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<div class="section" id="utility-functions">
<h1>Utility functions<a class="headerlink" href="#utility-functions" title="Permalink to this headline">¶</a></h1>
<p>This page lists functions that perform basic operations
on numbers or aid general programming.</p>
<div class="section" id="conversion-and-printing">
<h2>Conversion and printing<a class="headerlink" href="#conversion-and-printing" title="Permalink to this headline">¶</a></h2>
<div class="section" id="mpmathify-convert">
<h3><code class="xref py py-func docutils literal"><span class="pre">mpmathify()</span></code> / <code class="xref py py-func docutils literal"><span class="pre">convert()</span></code><a class="headerlink" href="#mpmathify-convert" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.mpmathify">
<code class="descclassname">mpmath.</code><code class="descname">mpmathify</code><span class="sig-paren">(</span><em>x</em>, <em>strings=True</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.mpmathify" title="Permalink to this definition">¶</a></dt>
<dd><p>Converts <em>x</em> to an <code class="docutils literal"><span class="pre">mpf</span></code> or <code class="docutils literal"><span class="pre">mpc</span></code>. If <em>x</em> is of type <code class="docutils literal"><span class="pre">mpf</span></code>,
<code class="docutils literal"><span class="pre">mpc</span></code>, <code class="docutils literal"><span class="pre">int</span></code>, <code class="docutils literal"><span class="pre">float</span></code>, <code class="docutils literal"><span class="pre">complex</span></code>, the conversion
will be performed losslessly.</p>
<p>If <em>x</em> is a string, the result will be rounded to the present
working precision. Strings representing fractions or complex
numbers are permitted.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">mpmathify</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
<span class="go">mpf('3.5')</span>
<span class="gp">>>> </span><span class="n">mpmathify</span><span class="p">(</span><span class="s1">'2.1'</span><span class="p">)</span>
<span class="go">mpf('2.1000000000000001')</span>
<span class="gp">>>> </span><span class="n">mpmathify</span><span class="p">(</span><span class="s1">'3/4'</span><span class="p">)</span>
<span class="go">mpf('0.75')</span>
<span class="gp">>>> </span><span class="n">mpmathify</span><span class="p">(</span><span class="s1">'2+3j'</span><span class="p">)</span>
<span class="go">mpc(real='2.0', imag='3.0')</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="nstr">
<h3><code class="xref py py-func docutils literal"><span class="pre">nstr()</span></code><a class="headerlink" href="#nstr" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.nstr">
<code class="descclassname">mpmath.</code><code class="descname">nstr</code><span class="sig-paren">(</span><em>x</em>, <em>n=6</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.nstr" title="Permalink to this definition">¶</a></dt>
<dd><p>Convert an <code class="docutils literal"><span class="pre">mpf</span></code> or <code class="docutils literal"><span class="pre">mpc</span></code> to a decimal string literal with <em>n</em>
significant digits. The small default value for <em>n</em> is chosen to
make this function useful for printing collections of numbers
(lists, matrices, etc).</p>
<p>If <em>x</em> is a list or tuple, <a class="reference internal" href="#mpmath.nstr" title="mpmath.nstr"><code class="xref py py-func docutils literal"><span class="pre">nstr()</span></code></a> is applied recursively
to each element. For unrecognized classes, <a class="reference internal" href="#mpmath.nstr" title="mpmath.nstr"><code class="xref py py-func docutils literal"><span class="pre">nstr()</span></code></a>
simply returns <code class="docutils literal"><span class="pre">str(x)</span></code>.</p>
<p>The companion function <a class="reference internal" href="#mpmath.nprint" title="mpmath.nprint"><code class="xref py py-func docutils literal"><span class="pre">nprint()</span></code></a> prints the result
instead of returning it.</p>
<p>The keyword arguments <em>strip_zeros</em>, <em>min_fixed</em>, <em>max_fixed</em>
and <em>show_zero_exponent</em> are forwarded to <code class="xref py py-func docutils literal"><span class="pre">to_str()</span></code>.</p>
<p>The number will be printed in fixed-point format if the position
of the leading digit is strictly between min_fixed
(default = min(-dps/3,-5)) and max_fixed (default = dps).</p>
<p>To force fixed-point format always, set min_fixed = -inf,
max_fixed = +inf. To force floating-point format, set
min_fixed >= max_fixed.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">nstr</span><span class="p">([</span><span class="o">+</span><span class="n">pi</span><span class="p">,</span> <span class="n">ldexp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">500</span><span class="p">)])</span>
<span class="go">'[3.14159, 3.05494e-151]'</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">([</span><span class="o">+</span><span class="n">pi</span><span class="p">,</span> <span class="n">ldexp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">500</span><span class="p">)])</span>
<span class="go">[3.14159, 3.05494e-151]</span>
<span class="gp">>>> </span><span class="n">nstr</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="s2">"5e-10"</span><span class="p">),</span> <span class="mi">5</span><span class="p">)</span>
<span class="go">'5.0e-10'</span>
<span class="gp">>>> </span><span class="n">nstr</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="s2">"5e-10"</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">strip_zeros</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="go">'5.0000e-10'</span>
<span class="gp">>>> </span><span class="n">nstr</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="s2">"5e-10"</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">strip_zeros</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">min_fixed</span><span class="o">=-</span><span class="mi">11</span><span class="p">)</span>
<span class="go">'0.00000000050000'</span>
<span class="gp">>>> </span><span class="n">nstr</span><span class="p">(</span><span class="n">mpf</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="n">show_zero_exponent</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="go">'0.0e+0'</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="nprint">
<h3><code class="xref py py-func docutils literal"><span class="pre">nprint()</span></code><a class="headerlink" href="#nprint" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.nprint">
<code class="descclassname">mpmath.</code><code class="descname">nprint</code><span class="sig-paren">(</span><em>x</em>, <em>n=6</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.nprint" title="Permalink to this definition">¶</a></dt>
<dd><p>Equivalent to <code class="docutils literal"><span class="pre">print(nstr(x,</span> <span class="pre">n))</span></code>.</p>
</dd></dl>
</div>
</div>
<div class="section" id="arithmetic-operations">
<h2>Arithmetic operations<a class="headerlink" href="#arithmetic-operations" title="Permalink to this headline">¶</a></h2>
<p>See also <a class="reference internal" href="functions/powers.html#mpmath.sqrt" title="mpmath.sqrt"><code class="xref py py-func docutils literal"><span class="pre">mpmath.sqrt()</span></code></a>, <a class="reference internal" href="functions/powers.html#mpmath.exp" title="mpmath.exp"><code class="xref py py-func docutils literal"><span class="pre">mpmath.exp()</span></code></a> etc., listed
in <a class="reference internal" href="functions/powers.html"><span class="doc">Powers and logarithms</span></a></p>
<div class="section" id="fadd">
<h3><code class="xref py py-func docutils literal"><span class="pre">fadd()</span></code><a class="headerlink" href="#fadd" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fadd">
<code class="descclassname">mpmath.</code><code class="descname">fadd</code><span class="sig-paren">(</span><em>x</em>, <em>y</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.fadd" title="Permalink to this definition">¶</a></dt>
<dd><p>Adds the numbers <em>x</em> and <em>y</em>, giving a floating-point result,
optionally using a custom precision and rounding mode.</p>
<p>The default precision is the working precision of the context.
