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<tr><td align="center" bgcolor="#8381de"><div class="skribetitle"><strong><big><big><big>R5Rs<br/>Scheme Revised(5) Report on the Algorithmic Language Scheme -- Example</big></big></big></strong></div><center>
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>Example</strong></font></th></tr>
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 <tr><td valign="top" align="left"></td><td colspan="4" width="100%"><a href="r5rs-1.html#Summary">Summary</a></td></tr>
 <tr><td valign="top" align="left"></td><td colspan="4" width="100%"><a href="r5rs-2.html#Introduction">Introduction</a></td></tr>
 <tr><td valign="top" align="left"></td><td colspan="4" width="100%"><a href="r5rs-3.html#Table-of-contents">Table of contents</a></td></tr>
 <tr><td valign="top" align="left">1</td><td colspan="4" width="100%"><a href="r5rs-4.html#Overview-of-Scheme">Overview of Scheme</a></td></tr>
 <tr><td valign="top" align="left">2</td><td colspan="4" width="100%"><a href="r5rs-5.html#Lexical-conventions">Lexical conventions</a></td></tr>
 <tr><td valign="top" align="left">3</td><td colspan="4" width="100%"><a href="r5rs-6.html#Basic-concepts">Basic concepts</a></td></tr>
 <tr><td valign="top" align="left">4</td><td colspan="4" width="100%"><a href="r5rs-7.html#Expressions">Expressions</a></td></tr>
 <tr><td valign="top" align="left">5</td><td colspan="4" width="100%"><a href="r5rs-8.html#Program-structure">Program structure</a></td></tr>
 <tr><td valign="top" align="left">6</td><td colspan="4" width="100%"><a href="r5rs-9.html#Standard-procedures">Standard procedures</a></td></tr>
 <tr><td valign="top" align="left">7</td><td colspan="4" width="100%"><a href="r5rs-10.html#Formal-syntax-and-semantics">Formal syntax and semantics</a></td></tr>
 <tr><td valign="top" align="left">8</td><td colspan="4" width="100%"><a href="r5rs-11.html#Concepts">Concepts</a></td></tr>
 <tr><td valign="top" align="left">9</td><td colspan="4" width="100%"><a href="r5rs-12.html#Variables-and-Procedures">Variables and Procedures</a></td></tr>
 <tr><td valign="top" align="left"></td><td colspan="4" width="100%"><a href="r5rs-13.html#Notes">Notes</a></td></tr>
 <tr><td valign="top" align="left"></td><td colspan="4" width="100%"><a href="r5rs-14.html#Additional-material">Additional material</a></td></tr>
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 <tr><td valign="top" align="left"></td><td colspan="4" width="100%"><a href="r5rs-16.html#Bibliography">Bibliography</a></td></tr>
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<td align="left" valign="top" class="skribe-body"><div class="skribe-body">
<a name="Example" class="mark"></a> 

<samp id='samp7053'
>Integrate-system</samp> integrates the system <br/><br/><center id='center7061'
>y'<sub id='sub7055'
>k</sub><sup id='sup7056'
> </sup>= f<sub id='sub7057'
>k</sub>(y<sub id='sub7058'
>1</sub>, y<sub id='sub7059'
>2</sub>, ..., y<sub id='sub7060'
>n</sub>),    k = 1, ..., n</center>

