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📁 scheme 标准（r5rs）。Scheme是MIT发布的基于Lambda运算的语言
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<!doctype html public "-//W3C//DTD HTML 4.0 Transitional//EN" "http://www.w3.org/TR/REC-html40/loose.dtd"><html><!-- Generated from TeX source by tex2page, v 4o4,      (c) Dorai Sitaram, http://www.cs.rice.edu/~dorai/tex2page --><head><title>Revised^5 Report on the Algorithmic Language Scheme</title><link rel="stylesheet" type="text/css" href="r5rs-Z-C.css" title=default><meta name=robots content="noindex,follow"></head><body><p><div class=navigation>[Go to <span><a href="r5rs.html">first</a>, <a href="r5rs-Z-H-12.html">previous</a></span><span>, <a href="r5rs-Z-H-14.html">next</a></span> page<span>; &nbsp;&nbsp;</span><span><a href="r5rs-Z-H-2.html#%_toc_start">contents</a></span><span><span>; &nbsp;&nbsp;</span><a href="r5rs-Z-H-15.html#%_index_start">index</a></span>]</div><p><a name="%_chap_Temp_9"></a><h1 class=chapter><div class=chapterheading>&nbsp;</div><p><a href="r5rs-Z-H-2.html#%_toc_%_chap_Temp_9">Example</a></h1><p><p><p> <tt>Integrate-system</tt> integrates the system <p><div align=left><img src="r5rs-Z-G-63.gif" border="0"></div><p>of differential equations with the method of Runge-Kutta.<p>The parameter <tt>system-derivative</tt> is a function that takes a systemstate (a vector of values for the state variables <em>y</em><sub>1</sub>, <tt>...</tt>, <em>y</em><sub><em>n</em></sub>)and produces a system derivative (the values <em>y</em><sub>1</sub><sup>/</sup>, <tt>...</tt>,<em>y</em><sub><em>n</em></sub><sup>/</sup>).  The parameter <tt>initial-state</tt> provides an initialsystem state, and <tt>h</tt> is an initial guess for the length of theintegration step.<p>The value returned by <tt>integrate-system</tt> is an infinite stream ofsystem states.<p><tt><p>(define&nbsp;integrate-system<br>&nbsp;&nbsp;(lambda&nbsp;(system-derivative&nbsp;initial-state&nbsp;h)<br>&nbsp;&nbsp;&nbsp;&nbsp;(let&nbsp;((next&nbsp;(runge-kutta-4&nbsp;system-derivative&nbsp;h)))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(letrec&nbsp;((states<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(cons&nbsp;initial-state<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(delay&nbsp;(map-streams&nbsp;next<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;states)))))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;states))))<p></tt><p><tt>Runge-Kutta-4</tt> takes a function, <tt>f</tt>, that produces asystem derivative from a system state.  <tt>Runge-Kutta-4</tt>produces a function that takes a system state andproduces a new system state.<p><tt><p>(define&nbsp;runge-kutta-4<br>&nbsp;&nbsp;(lambda&nbsp;(f&nbsp;h)<br>&nbsp;&nbsp;&nbsp;&nbsp;(let&nbsp;((*h&nbsp;(scale-vector&nbsp;h))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(*2&nbsp;(scale-vector&nbsp;2))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(*1/2&nbsp;(scale-vector&nbsp;(/&nbsp;1&nbsp;2)))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(*1/6&nbsp;(scale-vector&nbsp;(/&nbsp;1&nbsp;6))))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(lambda&nbsp;(y)<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;;;&nbsp;y&nbsp;is&nbsp;a&nbsp;system&nbsp;state<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(let*&nbsp;((k0&nbsp;(*h&nbsp;(f&nbsp;y)))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(k1&nbsp;(*h&nbsp;(f&nbsp;(add-vectors&nbsp;y&nbsp;(*1/2&nbsp;k0)))))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(k2&nbsp;(*h&nbsp;(f&nbsp;(add-vectors&nbsp;y&nbsp;(*1/2&nbsp;k1)))))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(k3&nbsp;(*h&nbsp;(f&nbsp;(add-vectors&nbsp;y&nbsp;k2)))))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(add-vectors&nbsp;y<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