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<a name="Introduction-to-orthogonal-polynomials"></a>
<div class="header">
<p>
Next: <a href="maxima_300.html#Functions-and-Variables-for-orthogonal-polynomials" accesskey="n" rel="next">Functions and Variables for orthogonal polynomials</a>, Previous: <a href="maxima_298.html#orthopoly_002dpkg" accesskey="p" rel="previous">orthopoly-pkg</a>, Up: <a href="maxima_298.html#orthopoly_002dpkg" accesskey="u" rel="up">orthopoly-pkg</a> &nbsp; [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_368.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Introduction-to-orthogonal-polynomials-1"></a>
<h3 class="section">79.1 Introduction to orthogonal polynomials</h3>

<p><code>orthopoly</code> is a package for symbolic and numerical evaluation of
several kinds of orthogonal polynomials, including Chebyshev,
Laguerre, Hermite, Jacobi, Legendre, and ultraspherical (Gegenbauer) 
polynomials. Additionally, <code>orthopoly</code> includes support for the spherical Bessel, 
spherical Hankel, and spherical harmonic functions.
</p>
<p>For the most part, <code>orthopoly</code> follows the conventions of Abramowitz and Stegun
<i>Handbook of Mathematical Functions</i>, Chapter 22 (10th printing, December 1972);
additionally, we use Gradshteyn and Ryzhik, 
<i>Table of Integrals, Series, and Products</i> (1980 corrected and 
enlarged edition), and Eugen Merzbacher <i>Quantum Mechanics</i> (2nd edition, 1970).
</p>


<p>Barton Willis of the University of Nebraska at Kearney (UNK) wrote
the <code>orthopoly</code> package and its documentation. The package 
is released under the GNU General Public License (GPL).
</p>
<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Orthogonal-polynomials">Orthogonal polynomials</a>
&middot;<a href="maxima_369.html#Category_003a-Share-packages">Share packages</a>
&middot;<a href="maxima_369.html#Category_003a-Package-orthopoly">Package orthopoly</a>
</div></p>
<a name="Getting-Started-with-orthopoly"></a>
<h4 class="subsection">79.1.1 Getting Started with orthopoly</h4>

<p><code>load (&quot;orthopoly&quot;)</code> loads the <code>orthopoly</code> package.
</p>
<p>To find the third-order Legendre polynomial,
</p>
<div class="example">
<pre class="example">(%i1) legendre_p (3, x);
                      3             2
             5 (1 - x)    15 (1 - x)
(%o1)      - ---------- + ----------- - 6 (1 - x) + 1
                 2             2
</pre></div>

<p>To express this as a sum of powers of <var>x</var>, apply <var>ratsimp</var> or <var>rat</var>
to the result.
</p>
<div class="example">
<pre class="example">(%i2) [ratsimp (%), rat (%)];
                        3           3
                     5 x  - 3 x  5 x  - 3 x
(%o2)/R/            [----------, ----------]
                         2           2
</pre></div>

<p>Alternatively, make the second argument to <code>legendre_p</code> (its &ldquo;main&rdquo; variable) 
a canonical rational expression (CRE).
</p>
<div class="example">
<pre class="example">(%i1) legendre_p (3, rat (x));
                              3
                           5 x  - 3 x
(%o1)/R/                   ----------
                               2
</pre></div>

<p>For floating point evaluation, <code>orthopoly</code> uses a running error analysis
to estimate an upper bound for the error. For example,
</p>
<div class="example">
<pre class="example">(%i1) jacobi_p (150, 2, 3, 0.2);
(%o1) interval(- 0.062017037936715, 1.533267919277521E-11)
</pre></div>

<p>Intervals have the form <code>interval (<var>c</var>, <var>r</var>)</code>, where <var>c</var> is the
center and <var>r</var> is the radius of the interval. Since Maxima
does not support arithmetic on intervals, in some situations, such
as graphics, you want to suppress the error and output only the 
center of the interval. To do this, set the option
variable <code>orthopoly_returns_intervals</code> to <code>false</code>.
</p>
<div class="example">
<pre class="example">(%i1) orthopoly_returns_intervals : false;
(%o1)                         false
(%i2) jacobi_p (150, 2, 3, 0.2);
(%o2)                  - 0.062017037936715
</pre></div>

