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Revised^6 Report on the Algorithmic Language Scheme
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<a name="node_chap_D"></a>
<h1 class=chapter>
<div class=chapterheading><a href="r6rs-Z-H-2.html#node_toc_node_chap_D">Appendix D</a></div><br>
<a href="r6rs-Z-H-2.html#node_toc_node_chap_D">Example </a></h1>
<p></p>
<p>
This section describes an example consisting of the
<tt>(runge-kutta)</tt> library, which provides an <tt>integrate-system</tt>
procedure that integrates the system
</p>
<div class=mathdisplay align=left><table><tr><td></td><td><table><tr><td align=center><em>y</em><sub><em>k</em></sub><sup>&frasl;</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>), <span style="margin-left: .27778em">&zwnj;</span> <em>k</em> = 1, <tt>...</tt>, <em>n</em></td></tr></table></td><td></td></tr></table></div>
<p class=noindent>
of differential equations with the method of Runge-Kutta.</p>
<p>
As the <tt>(runge-kutta)</tt> library makes use of the <tt>(rnrs base (6))</tt>
library, its skeleton is as follows:</p>
<p>
</p>
<tt>#!r6rs
<p class=nopadding>(library&nbsp;(runge-kutta)</p>
<p class=nopadding>&nbsp;&nbsp;(export&nbsp;integrate-system</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;head&nbsp;tail)</p>
<p class=nopadding>&nbsp;&nbsp;(import&nbsp;(rnrs&nbsp;base))</p>
<p class=nopadding>&nbsp;&nbsp;&lt;library&nbsp;body&gt;)</p>
<p class=nopadding></p>
<p></tt></p>
<p>
The procedure definitions described below go in the place of &lt;library body&gt;.</p>
<p>
The parameter <tt>system-derivative</tt> is a function that takes a system
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>)
and produces a system derivative (the values <em>y</em><sub>1</sub><sup>&frasl;</sup>, <tt>...</tt>,
<em>y</em><sub><em>n</em></sub><sup>&frasl;</sup>). The parameter <tt>initial-state</tt> provides an initial
system state, and <tt>h</tt> is an initial guess for the length of the
integration step.</p>
<p>
The value returned by <tt>integrate-system</tt> is an infinite stream of
system states.</p>
<p>
</p>
<tt>(define&nbsp;integrate-system
<p class=nopadding>&nbsp;&nbsp;(lambda&nbsp;(system-derivative&nbsp;initial-state&nbsp;h)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;(let&nbsp;((next&nbsp;(runge-kutta-4&nbsp;system-derivative&nbsp;h)))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(letrec&nbsp;((states</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(cons&nbsp;initial-state</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(lambda&nbsp;()</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(map-streams&nbsp;next&nbsp;states)))))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;states))))</p>
<p></tt></p>
<p>
The <tt>runge-kutta-4</tt> procedure takes a function, <tt>f</tt>, that produces a
system derivative from a system state. The <tt>runge-kutta-4</tt> procedure
produces a function that takes a system state and
produces a new system state.</p>
<p>
</p>
<tt>(define&nbsp;runge-kutta-4
<p class=nopadding>&nbsp;&nbsp;(lambda&nbsp;(f&nbsp;h)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;(let&nbsp;((*h&nbsp;(scale-vector&nbsp;h))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(*2&nbsp;(scale-vector&nbsp;2))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(*1/2&nbsp;(scale-vector&nbsp;(/&nbsp;1&nbsp;2)))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(*1/6&nbsp;(scale-vector&nbsp;(/&nbsp;1&nbsp;6))))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(lambda&nbsp;(y)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;;;&nbsp;y&nbsp;is&nbsp;a&nbsp;system&nbsp;state</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(let*&nbsp;((k0&nbsp;(*h&nbsp;(f&nbsp;y)))</p>
<p class=nopadding>&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)))))</p>
<p class=nopadding>&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)))))</p>
<p class=nopadding>&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)))))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(add-vectors&nbsp;y</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(*1/6&nbsp;(add-vectors&nbsp;k0</p>
<p class=nopadding>&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)</p>
<p class=nopadding>&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)</p>
<p class=nopadding>&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))))))))</p>
<p class=nopadding></p>
<p class=nopadding></p>
<p class=nopadding>(define&nbsp;elementwise</p>
<p class=nopadding>&nbsp;&nbsp;(lambda&nbsp;(f)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;(lambda&nbsp;vectors</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(generate-vector</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(vector-length&nbsp;(car&nbsp;vectors))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(lambda&nbsp;(i)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(apply&nbsp;f</p>
<p class=nopadding>&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))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;vectors)))))))</p>
<p class=nopadding></p>
<p class=nopadding>(define&nbsp;generate-vector</p>
