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<title>The Lanczos Approximation</title>
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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.lanczos"></a><a class="link" href="lanczos.html" title="The Lanczos Approximation">The Lanczos Approximation</a>
</h2></div></div></div>
<h5>
<a name="math_toolkit.lanczos.h0"></a>
      <span class="phrase"><a name="math_toolkit.lanczos.motivation"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.motivation">Motivation</a>
    </h5>
<p>
      <span class="emphasis"><em>Why base gamma and gamma-like functions on the Lanczos approximation?</em></span>
    </p>
<p>
      First of all I should make clear that for the gamma function over real numbers
      (as opposed to complex ones) the Lanczos approximation (See <a href="http://en.wikipedia.org/wiki/Lanczos_approximation" target="_top">Wikipedia
      or </a> <a href="http://mathworld.wolfram.com/LanczosApproximation.html" target="_top">Mathworld</a>)
      appears to offer no clear advantage over more traditional methods such as
      <a href="http://en.wikipedia.org/wiki/Stirling_approximation" target="_top">Stirling's
      approximation</a>. <a class="link" href="lanczos.html#pugh">Pugh</a> carried out an extensive
      comparison of the various methods available and discovered that they were all
      very similar in terms of complexity and relative error. However, the Lanczos
      approximation does have a couple of properties that make it worthy of further
      consideration:
    </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
          The approximation has an easy to compute truncation error that holds for
          all <span class="emphasis"><em>z &gt; 0</em></span>. In practice that means we can use the
          same approximation for all <span class="emphasis"><em>z &gt; 0</em></span>, and be certain
          that no matter how large or small <span class="emphasis"><em>z</em></span> is, the truncation
          error will <span class="emphasis"><em>at worst</em></span> be bounded by some finite value.
        </li>
<li class="listitem">
          The approximation has a form that is particularly amenable to analytic
          manipulation, in particular ratios of gamma or gamma-like functions are
          particularly easy to compute without resorting to logarithms.
        </li>
</ul></div>
<p>
      It is the combination of these two properties that make the approximation attractive:
      Stirling's approximation is highly accurate for large z, and has some of the
      same analytic properties as the Lanczos approximation, but can't easily be
      used across the whole range of z.
    </p>
<p>
      As the simplest example, consider the ratio of two gamma functions: one could
      compute the result via lgamma:
    </p>
<pre class="programlisting"><span class="identifier">exp</span><span class="special">(</span><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">b</span><span class="special">));</span>
</pre>
<p>
      However, even if lgamma is uniformly accurate to 0.5ulp, the worst case relative
      error in the above can easily be shown to be:
    </p>
<pre class="programlisting"><span class="identifier">Erel</span> <span class="special">&gt;</span> <span class="identifier">a</span> <span class="special">*</span> <span class="identifier">log</span><span class="special">(</span><span class="identifier">a</span><span class="special">)/</span><span class="number">2</span> <span class="special">+</span> <span class="identifier">b</span> <span class="special">*</span> <span class="identifier">log</span><span class="special">(</span><span class="identifier">b</span><span class="special">)/</span><span class="number">2</span>
</pre>
<p>
      For small <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> that's not a problem,
      but to put the relationship another way: <span class="emphasis"><em>each time a and b increase
      in magnitude by a factor of 10, at least one decimal digit of precision will
      be lost.</em></span>
    </p>
<p>
      In contrast, by analytically combining like power terms in a ratio of Lanczos
      approximation's, these errors can be virtually eliminated for small <span class="emphasis"><em>a</em></span>
      and <span class="emphasis"><em>b</em></span>, and kept under control for very large (or very
      small for that matter) <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>. Of
      course, computing large powers is itself a notoriously hard problem, but even
      so, analytic combinations of Lanczos approximations can make the difference
      between obtaining a valid result, or simply garbage. Refer to the implementation
      notes for the <a class="link" href="sf_beta/beta_function.html" title="Beta">beta</a>
      function for an example of this method in practice. The incomplete <a class="link" href="sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p
      gamma</a> and <a class="link" href="sf_beta/ibeta_function.html" title="Incomplete Beta Functions">beta</a>
      functions use similar analytic combinations of power terms, to combine gamma
      and beta functions divided by large powers into single (simpler) expressions.
    </p>
<h5>
<a name="math_toolkit.lanczos.h1"></a>
      <span class="phrase"><a name="math_toolkit.lanczos.the_approximation"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.the_approximation">The
      Approximation</a>
    </h5>
<p>
      The Lanczos Approximation to the Gamma Function is given by:
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        <span class="inlinemediaobject"><img src="../../equations/lanczos0.svg"></span>