<p>Refer to the section see <a href="#Floating-point-Evaluation">Floating point Evaluation</a> for more information.
</p>
<p>Most functions in <code>orthopoly</code> have a <code>gradef</code> property; thus
</p>
<div class="example">
<pre class="example">(%i1) diff (hermite (n, x), x);
(%o1)                     2 n H     (x)
                               n - 1
(%i2) diff (gen_laguerre (n, a, x), x);
              (a)               (a)
           n L   (x) - (n + a) L     (x) unit_step(n)
              n                 n - 1
(%o2)      ------------------------------------------
                               x
</pre></div>

<p>The unit step function in the second example prevents an error that would
otherwise arise by evaluating with <var>n</var> equal to 0.
</p>
<div class="example">
<pre class="example">(%i3) ev (%, n = 0);
(%o3)                           0
</pre></div>

<p>The <code>gradef</code> property only applies to the &ldquo;main&rdquo; variable; derivatives with 
respect other arguments usually result in an error message; for example
</p>
<div class="example">
<pre class="example">(%i1) diff (hermite (n, x), x);
(%o1)                     2 n H     (x)
                               n - 1
(%i2) diff (hermite (n, x), n);

Maxima doesn't know the derivative of hermite with respect the first
argument
 -- an error.  Quitting.  To debug this try debugmode(true);
</pre></div>

<p>Generally, functions in <code>orthopoly</code> map over lists and matrices. For
the mapping to fully evaluate, the option variables 
<code>doallmxops</code> and <code>listarith</code> must both be <code>true</code> (the defaults).
To illustrate the mapping over matrices, consider
</p>
<div class="example">
<pre class="example">(%i1) hermite (2, x);
                                     2
(%o1)                    - 2 (1 - 2 x )
(%i2) m : matrix ([0, x], [y, 0]);
                            [ 0  x ]
(%o2)                       [      ]
                            [ y  0 ]
(%i3) hermite (2, m);
               [                             2  ]
               [      - 2        - 2 (1 - 2 x ) ]
(%o3)          [                                ]
               [             2                  ]
               [ - 2 (1 - 2 y )       - 2       ]
</pre></div>

<p>In the second example, the <code>i, j</code> element of the value
is <code>hermite (2, m[i,j])</code>; this is not the same as computing
<code>-2 + 4 m . m</code>, as seen in the next example.
</p>
<div class="example">
<pre class="example">(%i4) -2 * matrix ([1, 0], [0, 1]) + 4 * m . m;
                    [ 4 x y - 2      0     ]
(%o4)               [                      ]
                    [     0      4 x y - 2 ]
</pre></div>

<p>If you evaluate a function at a point outside its domain, generally
<code>orthopoly</code> returns the function unevaluated. For example,
</p>
<div class="example">
<pre class="example">(%i1) legendre_p (2/3, x);
(%o1)                        P   (x)
                              2/3
</pre></div>

<p><code>orthopoly</code> supports translation into TeX; it also does two-dimensional
output on a terminal.
</p>
<div class="example">
<pre class="example">(%i1) spherical_harmonic (l, m, theta, phi);
                          m
(%o1)                    Y (theta, phi)
                          l
(%i2) tex (%);
$$Y_{l}^{m}\left(\vartheta,\varphi\right)$$
(%o2)                         false
(%i3) jacobi_p (n, a, a - b, x/2);
                          (a, a - b) x
(%o3)                    P          (-)
                          n          2
(%i4) tex (%);
$$P_{n}^{\left(a,a-b\right)}\left({{x}\over{2}}\right)$$
(%o4)                         false
</pre></div>

<a name="Limitations"></a>
<h4 class="subsection">79.1.2 Limitations</h4>

<p>When an expression involves several orthogonal polynomials with
symbolic orders, it&rsquo;s possible that the expression actually
vanishes, yet Maxima is unable to simplify it to zero. If you
divide by such a quantity, you&rsquo;ll be in trouble. For example,
the following expression vanishes for integers <var>n</var> greater than 1, yet Maxima
is unable to simplify it to zero.
</p>
<div class="example">
<pre class="example">(%i1) (2*n - 1) * legendre_p (n - 1, x) * x - n * legendre_p (n, x)
      + (1 - n) * legendre_p (n - 2, x);
(%o1)  (2 n - 1) P     (x) x - n P (x) + (1 - n) P     (x)
                  n - 1           n               n - 2
</pre></div>

