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<div class="section" id="function-approximation">
<h1>Function approximation<a class="headerlink" href="#function-approximation" title="Permalink to this headline">¶</a></h1>
<div class="section" id="taylor-series-taylor">
<h2>Taylor series (<tt class="docutils literal"><span class="pre">taylor</span></tt>)<a class="headerlink" href="#taylor-series-taylor" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="mpmath.taylor">
<tt class="descclassname">mpmath.</tt><tt class="descname">taylor</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x</em>, <em>n</em>, <em>**options</em><big>)</big><a class="headerlink" href="#mpmath.taylor" title="Permalink to this definition">¶</a></dt>
<dd><p>Produces a degree-<img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/> Taylor polynomial around the point <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/> of the
given function <img class="math" src="../_images/math/bb2c93730dbb48558bb3c4738c956c4e8f816437.png" alt="f"/>. The coefficients are returned as a list.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>
<span class="go">[0.0, 1.0, 0.0, -0.166667, 0.0, 0.00833333]</span>
</pre></div>
</div>
<p>The coefficients are computed using high-order numerical
differentiation. The function must be possible to evaluate
to arbitrary precision. See <a title="mpmath.diff" class="reference external" href="differentiation.html#mpmath.diff"><tt class="xref docutils literal"><span class="pre">diff()</span></tt></a> for additional details
and supported keyword options.</p>
<p>Note that to evaluate the Taylor polynomial as an approximation
of <img class="math" src="../_images/math/bb2c93730dbb48558bb3c4738c956c4e8f816437.png" alt="f"/>, e.g. with <a title="mpmath.polyval" class="reference external" href="polynomials.html#mpmath.polyval"><tt class="xref docutils literal"><span class="pre">polyval()</span></tt></a>, the coefficients must be reversed,
and the point of the Taylor expansion must be subtracted from
the argument:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">p</span> <span class="o">=</span> <span class="n">taylor</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">polyval</span><span class="p">(</span><span class="n">p</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mf">2.5</span> <span class="o">-</span> <span class="mf">2.0</span><span class="p">)</span>
<span class="go">12.1824939606092</span>
<span class="gp">>>> </span><span class="n">exp</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
<span class="go">12.1824939607035</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="pade-approximation-pade">
<h2>Pade approximation (<tt class="docutils literal"><span class="pre">pade</span></tt>)<a class="headerlink" href="#pade-approximation-pade" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="mpmath.pade">
<tt class="descclassname">mpmath.</tt><tt class="descname">pade</tt><big>(</big><em>ctx</em>, <em>a</em>, <em>L</em>, <em>M</em><big>)</big><a class="headerlink" href="#mpmath.pade" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes a Pade approximation of degree <img class="math" src="../_images/math/a2cc450ebacef8e05cc6ade46aea6a86708d6494.png" alt="(L, M)"/> to a function.
Given at least <img class="math" src="../_images/math/e73609e829d32a9e767c52e3dd564e4d3d8b66b7.png" alt="L+M+1"/> Taylor coefficients <img class="math" src="../_images/math/c7d457e388298246adb06c587bccd419ea67f7e8.png" alt="a"/> approximating
a function <img class="math" src="../_images/math/133abca2101afa88a51f60a6747c47e699a858e5.png" alt="A(x)"/>, <a title="mpmath.pade" class="reference internal" href="#mpmath.pade"><tt class="xref docutils literal"><span class="pre">pade()</span></tt></a> returns coefficients of
polynomials <img class="math" src="../_images/math/41bd81b0ae820e8ae1ecb70cb3f2ebfde5a42b4a.png" alt="P, Q"/> satisfying</p>
<div class="math">
<p><img src="../_images/math/c250a438f776950177437773eafd535d96f5f71b.png" alt="P = \sum_{k=0}^L p_k x^k
Q = \sum_{k=0}^M q_k x^k
Q_0 = 1
A(x) Q(x) = P(x) + O(x^{L+M+1})" /></p>
</div><p><img class="math" src="../_images/math/e1be778da9a55145063f22cf962c2f6b88befd61.png" alt="P(x)/Q(x)"/> can provide a good approximation to an analytic function
beyond the radius of convergence of its Taylor series (example
from G.A. Baker ‘Essentials of Pade Approximants’ Academic Press,
Ch.1A):</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">one</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">sqrt</span><span class="p">((</span><span class="n">one</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">one</span> <span class="o">+</span> <span class="n">x</span><span class="p">))</span>
<span class="gp">...</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">taylor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">pade</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="mi">10</span>