You can specify a custom precision in bits by passing the <em>prec</em> keyword
argument, or by providing an equivalent decimal precision with the <em>dps</em>
keyword argument. If the precision is set to <code class="docutils literal"><span class="pre">+inf</span></code>, or if the flag
<em>exact=True</em> is passed, an exact addition with no rounding is performed.</p>
<p>When the precision is finite, the optional <em>rounding</em> keyword argument
specifies the direction of rounding. Valid options are <code class="docutils literal"><span class="pre">'n'</span></code> for
nearest (default), <code class="docutils literal"><span class="pre">'f'</span></code> for floor, <code class="docutils literal"><span class="pre">'c'</span></code> for ceiling, <code class="docutils literal"><span class="pre">'d'</span></code>
for down, <code class="docutils literal"><span class="pre">'u'</span></code> for up.</p>
<p><strong>Examples</strong></p>
<p>Using <a class="reference internal" href="#mpmath.fadd" title="mpmath.fadd"><code class="xref py py-func docutils literal"><span class="pre">fadd()</span></code></a> with precision and rounding control:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">)</span>
<span class="go">mpf('2.0')</span>
<span class="gp">>>> </span><span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s1">'u'</span><span class="p">)</span>
<span class="go">mpf('2.0000000000000004')</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="mi">100</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
<span class="go">2.00000000000000000001</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">15</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
<span class="go">2.0</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">25</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
<span class="go">2.00000000000000000001</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">True</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
<span class="go">2.00000000000000000001</span>
</pre></div>
</div>
<p>Exact addition avoids cancellation errors, enforcing familiar laws
of numbers such as <span class="math">\(x+y-x = y\)</span>, which don’t hold in floating-point
arithmetic with finite precision:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">mpf</span><span class="p">(</span><span class="s1">'1e-1000'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">inf</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span>
<span class="go">1.0e-1000</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span>
<span class="go">1.0e-1000</span>
</pre></div>
</div>
<p>Exact addition can be inefficient and may be impossible to perform
with large magnitude differences:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">fadd</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="s1">'1e-100000000000000000000'</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">inf</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">OverflowError</span>: <span class="n">the exact result does not fit in memory</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="fsub">
<h3><code class="xref py py-func docutils literal"><span class="pre">fsub()</span></code><a class="headerlink" href="#fsub" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fsub">
<code class="descclassname">mpmath.</code><code class="descname">fsub</code><span class="sig-paren">(</span><em>x</em>, <em>y</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.fsub" title="Permalink to this definition">¶</a></dt>
<dd><p>Subtracts the numbers <em>x</em> and <em>y</em>, giving a floating-point result,
optionally using a custom precision and rounding mode.</p>
<p>See the documentation of <a class="reference internal" href="#mpmath.fadd" title="mpmath.fadd"><code class="xref py py-func docutils literal"><span class="pre">fadd()</span></code></a> for a detailed description
of how to specify precision and rounding.</p>
<p><strong>Examples</strong></p>
<p>Using <a class="reference internal" href="#mpmath.fsub" title="mpmath.fsub"><code class="xref py py-func docutils literal"><span class="pre">fsub()</span></code></a> with precision and rounding control:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">fsub</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">)</span>
<span class="go">mpf('2.0')</span>
<span class="gp">>>> </span><span class="n">fsub</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s1">'d'</span><span class="p">)</span>
<span class="go">mpf('1.9999999999999998')</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">fsub</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="mi">100</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
<span class="go">1.99999999999999999999</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">fsub</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">15</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
<span class="go">2.0</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">fsub</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">25</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
<span class="go">1.99999999999999999999</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">(</span><span class="n">fsub</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">True</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
<span class="go">1.99999999999999999999</span>
</pre></div>
</div>
<p>Exact subtraction avoids cancellation errors, enforcing familiar laws
of numbers such as <span class="math">\(x-y+y = x\)</span>, which don’t hold in floating-point
arithmetic with finite precision:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">mpf</span><span class="p">(</span><span class="s1">'1e1000'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">fsub</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">inf</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
<span class="go">2.0</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">fsub</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
<span class="go">2.0</span>
</pre></div>
</div>
<p>Exact addition can be inefficient and may be impossible to perform
with large magnitude differences:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">fsub</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="s1">'1e-100000000000000000000'</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">inf</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">OverflowError</span>: <span class="n">the exact result does not fit in memory</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="fneg">
<h3><code class="xref py py-func docutils literal"><span class="pre">fneg()</span></code><a class="headerlink" href="#fneg" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fneg">
<code class="descclassname">mpmath.</code><code class="descname">fneg</code><span class="sig-paren">(</span><em>x</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.fneg" title="Permalink to this definition">¶</a></dt>
<dd><p>Negates the number <em>x</em>, giving a floating-point result, optionally
using a custom precision and rounding mode.</p>
<p>See the documentation of <a class="reference internal" href="#mpmath.fadd" title="mpmath.fadd"><code class="xref py py-func docutils literal"><span class="pre">fadd()</span></code></a> for a detailed description
of how to specify precision and rounding.</p>
<p><strong>Examples</strong></p>
<p>An mpmath number is returned:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">fneg</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
<span class="go">mpf('-2.5')</span>
<span class="gp">>>> </span><span class="n">fneg</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="o">+</span><span class="mi">2</span><span class="n">j</span><span class="p">)</span>
<span class="go">mpc(real='5.0', imag='-2.0')</span>
</pre></div>
</div>
<p>Precise control over rounding is possible:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-100</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">fneg</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="go">mpf('-2.0')</span>
<span class="gp">>>> </span><span class="n">fneg</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s1">'f'</span><span class="p">)</span>
<span class="go">mpf('-2.0000000000000004')</span>
</pre></div>
</div>
<p>Negating with and without roundoff:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="mi">200000000000000000000001</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="o">-</span><span class="n">mpf</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
<span class="go">-200000000000000016777216</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fneg</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
<span class="go">-200000000000000016777216</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fneg</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">)))</span>
<span class="go">-200000000000000000000001</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fneg</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">)))</span>
<span class="go">-200000000000000000000001</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fneg</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">inf</span><span class="p">)))</span>
<span class="go">-200000000000000000000001</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fneg</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="n">inf</span><span class="p">)))</span>
<span class="go">-200000000000000000000001</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fneg</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">True</span><span class="p">)))</span>
<span class="go">-200000000000000000000001</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="fmul">
<h3><code class="xref py py-func docutils literal"><span class="pre">fmul()</span></code><a class="headerlink" href="#fmul" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fmul">
<code class="descclassname">mpmath.</code><code class="descname">fmul</code><span class="sig-paren">(</span><em>x</em>, <em>y</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.fmul" title="Permalink to this definition">¶</a></dt>
<dd><p>Multiplies the numbers <em>x</em> and <em>y</em>, giving a floating-point result,
optionally using a custom precision and rounding mode.</p>
<p>See the documentation of <a class="reference internal" href="#mpmath.fadd" title="mpmath.fadd"><code class="xref py py-func docutils literal"><span class="pre">fadd()</span></code></a> for a detailed description
of how to specify precision and rounding.</p>
<p><strong>Examples</strong></p>
<p>The result is an mpmath number:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">fmul</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">)</span>
<span class="go">mpf('10.0')</span>
<span class="gp">>>> </span><span class="n">fmul</span><span class="p">(</span><span class="mf">0.5</span><span class="n">j</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="go">mpc(real='0.0', imag='0.25')</span>
</pre></div>
</div>
<p>Avoiding roundoff:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">15</span><span class="o">+</span><span class="mi">1</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
<span class="go">10000000001000010000000001</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">mpf</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
<span class="go">1.0000000001e+25</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">mpf</span><span class="p">(</span><span class="n">y</span><span class="p">)))</span>
<span class="go">10000000001000011026399232</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)))</span>
<span class="go">10000000001000011026399232</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">25</span><span class="p">)))</span>
<span class="go">10000000001000010000000001</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">True</span><span class="p">)))</span>
<span class="go">10000000001000010000000001</span>
</pre></div>
</div>
<p>Exact multiplication with complex numbers can be inefficient and may
be impossible to perform with large magnitude differences between
real and imaginary parts:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2</span><span class="n">j</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="n">mpc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="s1">'1e-100000000000000000000'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">fmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="go">mpc(real='2.0', imag='4.0')</span>
<span class="gp">>>> </span><span class="n">fmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s1">'u'</span><span class="p">)</span>
<span class="go">mpc(real='2.0', imag='4.0000000000000009')</span>
<span class="gp">>>> </span><span class="n">fmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">OverflowError</span>: <span class="n">the exact result does not fit in memory</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="fdiv">
<h3><code class="xref py py-func docutils literal"><span class="pre">fdiv()</span></code><a class="headerlink" href="#fdiv" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fdiv">
<code class="descclassname">mpmath.</code><code class="descname">fdiv</code><span class="sig-paren">(</span><em>x</em>, <em>y</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.fdiv" title="Permalink to this definition">¶</a></dt>
<dd><p>Divides the numbers <em>x</em> and <em>y</em>, giving a floating-point result,
optionally using a custom precision and rounding mode.</p>
<p>See the documentation of <a class="reference internal" href="#mpmath.fadd" title="mpmath.fadd"><code class="xref py py-func docutils literal"><span class="pre">fadd()</span></code></a> for a detailed description
of how to specify precision and rounding.</p>
<p><strong>Examples</strong></p>
<p>The result is an mpmath number:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="go">mpf('1.5')</span>
<span class="gp">>>> </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="go">mpf('0.66666666666666663')</span>
<span class="gp">>>> </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="go">mpc(real='4.0', imag='8.0')</span>
</pre></div>
</div>
<p>The rounding direction and precision can be controlled:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span> <span class="c1"># Should be accurate to at least 3 digits</span>
<span class="go">mpf('0.6666259765625')</span>
<span class="gp">>>> </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s1">'d'</span><span class="p">)</span>
<span class="go">mpf('0.66666666666666663')</span>
<span class="gp">>>> </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="mi">60</span><span class="p">)</span>
<span class="go">mpf('0.66666666666666667')</span>
<span class="gp">>>> </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s1">'u'</span><span class="p">)</span>
<span class="go">mpf('0.66666666666666674')</span>
</pre></div>
</div>
<p>Checking the error of a division by performing it at higher precision:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
<span class="go">mpf('-3.7007434154172148e-17')</span>
</pre></div>
</div>
<p>Unlike <a class="reference internal" href="#mpmath.fadd" title="mpmath.fadd"><code class="xref py py-func docutils literal"><span class="pre">fadd()</span></code></a>, <a class="reference internal" href="#mpmath.fmul" title="mpmath.fmul"><code class="xref py py-func docutils literal"><span class="pre">fmul()</span></code></a>, etc., exact division is not
allowed since the quotient of two floating-point numbers generally
does not have an exact floating-point representation. (In the
future this might be changed to allow the case where the division
is actually exact.)</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">ValueError</span>: <span class="n">division is not an exact operation</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="fmod">
<h3><code class="xref py py-func docutils literal"><span class="pre">fmod()</span></code><a class="headerlink" href="#fmod" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fmod">
<code class="descclassname">mpmath.</code><code class="descname">fmod</code><span class="sig-paren">(</span><em>x</em>, <em>y</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.fmod" title="Permalink to this definition">¶</a></dt>
<dd><p>Converts <span class="math">\(x\)</span> and <span class="math">\(y\)</span> to mpmath numbers and returns <span class="math">\(x \mod y\)</span>.