<p>For a specific <var>n</var>, we can reduce the expression to zero.
</p>
<div class="example">
<pre class="example">(%i2) ev (% ,n = 10, ratsimp);
(%o2)                           0
</pre></div>

<p>Generally, the polynomial form of an orthogonal polynomial is ill-suited
for floating point evaluation. Here&rsquo;s an example.
</p>
<div class="example">
<pre class="example">(%i1) p : jacobi_p (100, 2, 3, x)$

(%i2) subst (0.2, x, p);
(%o2)                3.4442767023833592E+35
(%i3) jacobi_p (100, 2, 3, 0.2);
(%o3)  interval(0.18413609135169, 6.8990300925815987E-12)
(%i4) float(jacobi_p (100, 2, 3, 2/10));
(%o4)                   0.18413609135169
</pre></div>

<p>The true value is about 0.184; this calculation suffers from extreme
subtractive cancellation error. Expanding the polynomial and then
evaluating, gives a better result.
</p><div class="example">
<pre class="example">(%i5) p : expand(p)$
(%i6) subst (0.2, x, p);
(%o6) 0.18413609766122982
</pre></div>

<p>This isn&rsquo;t a general rule; expanding the polynomial does not always
result in an expression that is better suited for numerical evaluation.
By far, the best way to do numerical evaluation is to make one or more
of the function arguments floating point numbers. By doing that, 
specialized floating point algorithms are used for evaluation.
</p>
<p>Maxima&rsquo;s <code>float</code> function is somewhat indiscriminate; if you apply 
<code>float</code> to an expression involving an orthogonal polynomial with a
symbolic degree or order parameter, these parameters may be 
converted into floats; after that, the expression will not evaluate 
fully. Consider
</p>
<div class="example">
<pre class="example">(%i1) assoc_legendre_p (n, 1, x);
                               1
(%o1)                         P (x)
                               n
(%i2) float (%);
                              1.0
(%o2)                        P   (x)
                              n
(%i3) ev (%, n=2, x=0.9);
                             1.0
(%o3)                       P   (0.9)
                             2
</pre></div>

<p>The expression in (%o3) will not evaluate to a float; <code>orthopoly</code> doesn&rsquo;t
recognize floating point values where it requires an integer. Similarly, 
numerical evaluation of the <code>pochhammer</code> function for orders that
exceed <code>pochhammer_max_index</code> can be troublesome; consider
</p>
<div class="example">
<pre class="example">(%i1) x :  pochhammer (1, 10), pochhammer_max_index : 5;
(%o1)                         (1)
                                 10
</pre></div>

<p>Applying <code>float</code> doesn&rsquo;t evaluate <var>x</var> to a float
</p>
<div class="example">
<pre class="example">(%i2) float (x);
(%o2)                       (1.0)
                                 10.0
</pre></div>

<p>To evaluate <var>x</var> to a float, you&rsquo;ll need to bind
<code>pochhammer_max_index</code> to 11 or greater and apply <code>float</code> to <var>x</var>.
</p>
<div class="example">
<pre class="example">(%i3) float (x), pochhammer_max_index : 11;
(%o3)                       3628800.0
</pre></div>

<p>The default value of <code>pochhammer_max_index</code> is 100;
change its value after loading <code>orthopoly</code>.
</p>
<p>Finally, be aware that reference books vary on the definitions of the 
orthogonal polynomials; we&rsquo;ve generally used the conventions 
of Abramowitz and Stegun.
</p>
<p>Before you suspect a bug in orthopoly, check some special cases 
to determine if your definitions match those used by <code>orthopoly</code>. 
Definitions often differ by a normalization; occasionally, authors
use &ldquo;shifted&rdquo; versions of the functions that makes the family
orthogonal on an interval other than <em>(-1, 1)</em>. To define, for example,
a Legendre polynomial that is orthogonal on <em>(0, 1)</em>, define
</p>
<div class="example">
<pre class="example">(%i1) shifted_legendre_p (n, x) := legendre_p (n, 2*x - 1)$

(%i2) shifted_legendre_p (2, rat (x));
                            2
(%o2)/R/                 6 x  - 6 x + 1
(%i3) legendre_p (2, rat (x));
                               2
                            3 x  - 1
(%o3)/R/                    --------
                               2
</pre></div>

<a name="Floating-point-Evaluation"></a><a name="Floating-point-Evaluation-1"></a>
<h4 class="subsection">79.1.3 Floating point Evaluation</h4>