<span class="gp">>>> </span><span class="n">polyval</span><span class="p">(</span><span class="n">p</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">polyval</span><span class="p">(</span><span class="n">q</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
<span class="go">1.38169105566806</span>
<span class="gp">>>> </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">1.38169855941551</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="chebyshev-approximation-chebyfit">
<h2>Chebyshev approximation (<tt class="docutils literal"><span class="pre">chebyfit</span></tt>)<a class="headerlink" href="#chebyshev-approximation-chebyfit" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="mpmath.chebyfit">
<tt class="descclassname">mpmath.</tt><tt class="descname">chebyfit</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>interval</em>, <em>N</em>, <em>error=False</em><big>)</big><a class="headerlink" href="#mpmath.chebyfit" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes a polynomial of degree <img class="math" src="../_images/math/a256c70ad4c46ec1127c5be68f8bb3075e9ced31.png" alt="N-1"/> that approximates the
given function <img class="math" src="../_images/math/bb2c93730dbb48558bb3c4738c956c4e8f816437.png" alt="f"/> on the interval <img class="math" src="../_images/math/da2e551d2ca2155b8d8f4935d2e9757722c9bab6.png" alt="[a, b]"/>. With <tt class="docutils literal"><span class="pre">error=True</span></tt>,
<a title="mpmath.chebyfit" class="reference internal" href="#mpmath.chebyfit"><tt class="xref docutils literal"><span class="pre">chebyfit()</span></tt></a> also returns an accurate estimate of the
maximum absolute error; that is, the maximum value of
<img class="math" src="../_images/math/275078e7919d5cd6715724004e7013ddbe0b4973.png" alt="|f(x) - P(x)|"/> for <img class="math" src="../_images/math/ef57ada8ccf733fecb08daf2503f42216a801934.png" alt="x \in [a, b]"/>.</p>
<p><a title="mpmath.chebyfit" class="reference internal" href="#mpmath.chebyfit"><tt class="xref docutils literal"><span class="pre">chebyfit()</span></tt></a> uses the Chebyshev approximation formula,
which gives a nearly optimal solution: that is, the maximum
error of the approximating polynomial is very close to
the smallest possible for any polynomial of the same degree.</p>
<p>Chebyshev approximation is very useful if one needs repeated
evaluation of an expensive function, such as function defined
implicitly by an integral or a differential equation. (For
example, it could be used to turn a slow mpmath function
into a fast machine-precision version of the same.)</p>
<p><strong>Examples</strong></p>
<p>Here we use <a title="mpmath.chebyfit" class="reference internal" href="#mpmath.chebyfit"><tt class="xref docutils literal"><span class="pre">chebyfit()</span></tt></a> to generate a low-degree approximation
of <img class="math" src="../_images/math/c2644cc9ce7c9379a1398bed5ce9626ba5491d9f.png" alt="f(x) = \cos(x)"/>, valid on the interval <img class="math" src="../_images/math/f6e0fb93486b514b62f25bbf3c4724b4e53b3d36.png" alt="[1, 2]"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">poly</span><span class="p">,</span> <span class="n">err</span> <span class="o">=</span> <span class="n">chebyfit</span><span class="p">(</span><span class="n">cos</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">error</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
<span class="go">[0.00291682, 0.146166, -0.732491, 0.174141, 0.949553]</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">err</span><span class="p">,</span> <span class="mi">12</span><span class="p">)</span>
<span class="go">1.61351758081e-5</span>
</pre></div>
</div>
<p>The polynomial can be evaluated using <tt class="docutils literal"><span class="pre">polyval</span></tt>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">polyval</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="mf">1.6</span><span class="p">),</span> <span class="mi">12</span><span class="p">)</span>
<span class="go">-0.0291858904138</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="mf">1.6</span><span class="p">),</span> <span class="mi">12</span><span class="p">)</span>
<span class="go">-0.0291995223013</span>
</pre></div>
</div>
<p>Sampling the true error at 1000 points shows that the error
estimate generated by <tt class="docutils literal"><span class="pre">chebyfit</span></tt> is remarkably good:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">error</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">abs</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">polyval</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="nb">max</span><span class="p">([</span><span class="n">error</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">n</span><span class="o">/</span><span class="mf">1000.</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">)]),</span> <span class="mi">12</span><span class="p">)</span>
<span class="go">1.61349954245e-5</span>