For mpmath numbers, this is equivalent to <code class="docutils literal"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></code>.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">True</span>
<span class="gp">>>> </span><span class="n">fmod</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span>
<span class="go">2.61062773871641</span>
</pre></div>
</div>
<p>You can use <a class="reference internal" href="#mpmath.fmod" title="mpmath.fmod"><code class="xref py py-func docutils literal"><span class="pre">fmod()</span></code></a> to compute fractional parts of numbers:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">fmod</span><span class="p">(</span><span class="mf">10.25</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">0.25</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="fsum">
<h3><code class="xref py py-func docutils literal"><span class="pre">fsum()</span></code><a class="headerlink" href="#fsum" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fsum">
<code class="descclassname">mpmath.</code><code class="descname">fsum</code><span class="sig-paren">(</span><em>terms</em>, <em>absolute=False</em>, <em>squared=False</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.fsum" title="Permalink to this definition">¶</a></dt>
<dd><p>Calculates a sum containing a finite number of terms (for infinite
series, see <a class="reference internal" href="calculus/sums_limits.html#mpmath.nsum" title="mpmath.nsum"><code class="xref py py-func docutils literal"><span class="pre">nsum()</span></code></a>). The terms will be converted to
mpmath numbers. For len(terms) > 2, this function is generally
faster and produces more accurate results than the builtin
Python function <code class="xref py py-func docutils literal"><span class="pre">sum()</span></code>.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">fsum</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="mf">0.5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
<span class="go">mpf('10.5')</span>
</pre></div>
</div>
<p>With squared=True each term is squared, and with absolute=True
the absolute value of each term is used.</p>
</dd></dl>
</div>
<div class="section" id="fprod">
<h3><code class="xref py py-func docutils literal"><span class="pre">fprod()</span></code><a class="headerlink" href="#fprod" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fprod">
<code class="descclassname">mpmath.</code><code class="descname">fprod</code><span class="sig-paren">(</span><em>factors</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.fprod" title="Permalink to this definition">¶</a></dt>
<dd><p>Calculates a product containing a finite number of factors (for
infinite products, see <a class="reference internal" href="calculus/sums_limits.html#mpmath.nprod" title="mpmath.nprod"><code class="xref py py-func docutils literal"><span class="pre">nprod()</span></code></a>). The factors will be
converted to mpmath numbers.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">fprod</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="mf">0.5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
<span class="go">mpf('7.0')</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="fdot">
<h3><code class="xref py py-func docutils literal"><span class="pre">fdot()</span></code><a class="headerlink" href="#fdot" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fdot">
<code class="descclassname">mpmath.</code><code class="descname">fdot</code><span class="sig-paren">(</span><em>A</em>, <em>B=None</em>, <em>conjugate=False</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.fdot" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the dot product of the iterables <span class="math">\(A\)</span> and <span class="math">\(B\)</span>,</p>
<div class="math">
\[\sum_{k=0} A_k B_k.\]</div>
<p>Alternatively, <a class="reference internal" href="#mpmath.fdot" title="mpmath.fdot"><code class="xref py py-func docutils literal"><span class="pre">fdot()</span></code></a> accepts a single iterable of pairs.
In other words, <code class="docutils literal"><span class="pre">fdot(A,B)</span></code> and <code class="docutils literal"><span class="pre">fdot(zip(A,B))</span></code> are equivalent.
The elements are automatically converted to mpmath numbers.</p>
<p>With <code class="docutils literal"><span class="pre">conjugate=True</span></code>, the elements in the second vector
will be conjugated:</p>
<div class="math">
\[\sum_{k=0} A_k \overline{B_k}\]</div>
<p><strong>Examples</strong></p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">A</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">B</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">fdot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
<span class="go">mpf('6.5')</span>
<span class="gp">>>> </span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">))</span>
<span class="go">[(2, 1), (1.5, -1), (3, 2)]</span>
<span class="gp">>>> </span><span class="n">fdot</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
<span class="go">mpf('6.5')</span>
<span class="gp">>>> </span><span class="n">A</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">3</span><span class="n">j</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">B</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">fdot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
<span class="go">mpc(real='9.5', imag='-1.0')</span>
<span class="gp">>>> </span><span class="n">fdot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">conjugate</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="go">mpc(real='3.5', imag='-5.0')</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="complex-components">
<h2>Complex components<a class="headerlink" href="#complex-components" title="Permalink to this headline">¶</a></h2>
<div class="section" id="fabs">
<h3><code class="xref py py-func docutils literal"><span class="pre">fabs()</span></code><a class="headerlink" href="#fabs" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fabs">
<code class="descclassname">mpmath.</code><code class="descname">fabs</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.fabs" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the absolute value of <span class="math">\(x\)</span>, <span class="math">\(|x|\)</span>. Unlike <code class="xref py py-func docutils literal"><span class="pre">abs()</span></code>,
<a class="reference internal" href="#mpmath.fabs" title="mpmath.fabs"><code class="xref py py-func docutils literal"><span class="pre">fabs()</span></code></a> converts non-mpmath numbers (such as <code class="docutils literal"><span class="pre">int</span></code>)
into mpmath numbers:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">fabs</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">mpf('3.0')</span>
<span class="gp">>>> </span><span class="n">fabs</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
<span class="go">mpf('3.0')</span>
<span class="gp">>>> </span><span class="n">fabs</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">)</span>
<span class="go">mpf('5.0')</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="sign">
<h3><code class="xref py py-func docutils literal"><span class="pre">sign()</span></code><a class="headerlink" href="#sign" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.sign">
<code class="descclassname">mpmath.</code><code class="descname">sign</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.sign" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the sign of <span class="math">\(x\)</span>, defined as <span class="math">\(\mathrm{sign}(x) = x / |x|\)</span>
(with the special case <span class="math">\(\mathrm{sign}(0) = 0\)</span>):</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">sign</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="go">mpf('1.0')</span>
<span class="gp">>>> </span><span class="n">sign</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">)</span>
<span class="go">mpf('-1.0')</span>
<span class="gp">>>> </span><span class="n">sign</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">mpf('0.0')</span>
</pre></div>
</div>
<p>Note that the sign function is also defined for complex numbers,
for which it gives the projection onto the unit circle:</p>
<div class="highlight-default"><div class="highlight"><pre><span></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="kc">True</span>
<span class="gp">>>> </span><span class="n">sign</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">)</span>
<span class="go">(0.707106781186547 + 0.707106781186547j)</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="re">
<h3><code class="xref py py-func docutils literal"><span class="pre">re()</span></code><a class="headerlink" href="#re" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.re">
<code class="descclassname">mpmath.</code><code class="descname">re</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.re" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the real part of <span class="math">\(x\)</span>, <span class="math">\(\Re(x)\)</span>. <a class="reference internal" href="#mpmath.re" title="mpmath.re"><code class="xref py py-func docutils literal"><span class="pre">re()</span></code></a>
converts a non-mpmath number to an mpmath number:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">re</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">mpf('3.0')</span>
<span class="gp">>>> </span><span class="n">re</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">)</span>
<span class="go">mpf('-1.0')</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="im">
<h3><code class="xref py py-func docutils literal"><span class="pre">im()</span></code><a class="headerlink" href="#im" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.im">
<code class="descclassname">mpmath.</code><code class="descname">im</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.im" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the imaginary part of <span class="math">\(x\)</span>, <span class="math">\(\Im(x)\)</span>. <a class="reference internal" href="#mpmath.im" title="mpmath.im"><code class="xref py py-func docutils literal"><span class="pre">im()</span></code></a>
converts a non-mpmath number to an mpmath number:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">im</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">mpf('0.0')</span>
<span class="gp">>>> </span><span class="n">im</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">)</span>
<span class="go">mpf('4.0')</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="arg">
<h3><code class="xref py py-func docutils literal"><span class="pre">arg()</span></code><a class="headerlink" href="#arg" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.arg">
<code class="descclassname">mpmath.</code><code class="descname">arg</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.arg" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the complex argument (phase) of <span class="math">\(x\)</span>, defined as the
signed angle between the positive real axis and <span class="math">\(x\)</span> in the
complex plane:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">True</span>
<span class="gp">>>> </span><span class="n">arg</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">arg</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span>
<span class="go">0.785398163397448</span>
<span class="gp">>>> </span><span class="n">arg</span><span class="p">(</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span>
<span class="go">1.5707963267949</span>
<span class="gp">>>> </span><span class="n">arg</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
<span class="go">3.14159265358979</span>
<span class="gp">>>> </span><span class="n">arg</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span>
<span class="go">-1.5707963267949</span>
</pre></div>
</div>
<p>The angle is defined to satisfy <span class="math">\(-\pi < \arg(x) \le \pi\)</span> and
with the sign convention that a nonnegative imaginary part
results in a nonnegative argument.</p>
<p>The value returned by <a class="reference internal" href="#mpmath.arg" title="mpmath.arg"><code class="xref py py-func docutils literal"><span class="pre">arg()</span></code></a> is an <code class="docutils literal"><span class="pre">mpf</span></code> instance.</p>
</dd></dl>
</div>
<div class="section" id="conj">