      </p></blockquote></div>
<p>
      Where S<sub>g</sub>(z) is an infinite sum, that is convergent for all z &gt; 0, and <span class="emphasis"><em>g</em></span>
      is an arbitrary parameter that controls the "shape" of the terms
      in the sum which is given by:
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        <span class="inlinemediaobject"><img src="../../equations/lanczos0a.svg"></span>

      </p></blockquote></div>
<p>
      With individual coefficients defined in closed form by:
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        <span class="inlinemediaobject"><img src="../../equations/lanczos0b.svg"></span>

      </p></blockquote></div>
<p>
      However, evaluation of the sum in that form can lead to numerical instability
      in the computation of the ratios of rising and falling factorials (effectively
      we're multiplying by a series of numbers very close to 1, so roundoff errors
      can accumulate quite rapidly).
    </p>
<p>
      The Lanczos approximation is therefore often written in partial fraction form
      with the leading constants absorbed by the coefficients in the sum:
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        <span class="inlinemediaobject"><img src="../../equations/lanczos1.svg"></span>

      </p></blockquote></div>
<p>
      where:
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        <span class="inlinemediaobject"><img src="../../equations/lanczos2.svg"></span>

      </p></blockquote></div>
<p>
      Again parameter <span class="emphasis"><em>g</em></span> is an arbitrarily chosen constant, and
      <span class="emphasis"><em>N</em></span> is an arbitrarily chosen number of terms to evaluate
      in the "Lanczos sum" part.
    </p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
        Some authors choose to define the sum from k=1 to N, and hence end up with
        N+1 coefficients. This happens to confuse both the following discussion and
        the code (since C++ deals with half open array ranges, rather than the closed
        range of the sum). This convention is consistent with <a class="link" href="lanczos.html#godfrey">Godfrey</a>,
        but not <a class="link" href="lanczos.html#pugh">Pugh</a>, so take care when referring to
        the literature in this field.
      </p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.lanczos.h2"></a>
      <span class="phrase"><a name="math_toolkit.lanczos.computing_the_coefficients"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.computing_the_coefficients">Computing
      the Coefficients</a>
    </h5>
<p>
      The coefficients C0..CN-1 need to be computed from <span class="emphasis"><em>N</em></span> and
      <span class="emphasis"><em>g</em></span> at high precision, and then stored as part of the program.
      Calculation of the coefficients is performed via the method of <a class="link" href="lanczos.html#godfrey">Godfrey</a>;
      let the constants be contained in a column vector P, then:
    </p>
<p>
      P = D B C F
    </p>
<p>
      where B is an NxN matrix:
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        <span class="inlinemediaobject"><img src="../../equations/lanczos4.svg"></span>

      </p></blockquote></div>
<p>
      D is an NxN matrix:
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        <span class="inlinemediaobject"><img src="../../equations/lanczos3.svg"></span>

      </p></blockquote></div>
<p>
      C is an NxN matrix:
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        <span class="inlinemediaobject"><img src="../../equations/lanczos5.svg"></span>

      </p></blockquote></div>
<p>
      and F is an N element column vector:
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        <span class="inlinemediaobject"><img src="../../equations/lanczos6.svg"></span>

