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<title>Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions</title>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.roots_noderiv.TOMS748"></a><a class="link" href="TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">Algorithm TOMS 748:
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Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a>
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</h3></div></div></div>
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<pre class="programlisting"><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="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">toms748_solve</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">boost</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>
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<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="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">toms748_solve</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">boost</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>
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<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<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="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">toms748_solve</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">boost</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>
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<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="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">toms748_solve</span><span class="special">(</span>
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<span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span>
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<span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span>
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<span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
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<span class="identifier">boost</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>
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<span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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</pre>
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<p>
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These functions implement TOMS Algorithm 748: it uses a mixture of cubic,
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quadratic and linear (secant) interpolation to locate the root of <span class="emphasis"><em>f(x)</em></span>.
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The two pairs of functions differ only by whether values for <span class="emphasis"><em>f(a)</em></span>
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and <span class="emphasis"><em>f(b)</em></span> are already available.
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</p>
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<p>
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Generally speaking it is easier (and often more efficient) to use <a class="link" href="bracket_solve.html" title="Bracket and Solve Root">bracket
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and solve</a> rather than trying to bracket the root yourself as this
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function requires.
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</p>
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<p>
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This function is provided rather than <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent's
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method</a> as it is known to be more efficient in many cases (it is asymptotically
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the most efficient known, and has been shown to be optimal for a certain
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classes of smooth functions). It also has the useful property of decreasing
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the bracket size with each step, unlike Brent's method which only shrinks
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the enclosing interval in the final step. This makes it particularly useful
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when you need a result where the ends of the interval round to the same integer:
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as often happens in statistical applications, for example. In this situation
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the function is able to exit after a much smaller number of iterations than
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would otherwise be possible.
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</p>
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<p>
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The <a class="link" href="TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS 748 algorithm</a>
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parameters are:
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</p>
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<div class="variablelist">
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<p class="title"><b></b></p>
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<dl class="variablelist">
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<dt><span class="term">f</span></dt>
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<dd><p>
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A unary functor (or C++ lambda) that is the function whose root is
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to be solved. f(x) need not be uniformly increasing or decreasing on
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<span class="emphasis"><em>x</em></span> and may have multiple roots. However, the bounds
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given must bracket a single root.
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</p></dd>
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<dt><span class="term">a</span></dt>
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<dd><p>
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The lower bound for the initial bracket of the root.
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</p></dd>
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<dt><span class="term">b</span></dt>
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<dd><p>
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The upper bound for the initial bracket of the root. It is a precondition
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that <span class="emphasis"><em>a < b</em></span> and that <span class="emphasis"><em>a</em></span>
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and <span class="emphasis"><em>b</em></span> bracket the root to find so that <span class="emphasis"><em>f(a)
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* f(b) < 0</em></span>.
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</p></dd>
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<dt><span class="term">fa</span></dt>
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<dd><p>
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Optional: the value of <span class="emphasis"><em>f(a)</em></span>.
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</p></dd>
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<dt><span class="term">fb</span></dt>
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<dd><p>
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Optional: the value of <span class="emphasis"><em>f(b)</em></span>.
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</p></dd>
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<dt><span class="term">tol</span></dt>
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<dd><p>
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A binary functor (or C++ lambda) that determines the termination condition
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for the search for the root. <span class="emphasis"><em>tol</em></span> is passed the
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current brackets at each step, when it returns true, then the current
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brackets are returned as the result. See also <a class="link" href="root_termination.html" title="Termination Condition Functors">predefined
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termination functors</a>.
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</p></dd>
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<dt><span class="term">max_iter</span></dt>
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<dd><p>
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The maximum number of function invocations to perform in the search
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for the root. On exit, <span class="emphasis"><em>max_iter</em></span> is set to actual
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number of function invocations used.
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</p></dd>
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</dl>
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</div>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
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documentation for more details</a>.
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</p>
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<p>
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<code class="computeroutput"><span class="identifier">toms748_solve</span></code> returns: a
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pair of values <span class="emphasis"><em>r</em></span> that bracket the root so that:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="emphasis"><em>f(r.first) * f(r.second) <= 0</em></span>
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</p></blockquote></div>
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<p>
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and either
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="emphasis"><em>tol(r.first, r.second) == true</em></span>
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</p></blockquote></div>
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<p>
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or
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="emphasis"><em>max_iter >= m</em></span>
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</p></blockquote></div>
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<p>
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where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span>
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passed to the function.
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</p>
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<p>
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In other words, it's up to the caller to verify whether termination occurred
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as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations
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(easily done by checking the updated value of <span class="emphasis"><em>max_iter</em></span>
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against its previous value passed as parameter), rather than because the
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termination condition <span class="emphasis"><em>tol</em></span> was satisfied.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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</p>
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