<h3><code class="xref py py-func docutils literal"><span class="pre">conj()</span></code><a class="headerlink" href="#conj" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.conj">
<code class="descclassname">mpmath.</code><code class="descname">conj</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.conj" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the complex conjugate of <span class="math">\(x\)</span>, <span class="math">\(\overline{x}\)</span>. Unlike
<code class="docutils literal"><span class="pre">x.conjugate()</span></code>, <a class="reference internal" href="#mpmath.im" title="mpmath.im"><code class="xref py py-func docutils literal"><span class="pre">im()</span></code></a> converts <span class="math">\(x\)</span> to a mpmath number:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">conj</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">mpf('3.0')</span>
<span class="gp">>>> </span><span class="n">conj</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">)</span>
<span class="go">mpc(real='-1.0', imag='-4.0')</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="polar">
<h3><code class="xref py py-func docutils literal"><span class="pre">polar()</span></code><a class="headerlink" href="#polar" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.polar">
<code class="descclassname">mpmath.</code><code class="descname">polar</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.polar" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the polar representation of the complex number <span class="math">\(z\)</span>
as a pair <span class="math">\((r, \phi)\)</span> such that <span class="math">\(z = r e^{i \phi}\)</span>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">True</span>
<span class="gp">>>> </span><span class="n">polar</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="go">(2.0, 3.14159265358979)</span>
<span class="gp">>>> </span><span class="n">polar</span><span class="p">(</span><span class="mi">3</span><span class="o">-</span><span class="mi">4</span><span class="n">j</span><span class="p">)</span>
<span class="go">(5.0, -0.927295218001612)</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="rect">
<h3><code class="xref py py-func docutils literal"><span class="pre">rect()</span></code><a class="headerlink" href="#rect" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.rect">
<code class="descclassname">mpmath.</code><code class="descname">rect</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.rect" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the complex number represented by polar
coordinates <span class="math">\((r, \phi)\)</span>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">True</span>
<span class="gp">>>> </span><span class="n">chop</span><span class="p">(</span><span class="n">rect</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="p">))</span>
<span class="go">-2.0</span>
<span class="gp">>>> </span><span class="n">rect</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span>
<span class="go">(1.0 - 1.0j)</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="integer-and-fractional-parts">
<h2>Integer and fractional parts<a class="headerlink" href="#integer-and-fractional-parts" title="Permalink to this headline">¶</a></h2>
<div class="section" id="floor">
<h3><code class="xref py py-func docutils literal"><span class="pre">floor()</span></code><a class="headerlink" href="#floor" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.floor">
<code class="descclassname">mpmath.</code><code class="descname">floor</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.floor" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the floor of <span class="math">\(x\)</span>, <span class="math">\(\lfloor x \rfloor\)</span>, defined as
the largest integer less than or equal to <span class="math">\(x\)</span>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="kc">False</span>
<span class="gp">>>> </span><span class="n">floor</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
<span class="go">mpf('3.0')</span>
</pre></div>
</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last"><a class="reference internal" href="#mpmath.floor" title="mpmath.floor"><code class="xref py py-func docutils literal"><span class="pre">floor()</span></code></a>, <a class="reference internal" href="#mpmath.ceil" title="mpmath.ceil"><code class="xref py py-func docutils literal"><span class="pre">ceil()</span></code></a> and <a class="reference internal" href="#mpmath.nint" title="mpmath.nint"><code class="xref py py-func docutils literal"><span class="pre">nint()</span></code></a> return a
floating-point number, not a Python <code class="docutils literal"><span class="pre">int</span></code>. If <span class="math">\(\lfloor x \rfloor\)</span> is
too large to be represented exactly at the present working precision,
the result will be rounded, not necessarily in the direction
implied by the mathematical definition of the function.</p>
</div>
<p>To avoid rounding, use <em>prec=0</em>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></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="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">floor</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">30</span><span class="o">+</span><span class="mi">1</span><span class="p">)))</span>
<span class="go">1000000000000000019884624838656</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">floor</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">30</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="mi">0</span><span class="p">)))</span>
<span class="go">1000000000000000000000000000001</span>
</pre></div>
</div>
<p>The floor function is defined for complex numbers and
acts on the real and imaginary parts separately:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">floor</span><span class="p">(</span><span class="mf">3.25</span><span class="o">+</span><span class="mf">4.75</span><span class="n">j</span><span class="p">)</span>
<span class="go">mpc(real='3.0', imag='4.0')</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="ceil">
<h3><code class="xref py py-func docutils literal"><span class="pre">ceil()</span></code><a class="headerlink" href="#ceil" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.ceil">
<code class="descclassname">mpmath.</code><code class="descname">ceil</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.ceil" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the ceiling of <span class="math">\(x\)</span>, <span class="math">\(\lceil x \rceil\)</span>, defined as
the smallest integer greater than or equal to <span class="math">\(x\)</span>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="kc">False</span>
<span class="gp">>>> </span><span class="n">ceil</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
<span class="go">mpf('4.0')</span>
</pre></div>
</div>
<p>The ceiling function is defined for complex numbers and
acts on the real and imaginary parts separately:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">ceil</span><span class="p">(</span><span class="mf">3.25</span><span class="o">+</span><span class="mf">4.75</span><span class="n">j</span><span class="p">)</span>
<span class="go">mpc(real='4.0', imag='5.0')</span>
</pre></div>
</div>
<p>See notes about rounding for <a class="reference internal" href="#mpmath.floor" title="mpmath.floor"><code class="xref py py-func docutils literal"><span class="pre">floor()</span></code></a>.</p>
</dd></dl>
</div>
<div class="section" id="nint">
<h3><code class="xref py py-func docutils literal"><span class="pre">nint()</span></code><a class="headerlink" href="#nint" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.nint">
<code class="descclassname">mpmath.</code><code class="descname">nint</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.nint" title="Permalink to this definition">¶</a></dt>
<dd><p>Evaluates the nearest integer function, <span class="math">\(\mathrm{nint}(x)\)</span>.
This gives the nearest integer to <span class="math">\(x\)</span>; on a tie, it
gives the nearest even integer:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="kc">False</span>
<span class="gp">>>> </span><span class="n">nint</span><span class="p">(</span><span class="mf">3.2</span><span class="p">)</span>
<span class="go">mpf('3.0')</span>
<span class="gp">>>> </span><span class="n">nint</span><span class="p">(</span><span class="mf">3.8</span><span class="p">)</span>
<span class="go">mpf('4.0')</span>
<span class="gp">>>> </span><span class="n">nint</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
<span class="go">mpf('4.0')</span>
<span class="gp">>>> </span><span class="n">nint</span><span class="p">(</span><span class="mf">4.5</span><span class="p">)</span>
<span class="go">mpf('4.0')</span>
</pre></div>
</div>
<p>The nearest integer function is defined for complex numbers and
acts on the real and imaginary parts separately:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">nint</span><span class="p">(</span><span class="mf">3.25</span><span class="o">+</span><span class="mf">4.75</span><span class="n">j</span><span class="p">)</span>
<span class="go">mpc(real='3.0', imag='5.0')</span>
</pre></div>
</div>
<p>See notes about rounding for <a class="reference internal" href="#mpmath.floor" title="mpmath.floor"><code class="xref py py-func docutils literal"><span class="pre">floor()</span></code></a>.</p>
</dd></dl>
</div>
<div class="section" id="frac">
<h3><code class="xref py py-func docutils literal"><span class="pre">frac()</span></code><a class="headerlink" href="#frac" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.frac">
<code class="descclassname">mpmath.</code><code class="descname">frac</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.frac" title="Permalink to this definition">¶</a></dt>
<dd><p>Gives the fractional part of <span class="math">\(x\)</span>, defined as
<span class="math">\(\mathrm{frac}(x) = x - \lfloor x \rfloor\)</span> (see <a class="reference internal" href="#mpmath.floor" title="mpmath.floor"><code class="xref py py-func docutils literal"><span class="pre">floor()</span></code></a>).
In effect, this computes <span class="math">\(x\)</span> modulo 1, or <span class="math">\(x+n\)</span> where
<span class="math">\(n \in \mathbb{Z}\)</span> is such that <span class="math">\(x+n \in [0,1)\)</span>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="kc">False</span>
<span class="gp">>>> </span><span class="n">frac</span><span class="p">(</span><span class="mf">1.25</span><span class="p">)</span>
<span class="go">mpf('0.25')</span>
<span class="gp">>>> </span><span class="n">frac</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">mpf('0.0')</span>
<span class="gp">>>> </span><span class="n">frac</span><span class="p">(</span><span class="o">-</span><span class="mf">1.25</span><span class="p">)</span>
<span class="go">mpf('0.75')</span>
</pre></div>
</div>
<p>For a complex number, the fractional part function applies to
the real and imaginary parts separately:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">frac</span><span class="p">(</span><span class="mf">2.25</span><span class="o">+</span><span class="mf">3.75</span><span class="n">j</span><span class="p">)</span>
<span class="go">mpc(real='0.25', imag='0.75')</span>
</pre></div>
</div>
<p>Plotted, the fractional part function gives a sawtooth
wave. The Fourier series coefficients have a simple
form:</p>
<div class="highlight-default"><div class="highlight"><pre><span></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="gp">>>> </span><span class="n">nprint</span><span class="p">(</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">frac</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</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">4</span><span class="p">))</span>
<span class="go">([0.0, 0.0, 0.0, 0.0, 0.0], [0.0, -0.31831, -0.159155, -0.106103, -0.0795775])</span>
<span class="gp">>>> </span><span class="n">nprint</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">)])</span>
<span class="go">[-0.31831, -0.159155, -0.106103, -0.0795775]</span>
</pre></div>
</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">The fractional part is sometimes defined as a symmetric
function, i.e. returning <span class="math">\(-\mathrm{frac}(-x)\)</span> if <span class="math">\(x < 0\)</span>.
