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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.roots_deriv"></a><a class="link" href="roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley & Schröder">Root Finding With Derivatives:
Newton-Raphson, Halley & Schröder</a>
</h2></div></div></div>
<h5>
<a name="math_toolkit.roots_deriv.h0"></a>
<span class="phrase"><a name="math_toolkit.roots_deriv.synopsis"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.synopsis">Synopsis</a>
</h5>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">roots</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
</pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
<span class="keyword">namespace</span> <span class="identifier">tools</span> <span class="special">{</span> <span class="comment">// Note namespace boost::math::tools.</span>
<span class="comment">// Newton-Raphson</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">T</span> <span class="identifier">newton_raphson_iterate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">digits</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">T</span> <span class="identifier">newton_raphson_iterate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">digits</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
<span class="comment">// Halley</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">T</span> <span class="identifier">halley_iterate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">digits</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">T</span> <span class="identifier">halley_iterate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">digits</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
<span class="comment">// Schr'''&#xf6;'''der</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">T</span> <span class="identifier">schroder_iterate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">digits</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="identifier">T</span> <span class="identifier">schroder_iterate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">digits</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">ComplexType</span><span class="special">></span>
<span class="identifier">Complex</span> <span class="identifier">complex_newton</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Complex</span> <span class="identifier">guess</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">max_iterations</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">ComplexType</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">);</span>
<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<span class="keyword">auto</span> <span class="identifier">quadratic_roots</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">);</span>
<span class="special">}}}</span> <span class="comment">// namespaces boost::math::tools.</span>
</pre>
<h5>
<a name="math_toolkit.roots_deriv.h1"></a>
<span class="phrase"><a name="math_toolkit.roots_deriv.description"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.description">Description</a>
</h5>
<p>
These functions all perform iterative root-finding <span class="bold"><strong>using
derivatives</strong></span>:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<code class="computeroutput"><span class="identifier">newton_raphson_iterate</span></code>
performs second-order <a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.newton">Newton-Raphson
iteration</a>.
</li>
<li class="listitem">
<code class="computeroutput"><span class="identifier">halley_iterate</span></code> and <code class="computeroutput"><span class="identifier">schroder_iterate</span></code> perform third-order
<a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.halley">Halley</a> and <a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.schroder">Schröder</a> iteration.
</li>
<li class="listitem">
<code class="computeroutput"><span class="identifier">complex_newton</span></code> performs
Newton's method on complex-analytic functions.
</li>
<li class="listitem">
<code class="computeroutput"><span class="identifier">solve_quadratic</span></code> solves
quadratic equations using various tricks to keep catastrophic cancellation
from occurring in computation of the discriminant.
</li>
</ul></div>
<div class="variablelist">
<p class="title"><b>Parameters of the real-valued root finding functions</b></p>
<dl class="variablelist">
<dt><span class="term">F f</span></dt>
<dd>
<p>
Type F must be a callable function object (or C++ lambda) that accepts
one parameter and returns a <a class="link" href="internals/tuples.html" title="Tuples">std::pair,
std::tuple, boost::tuple or boost::fusion::tuple</a>:
</p>
<p>
For second-order iterative method (<a href="http://en.wikipedia.org/wiki/Newton_Raphson" target="_top">Newton
Raphson</a>) the <code class="computeroutput"><span class="identifier">tuple</span></code>
should have <span class="bold"><strong>two</strong></span> elements containing
the evaluation of the function and its first derivative.
</p>
<p>
For the third-order methods (<a href="http://en.wikipedia.org/wiki/Halley%27s_method" target="_top">Halley</a>
and Schröder) the <code class="computeroutput"><span class="identifier">tuple</span></code>
should have <span class="bold"><strong>three</strong></span> elements containing
the evaluation of the function and its first and second derivatives.
</p>
</dd>
<dt><span class="term">T guess</span></dt>
<dd><p>
The initial starting value. A good guess is crucial to quick convergence!
</p></dd>
<dt><span class="term">T min</span></dt>
<dd><p>
The minimum possible value for the result, this is used as an initial
lower bracket.
</p></dd>
<dt><span class="term">T max</span></dt>
<dd><p>
The maximum possible value for the result, this is used as an initial
upper bracket.
</p></dd>
<dt><span class="term">int digits</span></dt>
<dd><p>
The desired number of binary digits precision.
</p></dd>
<dt><span class="term">uintmax_t& max_iter</span></dt>
<dd><p>
An optional maximum number of iterations to perform. On exit, this is
updated to the actual number of iterations performed.
</p></dd>
</dl>
</div>
<p>
When using these functions you should note that:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Default <code class="computeroutput"><span class="identifier">max_iter</span> <span class="special">=</span>
<span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)()</span></code> is effectively 'iterate for ever'.
</li>
<li class="listitem">
They may be very sensitive to the initial guess, typically they converge
very rapidly if the initial guess has two or three decimal digits correct.