of differential equations with the method of Runge-Kutta.<br/><br/>The parameter <tt id='tt7063'
>system-derivative</tt> is a function that takes a system
state (a vector of values for the state variables y<sub id='sub7064'
>1</sub>, ..., 
y<sub id='sub7065'
>n</sub>)
and produces a system derivative (the values y'<sub id='sub7066'
>1</sub>, ..., 
y'<sub id='sub7067'
>n</sub>).  The parameter <tt id='tt7068'
>initial-state</tt> provides an initial
system state, and <tt id='tt7069'
>h</tt> is an initial guess for the length of the
integration step.<br/><br/>The value returned by <samp id='samp7071'
>integrate-system</samp> is an infinite stream of
system states.<br/><br/><center id='center7079'
><table cellspacing="0" class="color" cellpadding="0" width="95%"><tbody>
<tr><td bgcolor="#ffffcc"><pre class="prog" id='prog7077'
>(<font color="#6959cf"><strong id='bold10032'
>define</strong></font> <font color="#6959cf"><strong id='bold10034'
>integrate-system</strong></font>
  (<strong id='bold10036'
>lambda</strong> (system-derivative initial-state h)
    (<strong id='bold10037'
>let</strong> ((next (runge-kutta-4 system-derivative h)))
      (<strong id='bold10038'
>letrec</strong> ((states
                (cons initial-state
                      (delay (map-streams next
                                          states)))))
        states))))
</pre>
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<samp id='samp7080'
>Runge-Kutta-4</samp> takes a function, <tt id='tt7081'
>f</tt>, that produces a
system derivative from a system state.  <samp id='samp7082'
>Runge-Kutta-4</samp>
produces a function that takes a system state and
produces a new system state.<br/><br/><br/><center id='center7111'
><table cellspacing="0" class="color" cellpadding="0" width="95%"><tbody>
<tr><td bgcolor="#ffffcc"><pre class="prog" id='prog7109'
>(<font color="#6959cf"><strong id='bold10039'
>define</strong></font> <font color="#6959cf"><strong id='bold10041'
>runge-kutta-4</strong></font>
  (<strong id='bold10043'
>lambda</strong> (f h)
    (<strong id='bold10044'
>let</strong> ((*h (scale-vector h))
          (*2 (scale-vector 2))
          (*1/2 (scale-vector (/ 1 2)))
          (*1/6 (scale-vector (/ 1 6))))
      (<strong id='bold10045'
>lambda</strong> (y)
        <font color="#ffa600"><em id='it10046'
>;; y </em></font>is a system state
        (<strong id='bold10048'
>let*</strong> ((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))))))))<br/><br/>(<font color="#6959cf"><strong id='bold10049'
>define</strong></font> <font color="#6959cf"><strong id='bold10051'
>elementwise</strong></font>
  (<strong id='bold10053'
>lambda</strong> (f)
    (<strong id='bold10054'
>lambda</strong> vectors
      (generate-vector
        (vector-length (car vectors))
        (<strong id='bold10055'
>lambda</strong> (i)
          (<strong id='bold10056'
>apply</strong> f
                 (map (<strong id='bold10057'
>lambda</strong> (v) (vector-ref  v i))
                      vectors)))))))<br/><br/>(<font color="#6959cf"><strong id='bold10058'
>define</strong></font> <font color="#6959cf"><strong id='bold10060'
>generate-vector</strong></font>
  (<strong id='bold10062'
>lambda</strong> (size proc)
    (<strong id='bold10063'
>let</strong> ((ans (make-vector size)))
      (<strong id='bold10064'
>letrec</strong> ((loop
                (<strong id='bold10065'
>lambda</strong> (i)
                  (<strong id='bold10066'
>cond</strong> ((= i size) ans)
                        (else
                         (vector-set! ans i (proc i))
                         (loop (+ i 1)))))))
        (loop 0)))))<br/><br/>(<font color="#6959cf"><strong id='bold10067'
>define</strong></font> <font color="#6959cf"><strong id='bold10069'
>add-vectors</strong></font> (elementwise +))<br/><br/>(<font color="#6959cf"><strong id='bold10071'
>define</strong></font> <font color="#6959cf"><strong id='bold10073'
>scale-vector</strong></font>
  (<strong id='bold10075'
>lambda</strong> (s)
    (elementwise (<strong id='bold10076'
>lambda</strong> (x) (* x s)))))
</pre>
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<samp id='samp7112'
>Map-streams</samp> is analogous to <samp id='samp7113'
>map</samp>: it applies its first
argument (a procedure) to all the elements of its second argument (a
stream).<br/><br/><center id='center7119'
><table cellspacing="0" class="color" cellpadding="0" width="95%"><tbody>
<tr><td bgcolor="#ffffcc"><pre class="prog" id='prog7117'
>(<font color="#6959cf"><strong id='bold10077'
>define</strong></font> <font color="#6959cf"><strong id='bold10079'
>map-streams</strong></font>
  (<strong id='bold10081'
>lambda</strong> (f s)
    (cons (f (head s))
          (delay (map-streams f (tail s))))))
</pre>
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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.<br/><br/><center id='center7126'
><table cellspacing="0" class="color" cellpadding="0" width="95%"><tbody>
<tr><td bgcolor="#ffffcc"><pre class="prog" id='prog7124'
>(<font color="#6959cf"><strong id='bold10082'
>define</strong></font> <font color="#6959cf"><strong id='bold10084'
>head</strong></font> car)
(<font color="#6959cf"><strong id='bold10086'
>define</strong></font> <font color="#6959cf"><strong id='bold10088'
>tail</strong></font>
  (<strong id='bold10090'
>lambda</strong> (stream) (force (cdr stream))))
</pre>
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The following illustrates the use of <samp id='samp7127'
>integrate-system</samp> in
integrating the system<br/><br/><center id='center7132'
> C dv<sub id='sub7129'
>C</sub> / dt = -i<sub id='sub7130'
>L</sub> - v<sub id='sub7131'
>C</sub> / R</center>

<center id='center7135'
> L di<sub id='sub7133'
>L</sub> / dt = v<sub id='sub7134'
>C</sub></center>

which models a damped oscillator.<br/><br/><center id='center7145'
><table cellspacing="0" class="color" cellpadding="0" width="95%"><tbody>
<tr><td bgcolor="#ffffcc"><pre class="prog" id='prog7143'
>(<font color="#6959cf"><strong id='bold10091'
>define</strong></font> <font color="#6959cf"><strong id='bold10093'
>damped-oscillator</strong></font>
  (<strong id='bold10095'
>lambda</strong> (R L C)
    (<strong id='bold10096'
>lambda</strong> (state)
      (<strong id='bold10097'
>let</strong> ((Vc (vector-ref state 0))
            (Il (vector-ref state 1)))
        (vector (- 0 (+ (/ Vc (* R C)) (/ Il C)))
                (/ Vc L))))))<br/><br/>(<font color="#6959cf"><strong id='bold10098'
>define</strong></font> <font color="#6959cf"><strong id='bold10100'
>the-states</strong></font>
  (integrate-system
     (damped-oscillator 10000 1000 .001)
     '#(1 0)
     .01))
</pre>
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