This convention is used, for instance, by Mathematica’s
<code class="docutils literal"><span class="pre">FractionalPart</span></code>.</p>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="tolerances-and-approximate-comparisons">
<h2>Tolerances and approximate comparisons<a class="headerlink" href="#tolerances-and-approximate-comparisons" title="Permalink to this headline">¶</a></h2>
<div class="section" id="chop">
<h3><code class="xref py py-func docutils literal"><span class="pre">chop()</span></code><a class="headerlink" href="#chop" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.chop">
<code class="descclassname">mpmath.</code><code class="descname">chop</code><span class="sig-paren">(</span><em>x</em>, <em>tol=None</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.chop" title="Permalink to this definition">¶</a></dt>
<dd><p>Chops off small real or imaginary parts, or converts
numbers close to zero to exact zeros. The input can be a
single number or an iterable:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">chop</span><span class="p">(</span><span class="mi">5</span><span class="o">+</span><span class="mf">1e-10j</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-9</span><span class="p">)</span>
<span class="go">mpf('5.0')</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="mf">1.0</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mf">1e-18j</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">]))</span>
<span class="go">[1.0, 0.0, 3.0, -4.0, 2.0]</span>
</pre></div>
</div>
<p>The tolerance defaults to <code class="docutils literal"><span class="pre">100*eps</span></code>.</p>
</dd></dl>
</div>
<div class="section" id="almosteq">
<h3><code class="xref py py-func docutils literal"><span class="pre">almosteq()</span></code><a class="headerlink" href="#almosteq" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.almosteq">
<code class="descclassname">mpmath.</code><code class="descname">almosteq</code><span class="sig-paren">(</span><em>s</em>, <em>t</em>, <em>rel_eps=None</em>, <em>abs_eps=None</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.almosteq" title="Permalink to this definition">¶</a></dt>
<dd><p>Determine whether the difference between <span class="math">\(s\)</span> and <span class="math">\(t\)</span> is smaller
than a given epsilon, either relatively or absolutely.</p>
<p>Both a maximum relative difference and a maximum difference
(‘epsilons’) may be specified. The absolute difference is
defined as <span class="math">\(|s-t|\)</span> and the relative difference is defined
as <span class="math">\(|s-t|/\max(|s|, |t|)\)</span>.</p>
<p>If only one epsilon is given, both are set to the same value.
If none is given, both epsilons are set to <span class="math">\(2^{-p+m}\)</span> where
<span class="math">\(p\)</span> is the current working precision and <span class="math">\(m\)</span> is a small
integer. The default setting typically allows <a class="reference internal" href="#mpmath.almosteq" title="mpmath.almosteq"><code class="xref py py-func docutils literal"><span class="pre">almosteq()</span></code></a>
to be used to check for mathematical equality
in the presence of small rounding errors.</p>
<p><strong>Examples</strong></p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="gp">>>> </span><span class="n">almosteq</span><span class="p">(</span><span class="mf">3.141592653589793</span><span class="p">,</span> <span class="mf">3.141592653589790</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">almosteq</span><span class="p">(</span><span class="mf">3.141592653589793</span><span class="p">,</span> <span class="mf">3.141592653589700</span><span class="p">)</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">almosteq</span><span class="p">(</span><span class="mf">3.141592653589793</span><span class="p">,</span> <span class="mf">3.141592653589700</span><span class="p">,</span> <span class="mf">1e-10</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">almosteq</span><span class="p">(</span><span class="mf">1e-20</span><span class="p">,</span> <span class="mf">2e-20</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">almosteq</span><span class="p">(</span><span class="mf">1e-20</span><span class="p">,</span> <span class="mf">2e-20</span><span class="p">,</span> <span class="n">rel_eps</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">abs_eps</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">False</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="properties-of-numbers">
<h2>Properties of numbers<a class="headerlink" href="#properties-of-numbers" title="Permalink to this headline">¶</a></h2>
<div class="section" id="isinf">
<h3><code class="xref py py-func docutils literal"><span class="pre">isinf()</span></code><a class="headerlink" href="#isinf" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.isinf">
<code class="descclassname">mpmath.</code><code class="descname">isinf</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.isinf" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <em>True</em> if the absolute value of <em>x</em> is infinite;
otherwise return <em>False</em>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">isinf</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isinf</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isinf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isinf</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">)</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isinf</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">inf</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isinf</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="n">inf</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="isnan">
<h3><code class="xref py py-func docutils literal"><span class="pre">isnan()</span></code><a class="headerlink" href="#isnan" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.isnan">
<code class="descclassname">mpmath.</code><code class="descname">isnan</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.isnan" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <em>True</em> if <em>x</em> is a NaN (not-a-number), or for a complex
number, whether either the real or complex part is NaN;
otherwise return <em>False</em>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">isnan</span><span class="p">(</span><span class="mf">3.14</span><span class="p">)</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isnan</span><span class="p">(</span><span class="n">nan</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isnan</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mf">3.14</span><span class="p">,</span><span class="mf">2.72</span><span class="p">))</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isnan</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mf">3.14</span><span class="p">,</span><span class="n">nan</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="isnormal">
<h3><code class="xref py py-func docutils literal"><span class="pre">isnormal()</span></code><a class="headerlink" href="#isnormal" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.isnormal">
<code class="descclassname">mpmath.</code><code class="descname">isnormal</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.isnormal" title="Permalink to this definition">¶</a></dt>
<dd><p>Determine whether <em>x</em> is “normal” in the sense of floating-point
representation; that is, return <em>False</em> if <em>x</em> is zero, an
infinity or NaN; otherwise return <em>True</em>. By extension, a
complex number <em>x</em> is considered “normal” if its magnitude is
normal:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">isnormal</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isnormal</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isnormal</span><span class="p">(</span><span class="n">inf</span><span class="p">);</span> <span class="n">isnormal</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">);</span> <span class="n">isnormal</span><span class="p">(</span><span class="n">nan</span><span class="p">)</span>
<span class="go">False</span>
<span class="go">False</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isnormal</span><span class="p">(</span><span class="mi">0</span><span class="o">+</span><span class="mi">0</span><span class="n">j</span><span class="p">)</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isnormal</span><span class="p">(</span><span class="mi">0</span><span class="o">+</span><span class="mi">3</span><span class="n">j</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isnormal</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">nan</span><span class="p">))</span>
<span class="go">False</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="isfinite">
<h3><code class="xref py py-func docutils literal"><span class="pre">isfinite()</span></code><a class="headerlink" href="#isfinite" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.isfinite">
<code class="descclassname">mpmath.</code><code class="descname">isfinite</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.isfinite" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <em>True</em> if <em>x</em> is a finite number, i.e. neither
an infinity or a NaN.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">isfinite</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isfinite</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isfinite</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isfinite</span><span class="p">(</span><span class="n">nan</span><span class="p">)</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isfinite</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isfinite</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">inf</span><span class="p">))</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isfinite</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="n">nan</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
<span class="go">False</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="isint">
<h3><code class="xref py py-func docutils literal"><span class="pre">isint()</span></code><a class="headerlink" href="#isint" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.isint">
<code class="descclassname">mpmath.</code><code class="descname">isint</code><span class="sig-paren">(</span><em>x</em>, <em>gaussian=False</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.isint" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <em>True</em> if <em>x</em> is integer-valued; otherwise return
<em>False</em>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">isint</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isint</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isint</span><span class="p">(</span><span class="mf">3.2</span><span class="p">)</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isint</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
<span class="go">False</span>
</pre></div>
</div>
<p>Optionally, Gaussian integers can be checked for:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">isint</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">0</span><span class="n">j</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">isint</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">2</span><span class="n">j</span><span class="p">)</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">isint</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">2</span><span class="n">j</span><span class="p">,</span> <span class="n">gaussian</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="ldexp">
<h3><code class="xref py py-func docutils literal"><span class="pre">ldexp()</span></code><a class="headerlink" href="#ldexp" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.ldexp">
<code class="descclassname">mpmath.</code><code class="descname">ldexp</code><span class="sig-paren">(</span><em>x</em>, <em>n</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.ldexp" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes <span class="math">\(x 2^n\)</span> efficiently. No rounding is performed.