However convergence can be no better than <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>,
or in some rare cases, even worse than <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>
if the initial guess is a long way from the correct value and the derivatives
are close to zero.
</li>
<li class="listitem">
These functions include special cases to handle zero first (and second
where appropriate) derivatives, and fall back to <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>
in this case. However, it is helpful if functor F is defined to return
an arbitrarily small value <span class="emphasis"><em>of the correct sign</em></span> rather
than zero.
</li>
<li class="listitem">
The functions will raise an <a class="link" href="error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a>
if arguments <code class="computeroutput"><span class="identifier">min</span></code> and <code class="computeroutput"><span class="identifier">max</span></code> are the wrong way around or if they
converge to a local minima.
</li>
<li class="listitem">
If the derivative at the current best guess for the result is infinite
(or very close to being infinite) then these functions may terminate prematurely.
A large first derivative leads to a very small next step, triggering the
termination condition. Derivative based iteration may not be appropriate
in such cases.
</li>
<li class="listitem">
If the function is 'Really Well Behaved' (is monotonic and has only one
root) the bracket bounds <span class="emphasis"><em>min</em></span> and <span class="emphasis"><em>max</em></span>
may as well be set to the widest limits like zero and <code class="computeroutput"><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max</span><span class="special">()</span></code>.
</li>
<li class="listitem">
But if the function more complex and may have more than one root or a pole,
the choice of bounds is protection against jumping out to seek the 'wrong'
root.
</li>
<li class="listitem">
These functions fall back to <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>
if the next computed step would take the next value out of bounds. The
bounds are updated after each step to ensure this leads to convergence.
However, a good initial guess backed up by asymptotically-tight bounds
will improve performance no end - rather than relying on <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisection</a>.
</li>
<li class="listitem">
The value of <span class="emphasis"><em>digits</em></span> is crucial to good performance
of these functions, if it is set too high then at best you will get one
extra (unnecessary) iteration, and at worst the last few steps will proceed
by <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisection</a>.
Remember that the returned value can never be more accurate than <span class="emphasis"><em>f(x)</em></span>
can be evaluated, and that if <span class="emphasis"><em>f(x)</em></span> suffers from cancellation
errors as it tends to zero then the computed steps will be effectively
random. The value of <span class="emphasis"><em>digits</em></span> should be set so that
iteration terminates before this point: remember that for second and third
order methods the number of correct digits in the result is increasing
quite substantially with each iteration, <span class="emphasis"><em>digits</em></span> should
be set by experiment so that the final iteration just takes the next value
into the zone where <span class="emphasis"><em>f(x)</em></span> becomes inaccurate. A good
starting point for <span class="emphasis"><em>digits</em></span> would be 0.6*D for Newton
and 0.4*D for Halley or Shröder iteration, where D is <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span></code>.
</li>
<li class="listitem">
If you need some diagnostic output to see what is going on, you can <code class="computeroutput"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_INSTRUMENT</span></code>
before the <code class="computeroutput"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">roots</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></code>, and also ensure that display of all
the significant digits with <code class="computeroutput"> <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span></code>: or even possibly significant digits with
<code class="computeroutput"> <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">)</span></code>:
but be warned, this may produce copious output!
</li>
<li class="listitem">
Finally: you may well be able to do better than these functions by hand-coding
the heuristics used so that they are tailored to a specific function. You
may also be able to compute the ratio of derivatives used by these methods
more efficiently than computing the derivatives themselves. As ever, algebraic
simplification can be a big win.
</li>
</ul></div>
<h5>
<a name="math_toolkit.roots_deriv.h2"></a>
<span class="phrase"><a name="math_toolkit.roots_deriv.newton"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.newton">Newton
Raphson Method</a>
</h5>
<p>
Given an initial guess <span class="emphasis"><em>x0</em></span> the subsequent values are computed
using:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../equations/roots1.svg"></span>
</p></blockquote></div>
<p>
Out-of-bounds steps revert to <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisection</a>
of the current bounds.
</p>
<p>
Under ideal conditions, the number of correct digits doubles with each iteration.
</p>
<h5>
<a name="math_toolkit.roots_deriv.h3"></a>
<span class="phrase"><a name="math_toolkit.roots_deriv.halley"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.halley">Halley's
Method</a>
</h5>
<p>
Given an initial guess <span class="emphasis"><em>x0</em></span> the subsequent values are computed
using:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../equations/roots2.svg"></span>
</p></blockquote></div>
<p>
Over-compensation by the second derivative (one which would proceed in the
wrong direction) causes the method to revert to a Newton-Raphson step.
</p>
<p>
Out of bounds steps revert to bisection of the current bounds.
</p>
<p>
Under ideal conditions, the number of correct digits trebles with each iteration.
</p>
<h5>
<a name="math_toolkit.roots_deriv.h4"></a>
<span class="phrase"><a name="math_toolkit.roots_deriv.schroder"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.schroder">Schröder's
Method</a>
</h5>
<p>
Given an initial guess x0 the subsequent values are computed using:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../equations/roots3.svg"></span>
</p></blockquote></div>
<p>
Over-compensation by the second derivative (one which would proceed in the
wrong direction) causes the method to revert to a Newton-Raphson step. Likewise
a Newton step is used whenever that Newton step would change the next value
by more than 10%.
