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<h1 class="chapter"> 54. interpol </h1>
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<tr><td align="left" valign="top"><a href="#SEC199">54.1 Introduction to interpol</a></td><td> </td><td align="left" valign="top">
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<tr><td align="left" valign="top"><a href="#SEC200">54.2 Definitions for interpol</a></td><td> </td><td align="left" valign="top">
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<h2 class="section"> 54.1 Introduction to interpol </h2>
<p>Package <code>interpol</code> defines de 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>
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<h2 class="section"> 54.2 Definitions for interpol </h2>
<dl>
<dt><u>Function:</u> <b>lagrange</b><i> (<var>points</var>)</i>
<a name="IDX1718"></a>
</dt>
<dt><u>Function:</u> <b>lagrange</b><i> (<var>points</var>, <var>option</var>)</i>
<a name="IDX1719"></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>Examples:
</p>
<table><tr><td> </td><td><pre class="example">(%i1) load("interpol")$
(%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
(%i3) lagrange(p);
4 3 2
73 x - 1402 x + 8957 x - 21152 x + 15624
(%o3) -------------------------------------------
420
(%i4) f(x):=''%;
4 3 2
73 x - 1402 x + 8957 x - 21152 x + 15624
(%o4) f(x) := -------------------------------------------
420
(%i5) /* Evaluate the polynomial at some points */
map(f,[2.3,5/7,%pi]);
919062
(%o5) [- 1.567535000000005, ------,
84035
4 3 2
73 %pi - 1402 %pi + 8957 %pi - 21152 %pi + 15624
---------------------------------------------------]
420
(%i6) %,numer;
(%o6) [- 1.567535000000005, 10.9366573451538,
2.89319655125692]
(%i7) /* Plot the polynomial together with points */
plot2d([f(x),[discrete,p]],[x,0,10],
[gnuplot_curve_styles,
["with lines","with points pointsize 3"]])$
(%i8) /* Change variable name */
lagrange(p, varname=w);
4 3 2
73 w - 1402 w + 8957 w - 21152 w + 15624
(%o8) -------------------------------------------
420
</pre></td></tr></table>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>charfun2</b><i> (<var>x</var>, <var>a</var>, <var>b</var>)</i>
<a name="IDX1720"></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></dd></dl>
<dl>
<dt><u>Function:</u> <b>linearinterpol</b><i> (<var>points</var>)</i>
<a name="IDX1721"></a>
</dt>
<dt><u>Function:</u> <b>linearinterpol</b><i> (<var>points</var>, <var>option</var>)</i>
<a name="IDX1722"></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>Examples:
</p><table><tr><td> </td><td><pre class="example">(%i1) load("interpol")$
(%i2) p: matrix([7,2],[8,3],[1,5],[3,2],[6,7])$
(%i3) linearinterpol(p);
(%o3) - ((9 x - 39) charfun2(x, minf, 3)
+ (30 - 6 x) charfun2(x, 7, inf)
+ (30 x - 222) charfun2(x, 6, 7)
+ (18 - 10 x) charfun2(x, 3, 6))/6
(%i4) f(x):=''%;
(%o4) f(x) := - ((9 x - 39) charfun2(x, minf, 3)
+ (30 - 6 x) charfun2(x, 7, inf)
+ (30 x - 222) charfun2(x, 6, 7)
+ (18 - 10 x) charfun2(x, 3, 6))/6
(%i5) /* Evaluate the polynomial at some points */
map(f,[7.3,25/7,%pi]);
62 18 - 10 %pi
(%o5) [2.3, --, - -----------]
21 6
(%i6) %,numer;
(%o6) [2.3, 2.952380952380953, 2.235987755982988]
(%i7) /* Plot the polynomial together with points */
plot2d(['(f(x)),[discrete,args(p)]],[x,-5,20],
[gnuplot_curve_styles,
["with lines","with points pointsize 3"]])$
(%i8) /* Change variable name */
linearinterpol(p, varname='s);
(%o8) - ((9 s - 39) charfun2(s, minf, 3)
+ (30 - 6 s) charfun2(s, 7, inf)
+ (30 s - 222) charfun2(s, 6, 7)
+ (18 - 10 s) charfun2(s, 3, 6))/6
</pre></td></tr></table>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>cspline</b><i> (<var>points</var>)</i>
<a name="IDX1723"></a>
</dt>
<dt><u>Function:</u> <b>cspline</b><i> (<var>points</var>, <var>option1</var>, <var>option2</var>, ...)</i>
<a name="IDX1724"></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>Examples:
</p><table><tr><td> </td><td><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
(%o3) ((3477 x - 10431 x - 18273 x + 74547)
3 2
charfun2(x, minf, 3) + (- 15522 x + 372528 x - 2964702 x
+ 7842816) charfun2(x, 7, inf)
3 2
+ (28290 x - 547524 x + 3475662 x - 7184700)
3 2
charfun2(x, 6, 7) + (- 6574 x + 80028 x - 289650 x
+ 345924) charfun2(x, 3, 6))/9864
(%i4) f(x):=''%$
(%i5) /* Some evaluations */
map(f,[2.3,5/7,%pi]), numer;
(%o5) [1.991460766423358, 5.823200187269904,
2.227405312429501]
(%i6) /* Plotting interpolating function */
plot2d(['(f(x)),[discrete,p]],[x,0,10],
[gnuplot_curve_styles,
["with lines","with points pointsize 3"]])$
(%i7) /* New call, but giving values at the derivatives */
cspline(p,d1=0,dn=0);
3 2
(%o7) ((17541 x - 102933 x + 153243 x + 33669)
3 2
charfun2(x, minf, 3) + (- 55692 x + 1280916 x - 9801792 x
+ 24990624) charfun2(x, 7, inf)
3 2
+ (65556 x - 1265292 x + 8021664 x - 16597440)
3 2
charfun2(x, 6, 7) + (- 15580 x + 195156 x - 741024 x
+ 927936) charfun2(x, 3, 6))/20304
(%i8) /* Defining new interpolating function */
g(x):=''%$
(%i9) /* Plotting both functions together */
plot2d(['(f(x)),'(g(x)),[discrete,p]],[x,0,10],
[gnuplot_curve_styles,
["with lines","with lines","with points pointsize 3"]])$
</pre></td></tr></table>
</dd></dl>
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