The argument <span class="math">\(x\)</span> must be a real floating-point number (or
possible to convert into one) and <span class="math">\(n\)</span> must be a Python <code class="docutils literal"><span class="pre">int</span></code>.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">ldexp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="go">mpf('1024.0')</span>
<span class="gp">>>> </span><span class="n">ldexp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">)</span>
<span class="go">mpf('0.125')</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="frexp">
<h3><code class="xref py py-func docutils literal"><span class="pre">frexp()</span></code><a class="headerlink" href="#frexp" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.frexp">
<code class="descclassname">mpmath.</code><code class="descname">frexp</code><span class="sig-paren">(</span><em>x</em>, <em>n</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.frexp" title="Permalink to this definition">¶</a></dt>
<dd><p>Given a real number <span class="math">\(x\)</span>, returns <span class="math">\((y, n)\)</span> with <span class="math">\(y \in [0.5, 1)\)</span>,
<span class="math">\(n\)</span> a Python integer, and such that <span class="math">\(x = y 2^n\)</span>. No rounding is
performed.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">frexp</span><span class="p">(</span><span class="mf">7.5</span><span class="p">)</span>
<span class="go">(mpf('0.9375'), 3)</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="mag">
<h3><code class="xref py py-func docutils literal"><span class="pre">mag()</span></code><a class="headerlink" href="#mag" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.mag">
<code class="descclassname">mpmath.</code><code class="descname">mag</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.mag" title="Permalink to this definition">¶</a></dt>
<dd><p>Quick logarithmic magnitude estimate of a number. Returns an
integer or infinity <span class="math">\(m\)</span> such that <span class="math">\(|x| <= 2^m\)</span>. It is not
guaranteed that <span class="math">\(m\)</span> is an optimal bound, but it will never
be too large by more than 2 (and probably not more than 1).</p>
<p><strong>Examples</strong></p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="kc">True</span>
<span class="gp">>>> </span><span class="n">mag</span><span class="p">(</span><span class="mi">10</span><span class="p">),</span> <span class="n">mag</span><span class="p">(</span><span class="mf">10.0</span><span class="p">),</span> <span class="n">mag</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mi">10</span><span class="p">)),</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceil</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">2</span><span class="p">)))</span>
<span class="go">(4, 4, 4, 4)</span>
<span class="gp">>>> </span><span class="n">mag</span><span class="p">(</span><span class="mi">10</span><span class="n">j</span><span class="p">),</span> <span class="n">mag</span><span class="p">(</span><span class="mi">10</span><span class="o">+</span><span class="mi">10</span><span class="n">j</span><span class="p">)</span>
<span class="go">(4, 5)</span>
<span class="gp">>>> </span><span class="n">mag</span><span class="p">(</span><span class="mf">0.01</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceil</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="mf">0.01</span><span class="p">,</span><span class="mi">2</span><span class="p">)))</span>
<span class="go">(-6, -6)</span>
<span class="gp">>>> </span><span class="n">mag</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">mag</span><span class="p">(</span><span class="n">inf</span><span class="p">),</span> <span class="n">mag</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">),</span> <span class="n">mag</span><span class="p">(</span><span class="n">nan</span><span class="p">)</span>
<span class="go">(-inf, +inf, +inf, nan)</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="nint-distance">
<h3><code class="xref py py-func docutils literal"><span class="pre">nint_distance()</span></code><a class="headerlink" href="#nint-distance" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.nint_distance">
<code class="descclassname">mpmath.</code><code class="descname">nint_distance</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.nint_distance" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <span class="math">\((n,d)\)</span> where <span class="math">\(n\)</span> is the nearest integer to <span class="math">\(x\)</span> and <span class="math">\(d\)</span> is
an estimate of <span class="math">\(\log_2(|x-n|)\)</span>. If <span class="math">\(d < 0\)</span>, <span class="math">\(-d\)</span> gives the precision
(measured in bits) lost to cancellation when computing <span class="math">\(x-n\)</span>.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">nint_distance</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="nb">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="go">5</span>
<span class="go">-inf</span>
<span class="gp">>>> </span><span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">nint_distance</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="nb">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="go">5</span>
<span class="go">-inf</span>
<span class="gp">>>> </span><span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">nint_distance</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mf">5.00000001</span><span class="p">))</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="nb">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="go">5</span>
<span class="go">-26</span>
<span class="gp">>>> </span><span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">nint_distance</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mf">4.99999999</span><span class="p">))</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="nb">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="go">5</span>
<span class="go">-26</span>
<span class="gp">>>> </span><span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">nint_distance</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="nb">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="go">5</span>
<span class="go">4</span>
<span class="gp">>>> </span><span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">nint_distance</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mf">0.000001</span><span class="p">))</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="nb">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
<span class="go">5</span>
<span class="go">-19</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="number-generation">
<h2>Number generation<a class="headerlink" href="#number-generation" title="Permalink to this headline">¶</a></h2>
<div class="section" id="fraction">
<h3><code class="xref py py-func docutils literal"><span class="pre">fraction()</span></code><a class="headerlink" href="#fraction" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.fraction">
<code class="descclassname">mpmath.</code><code class="descname">fraction</code><span class="sig-paren">(</span><em>p</em>, <em>q</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.fraction" title="Permalink to this definition">¶</a></dt>
<dd><p>Given Python integers <span class="math">\((p, q)\)</span>, returns a lazy <code class="docutils literal"><span class="pre">mpf</span></code> representing
the fraction <span class="math">\(p/q\)</span>. The value is updated with the precision.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">b</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="o">/</span><span class="mi">100</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">a</span><span class="p">);</span> <span class="nb">print</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="go">0.01</span>
<span class="go">0.01</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">30</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">a</span><span class="p">);</span> <span class="nb">print</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="c1"># a will be accurate</span>
<span class="go">0.01</span>
<span class="go">0.0100000000000000002081668171172</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>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="rand">
<h3><code class="xref py py-func docutils literal"><span class="pre">rand()</span></code><a class="headerlink" href="#rand" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.rand">
<code class="descclassname">mpmath.</code><code class="descname">rand</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.rand" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns an <code class="docutils literal"><span class="pre">mpf</span></code> with value chosen randomly from <span class="math">\([0, 1)\)</span>.
The number of randomly generated bits in the mantissa is equal
to the working precision.</p>
</dd></dl>
</div>
<div class="section" id="arange">
<h3><code class="xref py py-func docutils literal"><span class="pre">arange()</span></code><a class="headerlink" href="#arange" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.arange">
<code class="descclassname">mpmath.</code><code class="descname">arange</code><span class="sig-paren">(</span><em>*args</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.arange" title="Permalink to this definition">¶</a></dt>
<dd><p>This is a generalized version of Python’s <code class="xref py py-func docutils literal"><span class="pre">range()</span></code> function
that accepts fractional endpoints and step sizes and
returns a list of <code class="docutils literal"><span class="pre">mpf</span></code> instances. Like <code class="xref py py-func docutils literal"><span class="pre">range()</span></code>,
<a class="reference internal" href="#mpmath.arange" title="mpmath.arange"><code class="xref py py-func docutils literal"><span class="pre">arange()</span></code></a> can be called with 1, 2 or 3 arguments:</p>
<dl class="docutils">
<dt><code class="docutils literal"><span class="pre">arange(b)</span></code></dt>
<dd><span class="math">\([0, 1, 2, \ldots, x]\)</span></dd>
<dt><code class="docutils literal"><span class="pre">arange(a,</span> <span class="pre">b)</span></code></dt>
<dd><span class="math">\([a, a+1, a+2, \ldots, x]\)</span></dd>
<dt><code class="docutils literal"><span class="pre">arange(a,</span> <span class="pre">b,</span> <span class="pre">h)</span></code></dt>
<dd><span class="math">\([a, a+h, a+h, \ldots, x]\)</span></dd>
</dl>
<p>where <span class="math">\(b-1 \le x < b\)</span> (in the third case, <span class="math">\(b-h \le x < b\)</span>).</p>
<p>Like Python’s <code class="xref py py-func docutils literal"><span class="pre">range()</span></code>, the endpoint is not included. To
produce ranges where the endpoint is included, <a class="reference internal" href="#mpmath.linspace" title="mpmath.linspace"><code class="xref py py-func docutils literal"><span class="pre">linspace()</span></code></a>
is more convenient.</p>
<p><strong>Examples</strong></p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">arange</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="go">[mpf('0.0'), mpf('1.0'), mpf('2.0'), mpf('3.0')]</span>
<span class="gp">>>> </span><span class="n">arange</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="mf">0.25</span><span class="p">)</span>
<span class="go">[mpf('1.0'), mpf('1.25'), mpf('1.5'), mpf('1.75')]</span>
<span class="gp">>>> </span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.75</span><span class="p">)</span>
<span class="go">[mpf('1.0'), mpf('0.25'), mpf('-0.5')]</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="linspace">
<h3><code class="xref py py-func docutils literal"><span class="pre">linspace()</span></code><a class="headerlink" href="#linspace" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.linspace">
<code class="descclassname">mpmath.</code><code class="descname">linspace</code><span class="sig-paren">(</span><em>*args</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.linspace" title="Permalink to this definition">¶</a></dt>
<dd><p><code class="docutils literal"><span class="pre">linspace(a,</span> <span class="pre">b,</span> <span class="pre">n)</span></code> returns a list of <span class="math">\(n\)</span> evenly spaced
samples from <span class="math">\(a\)</span> to <span class="math">\(b\)</span>. The syntax <code class="docutils literal"><span class="pre">linspace(mpi(a,b),</span> <span class="pre">n)</span></code>
is also valid.</p>
<p>This function is often more convenient than <a class="reference internal" href="#mpmath.arange" title="mpmath.arange"><code class="xref py py-func docutils literal"><span class="pre">arange()</span></code></a>
for partitioning an interval into subintervals, since
the endpoint is included:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">False</span>
<span class="gp">>>> </span><span class="n">linspace</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="go">[mpf('1.0'), mpf('2.0'), mpf('3.0'), mpf('4.0')]</span>
</pre></div>
</div>
<p>You may also provide the keyword argument <code class="docutils literal"><span class="pre">endpoint=False</span></code>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">linspace</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">endpoint</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="go">[mpf('1.0'), mpf('1.75'), mpf('2.5'), mpf('3.25')]</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="precision-management">
<h2>Precision management<a class="headerlink" href="#precision-management" title="Permalink to this headline">¶</a></h2>
<div class="section" id="autoprec">
<h3><code class="xref py py-func docutils literal"><span class="pre">autoprec()</span></code><a class="headerlink" href="#autoprec" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.autoprec">
<code class="descclassname">mpmath.</code><code class="descname">autoprec</code><span class="sig-paren">(</span><em>f</em>, <em>maxprec=None</em>, <em>catch=()</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.autoprec" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a wrapped copy of <em>f</em> that repeatedly evaluates <em>f</em>
with increasing precision until the result converges to the
full precision used at the point of the call.</p>
<p>This heuristically protects against rounding errors, at the cost of
roughly a 2x slowdown compared to manually setting the optimal
precision. This method can, however, easily be fooled if the results
from <em>f</em> depend “discontinuously” on the precision, for instance
if catastrophic cancellation can occur. Therefore, <a class="reference internal" href="#mpmath.autoprec" title="mpmath.autoprec"><code class="xref py py-func docutils literal"><span class="pre">autoprec()</span></code></a>
should be used judiciously.</p>
<p><strong>Examples</strong></p>
<p>Many functions are sensitive to perturbations of the input arguments.