</pre></div>
</div>
<p><strong>Choice of degree</strong></p>
<p>The degree <img class="math" src="../_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/> can be set arbitrarily high, to obtain an
arbitrarily good approximation. As a rule of thumb, an
<img class="math" src="../_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/>-term Chebyshev approximation is good to <img class="math" src="../_images/math/0c25ebe2a59d23cfa9536d78add3292711c79b16.png" alt="N/(b-a)"/> decimal
places on a unit interval (although this depends on how
well-behaved <img class="math" src="../_images/math/bb2c93730dbb48558bb3c4738c956c4e8f816437.png" alt="f"/> is). The cost grows accordingly: <tt class="docutils literal"><span class="pre">chebyfit</span></tt>
evaluates the function <img class="math" src="../_images/math/b7fcda37e581c55db282eb7e7939259bf2e02941.png" alt="(N^2)/2"/> times to compute the
coefficients and an additional <img class="math" src="../_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/> times to estimate the error.</p>
<p><strong>Possible issues</strong></p>
<p>One should be careful to use a sufficiently high working
precision both when calling <tt class="docutils literal"><span class="pre">chebyfit</span></tt> and when evaluating
the resulting polynomial, as the polynomial is sometimes
ill-conditioned. It is for example difficult to reach
15-digit accuracy when evaluating the polynomial using
machine precision floats, no matter the theoretical
accuracy of the polynomial. (The option to return the
coefficients in Chebyshev form should be made available
in the future.)</p>
<p>It is important to note the Chebyshev approximation works
poorly if <img class="math" src="../_images/math/bb2c93730dbb48558bb3c4738c956c4e8f816437.png" alt="f"/> is not smooth. A function containing singularities,
rapid oscillation, etc can be approximated more effectively by
multiplying it by a weight function that cancels out the
nonsmooth features, or by dividing the interval into several
segments.</p>
</dd></dl>
</div>
<div class="section" id="fourier-series-fourier-fourierval">
<h2>Fourier series (<tt class="docutils literal"><span class="pre">fourier</span></tt>, <tt class="docutils literal"><span class="pre">fourierval</span></tt>)<a class="headerlink" href="#fourier-series-fourier-fourierval" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="mpmath.fourier">
<tt class="descclassname">mpmath.</tt><tt class="descname">fourier</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>interval</em>, <em>N</em><big>)</big><a class="headerlink" href="#mpmath.fourier" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the Fourier series of degree <img class="math" src="../_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/> of the given function
on the interval <img class="math" src="../_images/math/da2e551d2ca2155b8d8f4935d2e9757722c9bab6.png" alt="[a, b]"/>. More precisely, <a title="mpmath.fourier" class="reference internal" href="#mpmath.fourier"><tt class="xref docutils literal"><span class="pre">fourier()</span></tt></a> returns
two lists <img class="math" src="../_images/math/6d3fe2ec7aa586eb870dffb5f076b05714ca91bb.png" alt="(c, s)"/> of coefficients (the cosine series and sine
series, respectively), such that</p>
<div class="math">
<p><img src="../_images/math/616dfedfc58ea503a6bb967e6dcaed6f745d8bc7.png" alt="f(x) \sim \sum_{k=0}^N
c_k \cos(k m) + s_k \sin(k m)" /></p>
</div><p>where <img class="math" src="../_images/math/b84bf9ebba38660860b353958230b555d0917674.png" alt="m = 2 \pi / (b-a)"/>.</p>
<p>Note that many texts define the first coefficient as <img class="math" src="../_images/math/f41ba6f0a01ad7e9a1451a2863d2f5e68bdce1ff.png" alt="2 c_0"/> instead
of <img class="math" src="../_images/math/0134af18ca53751c9f9ec73d48b7fa0ff8da9ea6.png" alt="c_0"/>. The easiest way to evaluate the computed series correctly
is to pass it to <a title="mpmath.fourierval" class="reference internal" href="#mpmath.fourierval"><tt class="xref docutils literal"><span class="pre">fourierval()</span></tt></a>.</p>
<p><strong>Examples</strong></p>
<p>The function <img class="math" src="../_images/math/427c0ee5f10895f708b681a16c4f5799b31f41a7.png" alt="f(x) = x"/> has a simple Fourier series on the standard
interval <img class="math" src="../_images/math/b35e10ec8f56c704d87c8b856a08c455e2f91ee3.png" alt="[-\pi, \pi]"/>. The cosine coefficients are all zero (because
the function has odd symmetry), and the sine coefficients are
rational numbers:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">c</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="n">fourier</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">pi</span><span class="p">,</span> <span class="n">pi</span><span class="p">],</span> <span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="go">[0.0, 0.0, 0.0, 0.0, 0.0, 0.0]</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="go">[0.0, 2.0, -1.0, 0.666667, -0.5, 0.4]</span>
