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<a name="interpol"></a>
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<h1 class="chapter"> 59. interpol </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC294">59.1 Introduction to interpol</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC295">59.2 Functions and Variables for interpol</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
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<p><a name="Item_003a-Introduction-to-interpol"></a>
</p><hr size="6">
<a name="Introduction-to-interpol"></a>
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<h2 class="section"> 59.1 Introduction to interpol </h2>

<p>Package <code>interpol</code> defines the Lagrangian, the linear and the cubic 
splines methods for polynomial interpolation.
</p>
<p>For comments, bugs or suggestions, please contact me at <var>'mario AT edu DOT xunta DOT es'</var>.
</p>
<div class=categorybox>
&middot;
<p>@ref{Category: Numerical methods} &middot;
@ref{Category: Share packages} &middot;
@ref{Category: Package interpol}
</div>
</p>
<p><a name="Item_003a-Functions-and-Variables-for-interpol"></a>
</p><hr size="6">
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<h2 class="section"> 59.2 Functions and Variables for interpol </h2>


<p><a name="Item_003a-lagrange"></a>
</p><dl>
<dt><u>Function:</u> <b>lagrange</b><i> (<var>points</var>)</i>
<a name="IDX2473"></a>
</dt>
<dt><u>Function:</u> <b>lagrange</b><i> (<var>points</var>, <var>option</var>)</i>
<a name="IDX2474"></a>
</dt>
<dd><p>Computes the polynomial interpolation by the Lagrangian method. Argument <var>points</var> must be either:
</p>
<ul>
<li>
a two column matrix, <code>p:matrix([2,4],[5,6],[9,3])</code>,
</li><li>
a list of pairs, <code>p: [[2,4],[5,6],[9,3]]</code>,
</li><li>
a list of numbers, <code>p: [4,6,3]</code>, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
</li></ul>

<p>In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
</p>
<p>With the <var>option</var> argument it is possible to select the name for the independent variable, which is <code>'x</code> by default; to define another one, write something like <code>varname='z</code>. 
</p>
<p>Note that when working with high degree polynomials, floating point evaluations are unstable.
</p>
<p>See also <code>linearinterpol</code>, <code>cspline</code>, and <code>ratinterpol</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) load(interpol)$
(%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
(%i3) lagrange(p);
       (x - 7) (x - 6) (x - 3) (x - 1)
(%o3)  -------------------------------
                     35
   (x - 8) (x - 6) (x - 3) (x - 1)
 - -------------------------------
                 12
   7 (x - 8) (x - 7) (x - 3) (x - 1)
 + ---------------------------------
                  30
   (x - 8) (x - 7) (x - 6) (x - 1)
 - -------------------------------
                 60
   (x - 8) (x - 7) (x - 6) (x - 3)
 + -------------------------------
                 84
(%i4) f(x):=''%;
               (x - 7) (x - 6) (x - 3) (x - 1)
(%o4)  f(x) := -------------------------------
                             35
   (x - 8) (x - 6) (x - 3) (x - 1)
 - -------------------------------
                 12
   7 (x - 8) (x - 7) (x - 3) (x - 1)
 + ---------------------------------
                  30
   (x - 8) (x - 7) (x - 6) (x - 1)
 - -------------------------------
                 60
   (x - 8) (x - 7) (x - 6) (x - 3)
 + -------------------------------
                 84
(%i5) /* Evaluate the polynomial at some points */
      expand(map(f,[2.3,5/7,%pi]));
                                  4          3           2
                    919062  73 %pi    701 %pi    8957 %pi
(%o5)  [- 1.567535, ------, ------- - -------- + ---------
                    84035     420       210         420
                                             5288 %pi   186
                                           - -------- + ---]
                                               105       5
(%i6) %,numer;
(%o6) [- 1.567535, 10.9366573451538, 2.89319655125692]
(%i7) load(draw)$  /* load draw package */
(%i8) /* Plot the polynomial together with points */
      draw2d(
        color      = red,
        key        = &quot;Lagrange polynomial&quot;,
        explicit(f(x),x,0,10),
        point_size = 3,
        color      = blue,
        key        = &quot;Sample points&quot;,
        points(p))$
(%i9) /* Change variable name */
      lagrange(p, varname=w);
       (w - 7) (w - 6) (w - 3) (w - 1)
(%o9)  -------------------------------
                     35
   (w - 8) (w - 6) (w - 3) (w - 1)
 - -------------------------------
                 12
   7 (w - 8) (w - 7) (w - 3) (w - 1)
 + ---------------------------------
                  30
   (w - 8) (w - 7) (w - 6) (w - 1)
 - -------------------------------
                 60
   (w - 8) (w - 7) (w - 6) (w - 3)
 + -------------------------------
                 84
</pre>
<div class=categorybox>
&middot;
<p>@ref{Category: Package interpol}
</div>
</p>
</dd></dl>