If the arguments are decimal numbers, they may have to be converted
to binary at a much higher precision. If the amount of required
extra precision is unknown, <a class="reference internal" href="#mpmath.autoprec" title="mpmath.autoprec"><code class="xref py py-func docutils literal"><span class="pre">autoprec()</span></code></a> is convenient:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="kc">True</span>
<span class="gp">>>> </span><span class="n">besselj</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">125</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**</span><span class="mi">28</span><span class="p">)</span> <span class="c1"># Exact input</span>
<span class="go">-8.03284785591801e-17</span>
<span class="gp">>>> </span><span class="n">besselj</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">'1.25e30'</span><span class="p">)</span> <span class="c1"># Bad</span>
<span class="go">7.12954868316652e-16</span>
<span class="gp">>>> </span><span class="n">autoprec</span><span class="p">(</span><span class="n">besselj</span><span class="p">)(</span><span class="mi">5</span><span class="p">,</span> <span class="s1">'1.25e30'</span><span class="p">)</span> <span class="c1"># Good</span>
<span class="go">-8.03284785591801e-17</span>
</pre></div>
</div>
<p>The following fails to converge because <span class="math">\(\sin(\pi) = 0\)</span> whereas all
finite-precision approximations of <span class="math">\(\pi\)</span> give nonzero values:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">autoprec</span><span class="p">(</span><span class="n">sin</span><span class="p">)(</span><span class="n">pi</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">NoConvergence</span>: <span class="n">autoprec: prec increased to 2910 without convergence</span>
</pre></div>
</div>
<p>As the following example shows, <a class="reference internal" href="#mpmath.autoprec" title="mpmath.autoprec"><code class="xref py py-func docutils literal"><span class="pre">autoprec()</span></code></a> can protect against
cancellation, but is fooled by too severe cancellation:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="mf">1e-10</span>
<span class="gp">>>> </span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span> <span class="n">expm1</span><span class="p">(</span><span class="n">x</span><span class="p">);</span> <span class="n">autoprec</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">t</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">1.00000008274037e-10</span>
<span class="go">1.00000000005e-10</span>
<span class="go">1.00000000005e-10</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="mf">1e-50</span>
<span class="gp">>>> </span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span> <span class="n">expm1</span><span class="p">(</span><span class="n">x</span><span class="p">);</span> <span class="n">autoprec</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">t</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">0.0</span>
<span class="go">1.0e-50</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p>With <em>catch</em>, an exception or list of exceptions to intercept
may be specified. The raised exception is interpreted
as signaling insufficient precision. This permits, for example,
evaluating a function where a too low precision results in a
division by zero:</p>
<div class="highlight-default"><div class="highlight"><pre><span></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="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">f</span><span class="p">(</span><span class="mf">1e-30</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">ZeroDivisionError</span>
<span class="gp">>>> </span><span class="n">autoprec</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">catch</span><span class="o">=</span><span class="ne">ZeroDivisionError</span><span class="p">)(</span><span class="mf">1e-30</span><span class="p">)</span>
<span class="go">1.0e+30</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="workprec">
<h3><code class="xref py py-func docutils literal"><span class="pre">workprec()</span></code><a class="headerlink" href="#workprec" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.workprec">
<code class="descclassname">mpmath.</code><code class="descname">workprec</code><span class="sig-paren">(</span><em>n</em>, <em>normalize_output=False</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.workprec" title="Permalink to this definition">¶</a></dt>
<dd><p>The block</p>
<blockquote>
<div><dl class="docutils">
<dt>with workprec(n):</dt>
<dd><code></dd>
</dl>
</div></blockquote>
<p>sets the precision to n bits, executes <code>, and then restores
the precision.</p>
<p>workprec(n)(f) returns a decorated version of the function f
that sets the precision to n bits before execution,
and restores the precision afterwards. With normalize_output=True,
it rounds the return value to the parent precision.</p>
</dd></dl>
</div>
<div class="section" id="workdps">
<h3><code class="xref py py-func docutils literal"><span class="pre">workdps()</span></code><a class="headerlink" href="#workdps" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.workdps">
<code class="descclassname">mpmath.</code><code class="descname">workdps</code><span class="sig-paren">(</span><em>n</em>, <em>normalize_output=False</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.workdps" title="Permalink to this definition">¶</a></dt>
<dd><p>This function is analogous to workprec (see documentation)
but changes the decimal precision instead of the number of bits.</p>
</dd></dl>
</div>
<div class="section" id="extraprec">
<h3><code class="xref py py-func docutils literal"><span class="pre">extraprec()</span></code><a class="headerlink" href="#extraprec" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.extraprec">
<code class="descclassname">mpmath.</code><code class="descname">extraprec</code><span class="sig-paren">(</span><em>n</em>, <em>normalize_output=False</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.extraprec" title="Permalink to this definition">¶</a></dt>
<dd><p>The block</p>
<blockquote>
<div><dl class="docutils">
<dt>with extraprec(n):</dt>
<dd><code></dd>
</dl>
</div></blockquote>
<p>increases the precision n bits, executes <code>, and then
restores the precision.</p>
<p>extraprec(n)(f) returns a decorated version of the function f
that increases the working precision by n bits before execution,
and restores the parent precision afterwards. With
normalize_output=True, it rounds the return value to the parent
precision.</p>
</dd></dl>
</div>
<div class="section" id="extradps">
<h3><code class="xref py py-func docutils literal"><span class="pre">extradps()</span></code><a class="headerlink" href="#extradps" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.extradps">
<code class="descclassname">mpmath.</code><code class="descname">extradps</code><span class="sig-paren">(</span><em>n</em>, <em>normalize_output=False</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.extradps" title="Permalink to this definition">¶</a></dt>
<dd><p>This function is analogous to extraprec (see documentation)
but changes the decimal precision instead of the number of bits.</p>
</dd></dl>
</div>
</div>
<div class="section" id="performance-and-debugging">
<h2>Performance and debugging<a class="headerlink" href="#performance-and-debugging" title="Permalink to this headline">¶</a></h2>
<div class="section" id="memoize">
<h3><code class="xref py py-func docutils literal"><span class="pre">memoize()</span></code><a class="headerlink" href="#memoize" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.memoize">
<code class="descclassname">mpmath.</code><code class="descname">memoize</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.memoize" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a wrapped copy of <em>f</em> that caches computed values, i.e.