</p>
<p>
Out of bounds steps revert to <a href="https://en.wikipedia.org/wiki/Bisection" target="_top">bisection</a>
of the current bounds.
</p>
<p>
Under ideal conditions, the number of correct digits trebles with each iteration.
</p>
<p>
This is Schröder's general result (equation 18 from <a href="http://drum.lib.umd.edu/handle/1903/577" target="_top">Stewart,
G. W. "On Infinitely Many Algorithms for Solving Equations." English
translation of Schröder's original paper. College Park, MD: University of Maryland,
Institute for Advanced Computer Studies, Department of Computer Science, 1993</a>.)
</p>
<p>
This method guarantees at least quadratic convergence (the same as Newton's
method), and is known to work well in the presence of multiple roots: something
that neither Newton nor Halley can do.
</p>
<p>
The complex Newton method works slightly differently than the rest of the methods:
Since there is no way to bracket roots in the complex plane, the <code class="computeroutput"><span class="identifier">min</span></code> and <code class="computeroutput"><span class="identifier">max</span></code>
arguments are not accepted. Failure to reach a root is communicated by returning
<code class="computeroutput"><span class="identifier">nan</span></code>s. Remember that if a function
has many roots, then which root the complex Newton's method converges to is
essentially impossible to predict a priori; see the Newton's fractal for more
information.
</p>
<p>
Finally, the derivative of <span class="emphasis"><em>f</em></span> must be continuous at the
root or else non-roots can be found; see <a href="https://math.stackexchange.com/questions/3017766/constructing-newton-iteration-converging-to-non-root" target="_top">here</a>
for an example.
</p>
<p>
An example usage of <code class="computeroutput"><span class="identifier">complex_newton</span></code>
is given in <code class="computeroutput"><span class="identifier">examples</span><span class="special">/</span><span class="identifier">daubechies_coefficients</span><span class="special">.</span><span class="identifier">cpp</span></code>.
</p>
<h5>
<a name="math_toolkit.roots_deriv.h5"></a>
<span class="phrase"><a name="math_toolkit.roots_deriv.quadratics"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.quadratics">Quadratics</a>
</h5>
<p>
To solve a quadratic <span class="emphasis"><em>ax</em></span><sup>2</sup> + <span class="emphasis"><em>bx</em></span> + <span class="emphasis"><em>c</em></span>
= 0, we may use
</p>
<pre class="programlisting"><span class="keyword">auto</span> <span class="special">[</span><span class="identifier">x0</span><span class="special">,</span> <span class="identifier">x1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">quadratic_roots</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">c</span><span class="special">);</span>
</pre>
<p>
If the roots are real, they are arranged so that <code class="computeroutput"><span class="identifier">x0</span></code>
≤ <code class="computeroutput"><span class="identifier">x1</span></code>. If the roots are
complex and the inputs are real, <code class="computeroutput"><span class="identifier">x0</span></code>
and <code class="computeroutput"><span class="identifier">x1</span></code> are both <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">()</span></code>. In this case we must cast <code class="computeroutput"><span class="identifier">a</span></code>, <code class="computeroutput"><span class="identifier">b</span></code>
and <code class="computeroutput"><span class="identifier">c</span></code> to a complex type to
extract the complex roots. If <code class="computeroutput"><span class="identifier">a</span></code>,
<code class="computeroutput"><span class="identifier">b</span></code> and <code class="computeroutput"><span class="identifier">c</span></code>
are integral, then the roots are of type double. The routine is much faster
if the fused-multiply-add instruction is available on your architecture. If
the fma is not available, the function resorts to slow emulation. Finally,
speed is improved if you compile for your particular architecture. For instance,
if you compile without any architecture flags, then the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">fma</span></code> call
compiles down to <code class="computeroutput"><span class="identifier">call</span> <span class="identifier">_fma</span></code>,
which dynamically chooses to emulate or execute the <code class="computeroutput"><span class="identifier">vfmadd132sd</span></code>
instruction based on the capabilities of the architecture. If instead, you
compile with (say) <code class="computeroutput"><span class="special">-</span><span class="identifier">march</span><span class="special">=</span><span class="identifier">native</span></code> then
no dynamic choice is made: The <code class="computeroutput"><span class="identifier">vfmadd132sd</span></code>
instruction is always executed if available and emulation is used if not.
</p>
<h5>
<a name="math_toolkit.roots_deriv.h6"></a>
<span class="phrase"><a name="math_toolkit.roots_deriv.examples"></a></span><a class="link" href="roots_deriv.html#math_toolkit.roots_deriv.examples">Examples</a>
</h5>
<p>
See <a class="link" href="root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)">root-finding examples</a>.
</p>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
</div>
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