(*1/6&nbsp;(add-vectors&nbsp;k0<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(*2&nbsp;k1)<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(*2&nbsp;k2)<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;k3))))))))<br><br>(define&nbsp;elementwise<br>&nbsp;&nbsp;(lambda&nbsp;(f)<br>&nbsp;&nbsp;&nbsp;&nbsp;(lambda&nbsp;vectors<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(generate-vector<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(vector-length&nbsp;(car&nbsp;vectors))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(lambda&nbsp;(i)<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(apply&nbsp;f<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(map&nbsp;(lambda&nbsp;(v)&nbsp;(vector-ref&nbsp;&nbsp;v&nbsp;i))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;vectors)))))))<br><br>(define&nbsp;generate-vector<br>&nbsp;&nbsp;(lambda&nbsp;(size&nbsp;proc)<br>&nbsp;&nbsp;&nbsp;&nbsp;(let&nbsp;((ans&nbsp;(make-vector&nbsp;size)))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(letrec&nbsp;((loop<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(lambda&nbsp;(i)<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(cond&nbsp;((=&nbsp;i&nbsp;size)&nbsp;ans)<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(else<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(vector-set!&nbsp;ans&nbsp;i&nbsp;(proc&nbsp;i))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(loop&nbsp;(+&nbsp;i&nbsp;1)))))))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(loop&nbsp;0)))))<br><br>(define&nbsp;add-vectors&nbsp;(elementwise&nbsp;+))<br><br>(define&nbsp;scale-vector<br>&nbsp;&nbsp;(lambda&nbsp;(s)<br>&nbsp;&nbsp;&nbsp;&nbsp;(elementwise&nbsp;(lambda&nbsp;(x)&nbsp;(*&nbsp;x&nbsp;s)))))<p></tt><p><tt>Map-streams</tt> is analogous to <tt>map</tt>: it applies its firstargument (a procedure) to all the elements of its second argument (astream).<p><tt><p>(define&nbsp;map-streams<br>&nbsp;&nbsp;(lambda&nbsp;(f&nbsp;s)<br>&nbsp;&nbsp;&nbsp;&nbsp;(cons&nbsp;(f&nbsp;(head&nbsp;s))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(delay&nbsp;(map-streams&nbsp;f&nbsp;(tail&nbsp;s))))))<p></tt><p>Infinite streams are implemented as pairs whose car holds the firstelement of the stream and whose cdr holds a promise to deliver the restof the stream.<p><tt><p>(define&nbsp;head&nbsp;car)<br>(define&nbsp;tail<br>&nbsp;&nbsp;(lambda&nbsp;(stream)&nbsp;(force&nbsp;(cdr&nbsp;stream))))<p></tt><p><p><br><p><br><p>The following illustrates the use of <tt>integrate-system</tt> inintegrating the system<p><div align=left><img src="r5rs-Z-G-64.gif" border="0"></div><p><p><div align=left><img src="r5rs-Z-G-65.gif" border="0"></div><p>which models a damped oscillator.<p><tt><p>(define&nbsp;damped-oscillator<br>&nbsp;&nbsp;(lambda&nbsp;(R&nbsp;L&nbsp;C)<br>&nbsp;&nbsp;&nbsp;&nbsp;(lambda&nbsp;(state)<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(let&nbsp;((Vc&nbsp;(vector-ref&nbsp;state&nbsp;0))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(Il&nbsp;(vector-ref&nbsp;state&nbsp;1)))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(vector&nbsp;(-&nbsp;0&nbsp;(+&nbsp;(/&nbsp;Vc&nbsp;(*&nbsp;R&nbsp;C))&nbsp;(/&nbsp;Il&nbsp;C)))<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(/&nbsp;Vc&nbsp;L))))))<br><br>(define&nbsp;the-states<br>&nbsp;&nbsp;(integrate-system<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(damped-oscillator&nbsp;10000&nbsp;1000&nbsp;.001)<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;'#(1&nbsp;0)<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;.01))<p></tt><p><p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<p><p><div class=navigation>[Go to <span><a href="r5rs.html">first</a>, <a 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