      </p></blockquote></div>
<p>
      Note than the matrices B, D and C contain all integer terms and depend only
      on <span class="emphasis"><em>N</em></span>, this product should be computed first, and then
      multiplied by <span class="emphasis"><em>F</em></span> as the last step.
    </p>
<h5>
<a name="math_toolkit.lanczos.h3"></a>
      <span class="phrase"><a name="math_toolkit.lanczos.choosing_the_right_parameters"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.choosing_the_right_parameters">Choosing
      the Right Parameters</a>
    </h5>
<p>
      The trick is to choose <span class="emphasis"><em>N</em></span> and <span class="emphasis"><em>g</em></span> to
      give the desired level of accuracy: choosing a small value for <span class="emphasis"><em>g</em></span>
      leads to a strictly convergent series, but one which converges only slowly.
      Choosing a larger value of <span class="emphasis"><em>g</em></span> causes the terms in the series
      to be large and/or divergent for about the first <span class="emphasis"><em>g-1</em></span> terms,
      and to then suddenly converge with a "crunch".
    </p>
<p>
      <a class="link" href="lanczos.html#pugh">Pugh</a> has determined the optimal value of <span class="emphasis"><em>g</em></span>
      for <span class="emphasis"><em>N</em></span> in the range <span class="emphasis"><em>1 &lt;= N &lt;= 60</em></span>:
      unfortunately in practice choosing these values leads to cancellation errors
      in the Lanczos sum as the largest term in the (alternating) series is approximately
      1000 times larger than the result. These optimal values appear not to be useful
      in practice unless the evaluation can be done with a number of guard digits
      <span class="emphasis"><em>and</em></span> the coefficients are stored at higher precision than
      that desired in the result. These values are best reserved for say, computing
      to float precision with double precision arithmetic.
    </p>
<div class="table">
<a name="math_toolkit.lanczos.optimal_choices_for_n_and_g_when"></a><p class="title"><b>Table&#160;22.1.&#160;Optimal choices for N and g when computing with guard digits (source:
      Pugh)</b></p>
<div class="table-contents"><table class="table" summary="Optimal choices for N and g when computing with guard digits (source:
      Pugh)">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              <p>
                Significand Size
              </p>
            </th>
<th>
              <p>
                N
              </p>
            </th>
<th>
              <p>
                g
              </p>
            </th>
<th>
              <p>
                Max Error
              </p>
            </th>
</tr></thead>
<tbody>
<tr>
<td>
              <p>
                24
              </p>
            </td>
<td>
              <p>
                6
              </p>
            </td>
<td>
              <p>
                5.581
              </p>
            </td>
<td>
              <p>
                9.51e-12
              </p>
            </td>
</tr>
<tr>
<td>
              <p>
                53
              </p>
            </td>
<td>
              <p>
                13
              </p>
            </td>
<td>
              <p>
                13.144565
              </p>
            </td>
<td>
              <p>
                9.2213e-23
              </p>
            </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
      The alternative described by <a class="link" href="lanczos.html#godfrey">Godfrey</a> is to perform
      an exhaustive search of the <span class="emphasis"><em>N</em></span> and <span class="emphasis"><em>g</em></span>
      parameter space to determine the optimal combination for a given <span class="emphasis"><em>p</em></span>
      digit floating-point type. Repeating this work found a good approximation for
      double precision arithmetic (close to the one <a class="link" href="lanczos.html#godfrey">Godfrey</a>
      found), but failed to find really good approximations for 80 or 128-bit long
      doubles. Further it was observed that the approximations obtained tended to
      optimised for the small values of z (1 &lt; z &lt; 200) used to test the implementation
      against the factorials. Computing ratios of gamma functions with large arguments
      were observed to suffer from error resulting from the truncation of the Lancozos
      series.
    </p>
<p>
      <a class="link" href="lanczos.html#pugh">Pugh</a> identified all the locations where the theoretical
      error of the approximation were at a minimum, but unfortunately has published
      only the largest of these minima. However, he makes the observation that the
      minima coincide closely with the location where the first neglected term (a<sub>N</sub>)
      in the Lanczos series S<sub>g</sub>(z) changes sign. These locations are quite easy to
      locate, albeit with considerable computer time. These "sweet spots"
      need only be computed once, tabulated, and then searched when required for
      an approximation that delivers the required precision for some fixed precision
      type.
    </p>
<p>
      Unfortunately, following this path failed to find a really good approximation
      for 128-bit long doubles, and those found for 64 and 80-bit reals required
      an excessive number of terms. There are two competing issues here: high precision
      requires a large value of <span class="emphasis"><em>g</em></span>, but avoiding cancellation
      errors in the evaluation requires a small <span class="emphasis"><em>g</em></span>.
    </p>
<p>
      At this point note that the Lanczos sum can be converted into rational form
      (a ratio of two polynomials, obtained from the partial-fraction form using
      polynomial arithmetic), and doing so changes the coefficients so that <span class="emphasis"><em>they
      are all positive</em></span>. That means that the sum in rational form can be
      evaluated without cancellation error, albeit with double the number of coefficients
      for a given N. Repeating the search of the "sweet spots", this time
      evaluating the Lanczos sum in rational form, and testing only those "sweet
      spots" whose theoretical error is less than the machine epsilon for the
      type being tested, yielded good approximations for all the types tested. The
      optimal values found were quite close to the best cases reported by <a class="link" href="lanczos.html#pugh">Pugh</a>
      (just slightly larger <span class="emphasis"><em>N</em></span> and slightly smaller <span class="emphasis"><em>g</em></span>
      for a given precision than <a class="link" href="lanczos.html#pugh">Pugh</a> reports), and even
      though converting to rational form doubles the number of stored coefficients,
      it should be noted that half of them are integers (and therefore require less
      storage space) and the approximations require a smaller <span class="emphasis"><em>N</em></span>
      than would otherwise be required, so fewer floating point operations may be
      required overall.
    </p>
<p>
      The following table shows the optimal values for <span class="emphasis"><em>N</em></span> and
      <span class="emphasis"><em>g</em></span> when computing at fixed precision. These should be taken
      as work in progress: there are no values for 106-bit significand machines (Darwin
      long doubles &amp; NTL quad_float), and further optimisation of the values
      of <span class="emphasis"><em>g</em></span> may be possible. Errors given in the table are estimates
      of the error due to truncation of the Lanczos infinite series to <span class="emphasis"><em>N</em></span>
      terms. They are calculated from the sum of the first five neglected terms -
      and are known to be rather pessimistic estimates - although it is noticeable
      that the best combinations of <span class="emphasis"><em>N</em></span> and <span class="emphasis"><em>g</em></span>
      occurred when the estimated truncation error almost exactly matches the machine
      epsilon for the type in question.
    </p>
<div class="table">
<a name="math_toolkit.lanczos.optimum_value_for_n_and_g_when_c"></a><p class="title"><b>Table&#160;22.2.&#160;Optimum value for N and g when computing at fixed precision</b></p>
<div class="table-contents"><table class="table" summary="Optimum value for N and g when computing at fixed precision">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              <p>
                Significand Size
              </p>
            </th>
<th>
              <p>
                Platform/Compiler Used
              </p>
            </th>
<th>
              <p>
                N
              </p>
            </th>
<th>
              <p>
                g
              </p>
            </th>
<th>
              <p>
                Max Truncation Error
              </p>
            </th>
</tr></thead>
<tbody>
<tr>
<td>
              <p>
                24
              </p>
            </td>
<td>
              <p>
                Win32, VC++ 7.1
              </p>
            </td>
<td>
              <p>
                6
              </p>
            </td>
<td>
              <p>
                1.428456135094165802001953125
              </p>
            </td>
<td>
              <p>
                9.41e-007
              </p>
            </td>
</tr>
<tr>
<td>
              <p>
                53
              </p>
            </td>
<td>
              <p>
                Win32, VC++ 7.1
              </p>
            </td>
<td>
              <p>
                13
              </p>
            </td>
<td>
              <p>
                6.024680040776729583740234375
              </p>
            </td>
<td>
              <p>
                3.23e-016
              </p>
            </td>
</tr>
<tr>
<td>
              <p>
                64
              </p>
            </td>
<td>
              <p>
                Suse Linux 9 IA64, gcc-3.3.3
              </p>
            </td>
<td>
              <p>
                17
              </p>
            </td>
<td>
              <p>
                12.2252227365970611572265625
              </p>
            </td>
<td>
              <p>
                2.34e-024
              </p>
            </td>
</tr>
<tr>
<td>
              <p>
                116
              </p>
            </td>
<td>
              <p>
                HP Tru64 Unix 5.1B / Alpha, Compaq C++ V7.1-006
              </p>
            </td>
<td>
              <p>
                24
              </p>
            </td>
<td>
              <p>
                20.3209821879863739013671875
              </p>
            </td>
<td>
              <p>
                4.75e-035
              </p>
            </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
      Finally note that the Lanczos approximation can be written as follows by removing
      a factor of exp(g) from the denominator, and then dividing all the coefficients
      by exp(g):
    </p>
<div class="blockquote"><blockquote class="blockquote"><p>
        <span class="inlinemediaobject"><img src="../../equations/lanczos7.svg"></span>