<p>Most functions in <code>orthopoly</code> use a running error analysis to 
estimate the error in floating point evaluation; the 
exceptions are the spherical Bessel functions and the associated Legendre 
polynomials of the second kind. For numerical evaluation, the spherical 
Bessel functions call SLATEC functions. No specialized method is used
for numerical evaluation of the associated Legendre polynomials of the
second kind.
</p>
<p>The running error analysis ignores errors that are second or higher order
in the machine epsilon (also known as unit roundoff). It also
ignores a few other errors. It&rsquo;s possible (although unlikely) 
that the actual error exceeds the estimate.
</p>
<p>Intervals have the form <code>interval (<var>c</var>, <var>r</var>)</code>, where <var>c</var> is the 
center of the interval and <var>r</var> is its radius. The 
center of an interval can be a complex number, and the radius is always a positive real number.
</p>
<p>Here is an example.
</p>

<div class="example">
<pre class="example">(%i1) fpprec : 50$

(%i2) y0 : jacobi_p (100, 2, 3, 0.2);
(%o2) interval(0.1841360913516871, 6.8990300925815987E-12)
(%i3) y1 : bfloat (jacobi_p (100, 2, 3, 1/5));
(%o3) 1.8413609135168563091370224958913493690868904463668b-1
</pre></div>

<p>Let&rsquo;s test that the actual error is smaller than the error estimate
</p>
<div class="example">
<pre class="example">(%i4) is (abs (part (y0, 1) - y1) &lt; part (y0, 2));
(%o4)                         true
</pre></div>

<p>Indeed, for this example the error estimate is an upper bound for the
true error.
</p>
<p>Maxima does not support arithmetic on intervals.
</p>
<div class="example">
<pre class="example">(%i1) legendre_p (7, 0.1) + legendre_p (8, 0.1);
(%o1) interval(0.18032072148437508, 3.1477135311021797E-15)
        + interval(- 0.19949294375000004, 3.3769353084291579E-15)
</pre></div>

<p>A user could define arithmetic operators that do interval math. To
define interval addition, we can define
</p>
<div class="example">
<pre class="example">(%i1) infix (&quot;@+&quot;)$

(%i2) &quot;@+&quot;(x,y) := interval (part (x, 1) + part (y, 1), part (x, 2)
      + part (y, 2))$

(%i3) legendre_p (7, 0.1) @+ legendre_p (8, 0.1);
(%o3) interval(- 0.019172222265624955, 6.5246488395313372E-15)
</pre></div>

<p>The special floating point routines get called when the arguments
are complex.  For example,
</p>
<div class="example">
<pre class="example">(%i1) legendre_p (10, 2 + 3.0*%i);
(%o1) interval(- 3.876378825E+7 %i - 6.0787748E+7, 
                                           1.2089173052721777E-6)
</pre></div>

<p>Let&rsquo;s compare this to the true value.
</p>
<div class="example">
<pre class="example">(%i1) float (expand (legendre_p (10, 2 + 3*%i)));
(%o1)          - 3.876378825E+7 %i - 6.0787748E+7
</pre></div>

<p>Additionally, when the arguments are big floats, the special floating point
routines get called; however, the big floats are converted into double floats
and the final result is a double.
</p>
<div class="example">
<pre class="example">(%i1) ultraspherical (150, 0.5b0, 0.9b0);
(%o1) interval(- 0.043009481257265, 3.3750051301228864E-14)
</pre></div>

<a name="Graphics-and-orthopoly"></a>
<h4 class="subsection">79.1.4 Graphics and <code>orthopoly</code></h4>

<p>To plot expressions that involve the orthogonal polynomials, you 
must do two things:
</p><ol>
<li> Set the option variable <code>orthopoly_returns_intervals</code> to <code>false</code>,
</li><li> Quote any calls to <code>orthopoly</code> functions.
</li></ol>
<p>If function calls aren&rsquo;t quoted, Maxima evaluates them to polynomials before 
plotting; consequently, the specialized floating point code doesn&rsquo;t get called.
Here is an example of how to plot an expression that involves
a Legendre polynomial.
</p>
<div class="example">
<pre class="example">(%i1) plot2d ('(legendre_p (5, x)), [x, 0, 1]),
                        orthopoly_returns_intervals : false;
(%o1)
</pre></div>

