces a new system state.</p>
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<p>
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</p>
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<tt>(define runge-kutta-4<br>
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(lambda (f h)<br>
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(let ((*h (scale-vector h))<br>
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(*2 (scale-vector 2))<br>
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(*1/2 (scale-vector (/ 1 2)))<br>
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(*1/6 (scale-vector (/ 1 6))))<br>
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(lambda (y)<br>
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;; y is a system state<br>
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(let* ((k0 (*h (f y)))<br>
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(k1 (*h (f (add-vectors y (*1/2 k0)))))<br>
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(k2 (*h (f (add-vectors y (*1/2 k1)))))<br>
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(k3 (*h (f (add-vectors y k2)))))<br>
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(add-vectors y<br>
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(*1/6 (add-vectors k0<br>
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(*2 k1)<br>
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(*2 k2)<br>
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k3))))))))<br>
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<br>
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<br>
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(define elementwise<br>
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(lambda (f)<br>
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(lambda vectors<br>
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(generate-vector<br>
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(vector-length (car vectors))<br>
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(lambda (i)<br>
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(apply f<br>
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(map (lambda (v) (vector-ref v i))<br>
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vectors)))))))<br>
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<br>
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(define generate-vector<br>
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(lambda (size proc)<br>
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(let ((ans (make-vector size)))<br>
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(letrec ((loop<br>
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(lambda (i)<br>
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(cond ((= i size) ans)<br>
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(else<br>
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(vector-set! ans i (proc i))<br>
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(loop (+ i 1)))))))<br>
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(loop 0)))))<br>
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<br>
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(define add-vectors (elementwise +))<br>
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<br>
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(define scale-vector<br>
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(lambda (s)<br>
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(elementwise (lambda (x) (* x s)))))<p></tt></p>
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<p>
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The <tt>map-streams</tt> procedure is analogous to <tt>map</tt>: it applies its first
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argument (a procedure) to all the elements of its second argument (a
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stream).</p>
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<p>
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</p>
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<tt>(define map-streams<br>
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(lambda (f s)<br>
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(cons (f (head s))<br>
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(lambda () (map-streams f (tail s))))))<p></tt></p>
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<p>
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Infinite streams are implemented as pairs whose car holds the first
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element of the stream and whose cdr holds a procedure that delivers the rest
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of the stream.</p>
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<p>
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</p>
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<tt>(define head car)<br>
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(define tail<br>
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(lambda (stream) ((cdr stream))))<p></tt></p>
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<p>
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</p>
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<div class=bigskip></div>
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<p style="margin-top: 0pt; margin-bottom: 0pt">
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The following program illustrates the use of <tt>integrate-system</tt> in
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integrating the system
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</p>
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<div class=mathdisplay align=left><table><tr><td> <em>C</em> <em>d</em><em>v</em><sub><em>C</em></sub> / <em>d</em><em>t</em> = <tt>-</tt> <em>i</em><sub><em>L</em></sub> <tt>-</tt> <em>v</em><sub><em>C</em></sub> / <em>R</em></td></tr></table></div>
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<p class=noindent></p>
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<div class=mathdisplay align=left><table><tr><td> <em>L</em> <em>d</em><em>i</em><sub><em>L</em></sub> / <em>d</em><em>t</em> = <em>v</em><sub><em>C</em></sub></td></tr></table></div>
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<p class=noindent>
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which models a damped oscillator.</p>
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<p>
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</p>
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<tt>#!r6rs<br>
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(import (rnrs base)<br>
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(rnrs io simple)<br>
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(runge-kutta))<br>
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<br>
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(define damped-oscillator<br>
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(lambda (R L C)<br>
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(lambda (state)<br>
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(let ((Vc (vector-ref state 0))<br>
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(Il (vector-ref state 1)))<br>
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(vector (- 0 (+ (/ Vc (* R C)) (/ Il C)))<br>
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(/ Vc L))))))<br>
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<br>
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(define the-states<br>
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(integrate-system<br>
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(damped-oscillator 10000 1000 .001)<br>
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’#(1 0)<br>
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.01))<br>
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<br>
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(letrec ((loop (lambda (s)<br>
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(newline)<br>
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(write (head s))<br>
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(loop (tail s)))))<br>
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(loop the-states))<p></tt></p>
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<p>
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This prints output like the following:</p>
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<p>
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</p>
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<tt>#(1 0)<br>
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#(0.99895054 9.994835e-6)<br>
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#(0.99780226 1.9978681e-5)<br>
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#(0.9965554 2.9950552e-5)<br>
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#(0.9952102 3.990946e-5)<br>
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#(0.99376684 4.985443e-5)<br>
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#(0.99222565 5.9784474e-5)<br>
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#(0.9905868 6.969862e-5)<br>
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#(0.9888506 7.9595884e-5)<br>
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#(0.9870173 8.94753e-5)<br>
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<p></tt></p>
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<p>
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</p>
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<p></p>
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<div class=smallskip></div>
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<p style="margin-top: 0pt; margin-bottom: 0pt">
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<div align=right class=navigation>[Go to <span><a href="r6rs.html">first</a>, <a href="r6rs-Z-H-17.html">previous</a></span><span>, <a href="r6rs-Z-H-19.html">next</a></span> page<span>; </span><span><a href="r6rs-Z-H-2.html#node_toc_start">contents</a></span><span><span>; </span><a href="r6rs-Z-H-21.html#node_index_start">index</a></span>]</div>
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</p>
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<p></p>
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