a memoized copy of <em>f</em>. Values are only reused if the cached precision
is equal to or higher than the working precision:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="kc">True</span>
<span class="gp">>>> </span><span class="n">f</span> <span class="o">=</span> <span class="n">memoize</span><span class="p">(</span><span class="n">maxcalls</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">0.909297426825682</span>
<span class="gp">>>> </span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">0.909297426825682</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">25</span>
<span class="gp">>>> </span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">NoConvergence</span>: <span class="n">maxcalls: function evaluated 1 times</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="maxcalls">
<h3><code class="xref py py-func docutils literal"><span class="pre">maxcalls()</span></code><a class="headerlink" href="#maxcalls" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.maxcalls">
<code class="descclassname">mpmath.</code><code class="descname">maxcalls</code><span class="sig-paren">(</span><em>f</em>, <em>N</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.maxcalls" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a wrapped copy of <em>f</em> that raises <code class="docutils literal"><span class="pre">NoConvergence</span></code> when <em>f</em>
has been called more than <em>N</em> times:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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="gp">>>> </span><span class="n">f</span> <span class="o">=</span> <span class="n">maxcalls</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">n</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">10</span><span class="p">)))</span>
<span class="go">1.95520948210738</span>
<span class="gp">>>> </span><span class="n">f</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">NoConvergence</span>: <span class="n">maxcalls: function evaluated 10 times</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="monitor">
<h3><code class="xref py py-func docutils literal"><span class="pre">monitor()</span></code><a class="headerlink" href="#monitor" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.monitor">
<code class="descclassname">mpmath.</code><code class="descname">monitor</code><span class="sig-paren">(</span><em>f</em>, <em>input='print'</em>, <em>output='print'</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.monitor" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns a wrapped copy of <em>f</em> that monitors evaluation by calling
<em>input</em> with every input (<em>args</em>, <em>kwargs</em>) passed to <em>f</em> and
<em>output</em> with every value returned from <em>f</em>. The default action
(specify using the special string value <code class="docutils literal"><span class="pre">'print'</span></code>) is to print
inputs and outputs to stdout, along with the total evaluation
count:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="k">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">5</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="kc">False</span>
<span class="gp">>>> </span><span class="n">diff</span><span class="p">(</span><span class="n">monitor</span><span class="p">(</span><span class="n">exp</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># diff will eval f(x-h) and f(x+h)</span>
<span class="go">in 0 (mpf('0.99999999906867742538452148'),) {}</span>
<span class="go">out 0 mpf('2.7182818259274480055282064')</span>
<span class="go">in 1 (mpf('1.0000000009313225746154785'),) {}</span>
<span class="go">out 1 mpf('2.7182818309906424675501024')</span>
<span class="go">mpf('2.7182808')</span>
</pre></div>
</div>
<p>To disable either the input or the output handler, you may
pass <em>None</em> as argument.</p>
<p>Custom input and output handlers may be used e.g. to store
results for later analysis:</p>
<div class="highlight-default"><div class="highlight"><pre><span></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="gp">>>> </span><span class="nb">input</span> <span class="o">=</span> <span class="p">[]</span>
<span class="gp">>>> </span><span class="n">output</span> <span class="o">=</span> <span class="p">[]</span>
<span class="gp">>>> </span><span class="n">findroot</span><span class="p">(</span><span class="n">monitor</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="nb">input</span><span class="o">.</span><span class="n">append</span><span class="p">,</span> <span class="n">output</span><span class="o">.</span><span class="n">append</span><span class="p">),</span> <span class="mf">3.0</span><span class="p">)</span>
<span class="go">mpf('3.1415926535897932')</span>
<span class="gp">>>> </span><span class="nb">len</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span> <span class="c1"># Count number of evaluations</span>
<span class="go">9</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">input</span><span class="p">[</span><span class="mi">3</span><span class="p">]);</span> <span class="nb">print</span><span class="p">(</span><span class="n">output</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span>
<span class="go">((mpf('3.1415076583334066'),), {})</span>
<span class="go">8.49952562843408e-5</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="nb">input</span><span class="p">[</span><span class="mi">4</span><span class="p">]);</span> <span class="nb">print</span><span class="p">(</span><span class="n">output</span><span class="p">[</span><span class="mi">4</span><span class="p">])</span>
<span class="go">((mpf('3.1415928201669122'),), {})</span>
<span class="go">-1.66577118985331e-7</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="timing">
<h3><code class="xref py py-func docutils literal"><span class="pre">timing()</span></code><a class="headerlink" href="#timing" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.timing">
<code class="descclassname">mpmath.</code><code class="descname">timing</code><span class="sig-paren">(</span><em>f</em>, <em>*args</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#mpmath.timing" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns time elapsed for evaluating <code class="docutils literal"><span class="pre">f()</span></code>. Optionally arguments
may be passed to time the execution of <code class="docutils literal"><span class="pre">f(*args,</span> <span class="pre">**kwargs)</span></code>.</p>
<p>If the first call is very quick, <code class="docutils literal"><span class="pre">f</span></code> is called
repeatedly and the best time is returned.</p>
</dd></dl>
</div>
</div>
</div>
</div>
</div>
</div>
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<h3><a href="index.html">Table Of Contents</a></h3>
<ul>
<li><a class="reference internal" href="#">Utility functions</a><ul>
<li><a class="reference internal" href="#conversion-and-printing">Conversion and printing</a><ul>
<li><a class="reference internal" href="#mpmathify-convert"><code class="docutils literal"><span class="pre">mpmathify()</span></code> / <code class="docutils literal"><span class="pre">convert()</span></code></a></li>
<li><a class="reference internal" href="#nstr"><code class="docutils literal"><span class="pre">nstr()</span></code></a></li>
<li><a class="reference internal" href="#nprint"><code class="docutils literal"><span class="pre">nprint()</span></code></a></li>
</ul>
</li>
<li><a class="reference internal" href="#arithmetic-operations">Arithmetic operations</a><ul>
<li><a class="reference internal" href="#fadd"><code class="docutils literal"><span class="pre">fadd()</span></code></a></li>
<li><a class="reference internal" href="#fsub"><code class="docutils literal"><span class="pre">fsub()</span></code></a></li>
<li><a class="reference internal" href="#fneg"><code class="docutils literal"><span class="pre">fneg()</span></code></a></li>
<li><a class="reference internal" href="#fmul"><code class="docutils literal"><span class="pre">fmul()</span></code></a></li>
<li><a class="reference internal" href="#fdiv"><code class="docutils literal"><span class="pre">fdiv()</span></code></a></li>
<li><a class="reference internal" href="#fmod"><code class="docutils literal"><span class="pre">fmod()</span></code></a></li>
<li><a class="reference internal" href="#fsum"><code class="docutils literal"><span class="pre">fsum()</span></code></a></li>
<li><a class="reference internal" href="#fprod"><code class="docutils literal"><span class="pre">fprod()</span></code></a></li>
<li><a class="reference internal" href="#fdot"><code class="docutils literal"><span class="pre">fdot()</span></code></a></li>
</ul>
</li>
<li><a class="reference internal" href="#complex-components">Complex components</a><ul>
<li><a class="reference internal" href="#fabs"><code class="docutils literal"><span class="pre">fabs()</span></code></a></li>
<li><a class="reference internal" href="#sign"><code class="docutils literal"><span class="pre">sign()</span></code></a></li>
<li><a class="reference internal" href="#re"><code class="docutils literal"><span class="pre">re()</span></code></a></li>
<li><a class="reference internal" href="#im"><code class="docutils literal"><span class="pre">im()</span></code></a></li>
<li><a class="reference internal" href="#arg"><code class="docutils literal"><span class="pre">arg()</span></code></a></li>
<li><a class="reference internal" href="#conj"><code class="docutils literal"><span class="pre">conj()</span></code></a></li>
<li><a class="reference internal" href="#polar"><code class="docutils literal"><span class="pre">polar()</span></code></a></li>
<li><a class="reference internal" href="#rect"><code class="docutils literal"><span class="pre">rect()</span></code></a></li>
</ul>
</li>
<li><a class="reference internal" href="#integer-and-fractional-parts">Integer and fractional parts</a><ul>
<li><a class="reference internal" href="#floor"><code class="docutils literal"><span class="pre">floor()</span></code></a></li>
<li><a class="reference internal" href="#ceil"><code class="docutils literal"><span class="pre">ceil()</span></code></a></li>
<li><a class="reference internal" href="#nint"><code class="docutils literal"><span class="pre">nint()</span></code></a></li>
<li><a class="reference internal" href="#frac"><code class="docutils literal"><span class="pre">frac()</span></code></a></li>
</ul>
</li>
<li><a class="reference internal" href="#tolerances-and-approximate-comparisons">Tolerances and approximate comparisons</a><ul>
<li><a class="reference internal" href="#chop"><code class="docutils literal"><span class="pre">chop()</span></code></a></li>
<li><a class="reference internal" href="#almosteq"><code class="docutils literal"><span class="pre">almosteq()</span></code></a></li>
</ul>
</li>
<li><a class="reference internal" href="#properties-of-numbers">Properties of numbers</a><ul>
<li><a class="reference internal" href="#isinf"><code class="docutils literal"><span class="pre">isinf()</span></code></a></li>
<li><a class="reference internal" href="#isnan"><code class="docutils literal"><span class="pre">isnan()</span></code></a></li>
<li><a class="reference internal" href="#isnormal"><code class="docutils literal"><span class="pre">isnormal()</span></code></a></li>
<li><a class="reference internal" href="#isfinite"><code class="docutils literal"><span class="pre">isfinite()</span></code></a></li>
<li><a class="reference internal" href="#isint"><code class="docutils literal"><span class="pre">isint()</span></code></a></li>
<li><a class="reference internal" href="#ldexp"><code class="docutils literal"><span class="pre">ldexp()</span></code></a></li>
<li><a class="reference internal" href="#frexp"><code class="docutils literal"><span class="pre">frexp()</span></code></a></li>
<li><a class="reference internal" href="#mag"><code class="docutils literal"><span class="pre">mag()</span></code></a></li>
<li><a class="reference internal" href="#nint-distance"><code class="docutils literal"><span class="pre">nint_distance()</span></code></a></li>
</ul>
</li>
<li><a class="reference internal" href="#number-generation">Number generation</a><ul>
<li><a class="reference internal" href="#fraction"><code class="docutils literal"><span class="pre">fraction()</span></code></a></li>
<li><a class="reference internal" href="#rand"><code class="docutils literal"><span class="pre">rand()</span></code></a></li>
<li><a class="reference internal" href="#arange"><code class="docutils literal"><span class="pre">arange()</span></code></a></li>
<li><a class="reference internal" href="#linspace"><code class="docutils literal"><span class="pre">linspace()</span></code></a></li>
</ul>
</li>
<li><a class="reference internal" href="#precision-management">Precision management</a><ul>
<li><a class="reference internal" href="#autoprec"><code class="docutils literal"><span class="pre">autoprec()</span></code></a></li>
<li><a class="reference internal" href="#workprec"><code class="docutils literal"><span class="pre">workprec()</span></code></a></li>
<li><a class="reference internal" href="#workdps"><code class="docutils literal"><span class="pre">workdps()</span></code></a></li>
<li><a class="reference internal" href="#extraprec"><code class="docutils literal"><span class="pre">extraprec()</span></code></a></li>
<li><a class="reference internal" href="#extradps"><code class="docutils literal"><span class="pre">extradps()</span></code></a></li>
</ul>
</li>
<li><a class="reference internal" href="#performance-and-debugging">Performance and debugging</a><ul>
<li><a class="reference internal" href="#memoize"><code class="docutils literal"><span class="pre">memoize()</span></code></a></li>
<li><a class="reference internal" href="#maxcalls"><code class="docutils literal"><span class="pre">maxcalls()</span></code></a></li>
<li><a class="reference internal" href="#monitor"><code class="docutils literal"><span class="pre">monitor()</span></code></a></li>
<li><a class="reference internal" href="#timing"><code class="docutils literal"><span class="pre">timing()</span></code></a></li>
</ul>
</li>
</ul>
</li>
</ul>
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