</pre></div>
</div>
<p>This computes a Fourier series of a nonsymmetric function on
a nonstandard interval:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">I</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span>
<span class="gp">>>> </span><span class="n">cs</span> <span class="o">=</span> <span class="n">fourier</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">cs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="go">[0.583333, 1.12479, -1.27552, 0.904708, -0.441296]</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">cs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="go">[0.0, -2.6255, 0.580905, 0.219974, -0.540057]</span>
</pre></div>
</div>
<p>It is instructive to plot a function along with its truncated
Fourier series:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">plot</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">fourierval</span><span class="p">(</span><span class="n">cs</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">x</span><span class="p">)],</span> <span class="n">I</span><span class="p">)</span> <span class="c">#doctest: +SKIP</span>
</pre></div>
</div>
<p>Fourier series generally converge slowly (and may not converge
pointwise). For example, if <img class="math" src="../_images/math/98b7d89d080d3b48db075f5288db991f97bb0380.png" alt="f(x) = \cosh(x)"/>, a 10-term Fourier
series gives an <img class="math" src="../_images/math/07c3de5211dc08b13546363f8b139700a16dd963.png" alt="L^2"/> error corresponding to 2-digit accuracy:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">I</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">cs</span> <span class="o">=</span> <span class="n">fourier</span><span class="p">(</span><span class="n">cosh</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">g</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="n">cosh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">fourierval</span><span class="p">(</span><span class="n">cs</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">I</span><span class="p">)))</span>
<span class="go">0.00467963</span>
</pre></div>
</div>
<p><a title="mpmath.fourier" class="reference internal" href="#mpmath.fourier"><tt class="xref docutils literal"><span class="pre">fourier()</span></tt></a> uses numerical quadrature. For nonsmooth functions,
the accuracy (and speed) can be improved by including all singular
points in the interval specification:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">fourier</span><span class="p">(</span><span class="nb">abs</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">10</span><span class="p">)</span>
<span class="go">([0.5000441648], [0.0])</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">fourier</span><span class="p">(</span><span class="nb">abs</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">10</span><span class="p">)</span>
<span class="go">([0.5], [0.0])</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="mpmath.fourierval">
<tt class="descclassname">mpmath.</tt><tt class="descname">fourierval</tt><big>(</big><em>ctx</em>, <em>series</em>, <em>interval</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.fourierval" title="Permalink to this definition">¶</a></dt>
<dd><p>Evaluates a Fourier series (in the format computed by
by <a title="mpmath.fourier" class="reference internal" href="#mpmath.fourier"><tt class="xref docutils literal"><span class="pre">fourier()</span></tt></a> for the given interval) at the point <img class="math" src="../_images/math/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" alt="x"/>.</p>
<p>The series should be a pair <img class="math" src="../_images/math/6d3fe2ec7aa586eb870dffb5f076b05714ca91bb.png" alt="(c, s)"/> where <img class="math" src="../_images/math/3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc.png" alt="c"/> is the
cosine series and <img class="math" src="../_images/math/f37bba504894945c07a32f5496d74299a37aa51c.png" alt="s"/> is the sine series. The two lists
need not have the same length.</p>
</dd></dl>
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<ul>
<li><a class="reference external" href="#">Function approximation</a><ul>
<li><a class="reference external" href="#taylor-series-taylor">Taylor series (<tt class="docutils literal"><span class="pre">taylor</span></tt>)</a></li>
<li><a class="reference external" href="#pade-approximation-pade">Pade approximation (<tt class="docutils literal"><span class="pre">pade</span></tt>)</a></li>
<li><a class="reference external" href="#chebyshev-approximation-chebyfit">Chebyshev approximation (<tt class="docutils literal"><span class="pre">chebyfit</span></tt>)</a></li>
<li><a class="reference external" href="#fourier-series-fourier-fourierval">Fourier series (<tt class="docutils literal"><span class="pre">fourier</span></tt>, <tt class="docutils literal"><span class="pre">fourierval</span></tt>)</a></li>
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