<p class=nopadding>&nbsp;&nbsp;(lambda&nbsp;(size&nbsp;proc)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;(let&nbsp;((ans&nbsp;(make-vector&nbsp;size)))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(letrec&nbsp;((loop</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(lambda&nbsp;(i)</p>
<p class=nopadding>&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)</p>
<p class=nopadding>&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</p>
<p class=nopadding>&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))</p>
<p class=nopadding>&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)))))))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(loop&nbsp;0)))))</p>
<p class=nopadding></p>
<p class=nopadding>(define&nbsp;add-vectors&nbsp;(elementwise&nbsp;+))</p>
<p class=nopadding></p>
<p class=nopadding>(define&nbsp;scale-vector</p>
<p class=nopadding>&nbsp;&nbsp;(lambda&nbsp;(s)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;(elementwise&nbsp;(lambda&nbsp;(x)&nbsp;(*&nbsp;x&nbsp;s)))))</p>
<p></tt></p>
<p>
The <tt>map-streams</tt> procedure is analogous to <tt>map</tt>: it applies its first
argument (a procedure) to all the elements of its second argument (a
stream).</p>
<p>
</p>
<tt>(define&nbsp;map-streams
<p class=nopadding>&nbsp;&nbsp;(lambda&nbsp;(f&nbsp;s)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;(cons&nbsp;(f&nbsp;(head&nbsp;s))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(lambda&nbsp;()&nbsp;(map-streams&nbsp;f&nbsp;(tail&nbsp;s))))))</p>
<p></tt></p>
<p>
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.</p>
<p>
</p>
<tt>(define&nbsp;head&nbsp;car)
<p class=nopadding>(define&nbsp;tail</p>
<p class=nopadding>&nbsp;&nbsp;(lambda&nbsp;(stream)&nbsp;((cdr&nbsp;stream))))</p>
<p></tt></p>
<p>
</p>
<div class=bigskip></div>
<p style="margin-top: 0pt; margin-bottom: 0pt">
The following program illustrates the use of <tt>integrate-system</tt> in
integrating the system
</p>
<div class=mathdisplay align=left><table><tr><td></td><td><table><tr><td align=center> <em>C</em> </td><td><table><tr><td align=center><em>d</em><em>v</em><sub><em>C</em></sub> </td></tr><tr><td style="height=1pt; background-color: black"></td></tr><tr><td align=center> <em>d</em><em>t</em></td></tr></table></td><td> = &minus;<em>i</em><sub><em>L</em></sub> &minus; </td><td><table><tr><td align=center><em>v</em><sub><em>C</em></sub> </td></tr><tr><td style="height=1pt; background-color: black"></td></tr><tr><td align=center> <em>R</em></td></tr></table></td><td></td></tr></table></td><td></td></tr></table></div>
<p class=noindent></p>
<div class=mathdisplay align=left><table><tr><td></td><td><table><tr><td align=center> <em>L</em> </td><td><table><tr><td align=center><em>d</em><em>i</em><sub><em>L</em></sub> </td></tr><tr><td style="height=1pt; background-color: black"></td></tr><tr><td align=center> <em>d</em><em>t</em></td></tr></table></td><td> = <em>v</em><sub><em>C</em></sub></td></tr></table></td><td></td></tr></table></div>
<p class=noindent>
which models a damped oscillator.</p>
<p>
</p>
<tt>#!r6rs
<p class=nopadding>(import&nbsp;(rnrs&nbsp;base)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(rnrs&nbsp;io&nbsp;simple)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(runge-kutta))</p>
<p class=nopadding></p>
<p class=nopadding>(define&nbsp;damped-oscillator</p>
<p class=nopadding>&nbsp;&nbsp;(lambda&nbsp;(R&nbsp;L&nbsp;C)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;(lambda&nbsp;(state)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(let&nbsp;((Vc&nbsp;(vector-ref&nbsp;state&nbsp;0))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(Il&nbsp;(vector-ref&nbsp;state&nbsp;1)))</p>
<p class=nopadding>&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)))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(/&nbsp;Vc&nbsp;L))))))</p>
<p class=nopadding></p>
<p class=nopadding>(define&nbsp;the-states</p>
<p class=nopadding>&nbsp;&nbsp;(integrate-system</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(damped-oscillator&nbsp;10000&nbsp;1000&nbsp;.001)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;'#(1&nbsp;0)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;.01))</p>
<p class=nopadding></p>
<p class=nopadding>(letrec&nbsp;((loop&nbsp;(lambda&nbsp;(s)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(newline)</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(write&nbsp;(head&nbsp;s))</p>
<p class=nopadding>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(loop&nbsp;(tail&nbsp;s)))))</p>
<p class=nopadding>&nbsp;&nbsp;(loop&nbsp;the-states))</p>
<p></tt></p>
<p>
This prints output like the following:</p>
<p>
</p>
<tt>#(1&nbsp;0)
<p class=nopadding>#(0.99895054&nbsp;9.994835e-6)</p>
<p class=nopadding>#(0.99780226&nbsp;1.9978681e-5)</p>
<p class=nopadding>#(0.9965554&nbsp;2.9950552e-5)</p>
<p class=nopadding>#(0.9952102&nbsp;3.990946e-5)</p>
<p class=nopadding>#(0.99376684&nbsp;4.985443e-5)</p>
<p class=nopadding>#(0.99222565&nbsp;5.9784474e-5)</p>
<p class=nopadding>#(0.9905868&nbsp;6.969862e-5)</p>
<p class=nopadding>#(0.9888506&nbsp;7.9595884e-5)</p>
<p class=nopadding>#(0.9870173&nbsp;8.94753e-5)</p>
<p class=nopadding></p>
<p></tt></p>
<p>
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