      </p></blockquote></div>
<p>
      This form is more convenient for calculating lgamma, but for the gamma function
      the division by <span class="emphasis"><em>e</em></span> turns a possibly exact quality into
      an inexact value: this reduces accuracy in the common case that the input is
      exact, and so isn't used for the gamma function.
    </p>
<h5>
<a name="math_toolkit.lanczos.h4"></a>
      <span class="phrase"><a name="math_toolkit.lanczos.references"></a></span><a class="link" href="lanczos.html#math_toolkit.lanczos.references">References</a>
    </h5>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
          <a name="godfrey"></a>Paul Godfrey, <a href="http://my.fit.edu/~gabdo/gamma.txt" target="_top">"A
          note on the computation of the convergent Lanczos complex Gamma approximation"</a>.
        </li>
<li class="listitem">
          <a name="pugh"></a>Glendon Ralph Pugh, <a href="http://bh0.physics.ubc.ca/People/matt/Doc/ThesesOthers/Phd/pugh.pdf" target="_top">"An
          Analysis of the Lanczos Gamma Approximation"</a>, PhD Thesis November
          2004.
        </li>
<li class="listitem">
          Viktor T. Toth, <a href="http://www.rskey.org/gamma.htm" target="_top">"Calculators
          and the Gamma Function"</a>.
        </li>
<li class="listitem">
          Mathworld, <a href="http://mathworld.wolfram.com/LanczosApproximation.html" target="_top">The
          Lanczos Approximation</a>.
        </li>
</ol></div>
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