<p><a name="Item_003a-charfun2"></a>
</p><dl>
<dt><u>Function:</u> <b>charfun2</b><i> (<var>x</var>, <var>a</var>, <var>b</var>)</i>
<a name="IDX2475"></a>
</dt>
<dd><p>Returns <code>true</code> if number <var>x</var> belongs to the interval <em>[a, b)</em>, and <code>false</code> otherwise.
</p>
<div class=categorybox>
&middot;
<p>@ref{Category: Package interpol}
</div>
</p>
</dd></dl>


<p><a name="Item_003a-linearinterpol"></a>
</p><dl>
<dt><u>Function:</u> <b>linearinterpol</b><i> (<var>points</var>)</i>
<a name="IDX2476"></a>
</dt>
<dt><u>Function:</u> <b>linearinterpol</b><i> (<var>points</var>, <var>option</var>)</i>
<a name="IDX2477"></a>
</dt>
<dd><p>Computes the polynomial interpolation by the linear method. Argument <var>points</var> must be either:
</p>
<ul>
<li>
a two column matrix, <code>p:matrix([2,4],[5,6],[9,3])</code>,
</li><li>
a list of pairs, <code>p: [[2,4],[5,6],[9,3]]</code>,
</li><li>
a list of numbers, <code>p: [4,6,3]</code>, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
</li></ul>

<p>In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
</p>
<p>With the <var>option</var> argument it is possible to select the name for the independent variable, which is <code>'x</code> by default; to define another one, write something like <code>varname='z</code>. 
</p>
<p>See also <code>lagrange</code>, <code>cspline</code>, and <code>ratinterpol</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) load(interpol)$
(%i2) p: matrix([7,2],[8,3],[1,5],[3,2],[6,7])$
(%i3) linearinterpol(p);
        13   3 x
(%o3)  (-- - ---) charfun2(x, minf, 3)
        2     2
 + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
    5 x
 + (--- - 3) charfun2(x, 3, 6)
     3

(%i4) f(x):=''%;
                13   3 x
(%o4)  f(x) := (-- - ---) charfun2(x, minf, 3)
                2     2
 + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
    5 x
 + (--- - 3) charfun2(x, 3, 6)
     3
(%i5)  /* Evaluate the polynomial at some points */
       map(f,[7.3,25/7,%pi]);
                            62  5 %pi
(%o5)                 [2.3, --, ----- - 3]
                            21    3
(%i6) %,numer;
(%o6)  [2.3, 2.952380952380953, 2.235987755982989]
(%i7) load(draw)$  /* load draw package */
(%i8)  /* Plot the polynomial together with points */
       draw2d(
         color      = red,
         key        = &quot;Linear interpolator&quot;,
         explicit(f(x),x,-5,20),
         point_size = 3,
         color      = blue,
         key        = &quot;Sample points&quot;,
         points(args(p)))$
(%i9)  /* Change variable name */
       linearinterpol(p, varname='s);
       13   3 s
(%o9) (-- - ---) charfun2(s, minf, 3)
       2     2
 + (s - 5) charfun2(s, 7, inf) + (37 - 5 s) charfun2(s, 6, 7)
    5 s
 + (--- - 3) charfun2(s, 3, 6)
     3
</pre>
<div class=categorybox>
&middot;
<p>@ref{Category: Package interpol}
</div>
</p>
</dd></dl>



<p><a name="Item_003a-cspline"></a>
</p><dl>
<dt><u>Function:</u> <b>cspline</b><i> (<var>points</var>)</i>
<a name="IDX2478"></a>
</dt>
<dt><u>Function:</u> <b>cspline</b><i> (<var>points</var>, <var>option1</var>, <var>option2</var>, ...)</i>
<a name="IDX2479"></a>
</dt>
<dd><p>Computes the polynomial interpolation by the cubic splines method. Argument <var>points</var> must be either:
</p>
<ul>
<li>
a two column matrix, <code>p:matrix([2,4],[5,6],[9,3])</code>,
</li><li>
a list of pairs, <code>p: [[2,4],[5,6],[9,3]]</code>,
</li><li>
a list of numbers, <code>p: [4,6,3]</code>, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
</li></ul>

<p>In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
</p>
<p>There are three options to fit specific needs:
</p><ul>
<li>
<code>'d1</code>, default <code>'unknown</code>, is the first derivative at <em>x_1</em>; if it is <code>'unknown</code>, the second derivative at <em>x_1</em> is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.

</li><li>
<code>'dn</code>, default <code>'unknown</code>, is the first derivative at <em>x_n</em>; if it is <code>'unknown</code>, the second derivative at <em>x_n</em> is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.