<img src="figures/orthopoly1.png" alt="figures/orthopoly1">

<p>The <i>entire</i> expression <code>legendre_p (5, x)</code> is quoted; this is 
different than just quoting the function name using <code>'legendre_p (5, <var>x</var>)</code>.
</p>
<div class=categorybox>&middot;<p>Categories:&nbsp;&nbsp;<a href="maxima_369.html#Category_003a-Plotting">Plotting</a>
</div></p>

<a name="Miscellaneous-Functions"></a>
<h4 class="subsection">79.1.5 Miscellaneous Functions</h4>

<p>The <code>orthopoly</code> package defines the
Pochhammer symbol and an unit step function. <code>orthopoly</code> uses 
the Kronecker delta function and the unit step function in
<code>gradef</code> statements.
</p>
<p>To convert Pochhammer symbols into quotients of gamma functions,
use <code>makegamma</code>.
</p>
<div class="example">
<pre class="example">(%i1) makegamma (pochhammer (x, n));
                          gamma(x + n)
(%o1)                     ------------
                            gamma(x)
(%i2) makegamma (pochhammer (1/2, 1/2));
                                1
(%o2)                       ---------
                            sqrt(%pi)
</pre></div>

<p>Derivatives of the Pochhammer symbol are given in terms of the <code>psi</code>
function.
</p>
<div class="example">
<pre class="example">(%i1) diff (pochhammer (x, n), x);
(%o1)             (x)  (psi (x + n) - psi (x))
                     n     0             0
(%i2) diff (pochhammer (x, n), n);
(%o2)                   (x)  psi (x + n)
                           n    0
</pre></div>

<p>You need to be careful with the expression in (%o1); the difference of the
<code>psi</code> functions has polynomials when <code><var>x</var> = -1, -2, .., -<var>n</var></code>. These polynomials
cancel with factors in <code>pochhammer (<var>x</var>, <var>n</var>)</code> making the derivative a degree
<code><var>n</var> - 1</code> polynomial when <var>n</var> is a positive integer.
</p>
<p>The Pochhammer symbol is defined for negative orders through its
representation as a quotient of gamma functions. Consider
</p>
<div class="example">
<pre class="example">(%i1) q : makegamma (pochhammer (x, n));
                          gamma(x + n)
(%o1)                     ------------
                            gamma(x)
(%i2) sublis ([x=11/3, n= -6], q);
                               729
(%o2)                        - ----
                               2240
</pre></div>

<p>Alternatively, we can get this result directly.
</p>
<div class="example">
<pre class="example">(%i1) pochhammer (11/3, -6);
                               729
(%o1)                        - ----
                               2240
</pre></div>

<p>The unit step function is left-continuous; thus
</p>
<div class="example">
<pre class="example">(%i1) [unit_step (-1/10), unit_step (0), unit_step (1/10)];
(%o1)                       [0, 0, 1]
</pre></div>

<p>If you need an unit step function that is neither left or right continuous
at zero, define your own using <code>signum</code>; for example,
</p>
<div class="example">
<pre class="example">(%i1) xunit_step (x) := (1 + signum (x))/2$

(%i2) [xunit_step (-1/10), xunit_step (0), xunit_step (1/10)];
                                1
(%o2)                       [0, -, 1]
                                2
</pre></div>

<p>Do not redefine <code>unit_step</code> itself; some code in <code>orthopoly</code>
requires that the unit step function be left-continuous.
</p>
<a name="Algorithms"></a>
<h4 class="subsection">79.1.6 Algorithms</h4>

<p>Generally, <code>orthopoly</code> does symbolic evaluation by using a hypergeometic 
representation of the orthogonal polynomials. The hypergeometic 
functions are evaluated using the (undocumented) functions <code>hypergeo11</code> 
and <code>hypergeo21</code>. The exceptions are the half-integer Bessel functions 
and the associated Legendre function of the second kind. The half-integer Bessel functions are
evaluated using an explicit representation, and the associated Legendre 
function of the second kind is evaluated using recursion.
</p>
<p>For floating point evaluation, we again convert most functions into
a hypergeometic form; we evaluate the hypergeometic functions using 
forward recursion. Again, the exceptions are the half-integer Bessel functions 
and the associated Legendre function of the second kind. Numerically, 
the half-integer Bessel functions are evaluated using the SLATEC code.
</p>

<a name="Item_003a-orthopoly_002fnode_002fFunctions-and-Variables-for-orthogonal-polynomials"></a><hr>
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