</li><li>
<code>'varname</code>, default <code>'x</code>, is the name of the independent variable.
</li></ul>

<p>See also <code>lagrange</code>, <code>linearinterpol</code>, and <code>ratinterpol</code>.
</p>
<p>Examples:
</p><pre class="example">(%i1) load(interpol)$
(%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
(%i3) /* Unknown first derivatives at the extremes
         is equivalent to natural cubic splines */
      cspline(p);
              3         2
        1159 x    1159 x    6091 x   8283
(%o3)  (------- - ------- - ------ + ----) charfun2(x, minf, 3)
         3288      1096      3288    1096
            3         2
      2587 x    5174 x    494117 x   108928
 + (- ------- + ------- - -------- + ------) charfun2(x, 7, inf)
       1644       137       1644      137
          3          2
    4715 x    15209 x    579277 x   199575
 + (------- - -------- + -------- - ------) charfun2(x, 6, 7)
     1644       274        1644      274
            3         2
      3287 x    2223 x    48275 x   9609
 + (- ------- + ------- - ------- + ----) charfun2(x, 3, 6)
       4932       274      1644     274

(%i4) f(x):=''%$
(%i5) /* Some evaluations */
      map(f,[2.3,5/7,%pi]), numer;
(%o5) [1.991460766423356, 5.823200187269903, 2.227405312429507]
(%i6) load(draw)$  /* load draw package */
(%i7) /* Plotting interpolating function */
      draw2d(
        color      = red,
        key        = &quot;Cubic splines&quot;,
        explicit(f(x),x,0,10),
        point_size = 3,
        color      = blue,
        key        = &quot;Sample points&quot;,
        points(p))$
(%i8) /* New call, but giving values at the derivatives */
      cspline(p,d1=0,dn=0);
              3          2
        1949 x    11437 x    17027 x   1247
(%o8)  (------- - -------- + ------- + ----) charfun2(x, minf, 3)
         2256       2256      2256     752
            3          2
      1547 x    35581 x    68068 x   173546
 + (- ------- + -------- - ------- + ------) charfun2(x, 7, inf)
        564       564        141      141
         3          2
    607 x    35147 x    55706 x   38420
 + (------ - -------- + ------- - -----) charfun2(x, 6, 7)
     188       564        141      47
            3         2
      3895 x    1807 x    5146 x   2148
 + (- ------- + ------- - ------ + ----) charfun2(x, 3, 6)
       5076       188      141      47
(%i8) /* Defining new interpolating function */
      g(x):=''%$
(%i9) /* Plotting both functions together */
      draw2d(
        color      = black,
        key        = &quot;Cubic splines (default)&quot;,
        explicit(f(x),x,0,10),
        color      = red,
        key        = &quot;Cubic splines (d1=0,dn=0)&quot;,
        explicit(g(x),x,0,10),
        point_size = 3,
        color      = blue,
        key        = &quot;Sample points&quot;,
        points(p))$
</pre>
<div class=categorybox>
&middot;
<p>@ref{Category: Package interpol}
</div>
</p>
</dd></dl>

<p><a name="Item_003a-ratinterpol"></a>
</p><dl>
<dt><u>Function:</u> <b>ratinterpol</b><i> (<var>points</var>, <var>numdeg</var>)</i>
<a name="IDX2480"></a>
</dt>
<dt><u>Function:</u> <b>ratinterpol</b><i> (<var>points</var>, <var>numdeg</var>, <var>option1</var>)</i>
<a name="IDX2481"></a>
</dt>
<dd><p>Generates a rational interpolator for data given by <var>points</var> and the degree of the numerator
being equal to <var>numdeg</var>; the degree of the denominator is calculated
automatically. Argument <var>points</var> must be either:
</p>
<ul>
<li>
a two column matrix, <code>p:matrix([2,4],[5,6],[9,3])</code>,
</li><li>
a list of pairs, <code>p: [[2,4],[5,6],[9,3]]</code>,
</li><li>
a list of numbers, <code>p: [4,6,3]</code>, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
</li></ul>

<p>In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
</p>
<p>There is one option to fit specific needs:
</p><ul>
<li>
<code>'varname</code>, default <code>'x</code>, is the name of the independent variable.
</li></ul>

<p>See also <code>lagrange</code>, <code>linearinterpol</code>, <code>cspline</code>, <code>minpack_lsquares</code>, and <code>lbfgs</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) load(interpol)$
(%i2) load(draw)$
(%i3) p:[[7.2,2.5],[8.5,2.1],[1.6,5.1],[3.4,2.4],[6.7,7.9]]$
(%i4) for k:0 thru length(p)-1 do                                     
        draw2d(
          explicit(ratinterpol(p,k),x,0,9),                      
          point_size = 3,                                        
          points(p),                                             
          title = concat(&quot;Degree of numerator = &quot;,k),            
          yrange=[0,10])$
</pre>
<div class=categorybox>
&middot;
<p>@ref{Category: Package interpol}
</div>
</p>
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