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| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.internals1.roots2"></a><a class="link" href="roots2.html" title="Root Finding Without Derivatives: Bisection, Bracket and TOMS748">Root Finding Without |
| Derivatives: Bisection, Bracket and TOMS748</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.internals1.roots2.h0"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.roots2.synopsis"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.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">// Bisection</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="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span> |
| <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> |
| <span class="identifier">bisect</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">min</span><span class="special">,</span> |
| <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| |
| <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="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> |
| <span class="identifier">bisect</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">min</span><span class="special">,</span> |
| <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</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="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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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> |
| <span class="identifier">bisect</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">min</span><span class="special">,</span> |
| <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| |
| <span class="comment">// Bracket and Solve Root</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="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span> |
| <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> |
| <span class="identifier">bracket_and_solve_root</span><span class="special">(</span> |
| <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span> |
| <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| |
| <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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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> |
| <span class="identifier">bracket_and_solve_root</span><span class="special">(</span> |
| <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span> |
| <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| |
| <span class="comment">// TOMS 748 algorithm</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="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span> |
| <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> |
| <span class="identifier">toms748_solve</span><span class="special">(</span> |
| <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| |
| <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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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> |
| <span class="identifier">toms748_solve</span><span class="special">(</span> |
| <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><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="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span> |
| <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> |
| <span class="identifier">toms748_solve</span><span class="special">(</span> |
| <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| |
| <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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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> |
| <span class="identifier">toms748_solve</span><span class="special">(</span> |
| <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| |
| <span class="comment">// Termination conditions:</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">struct</span> <span class="identifier">eps_tolerance</span><span class="special">;</span> |
| |
| <span class="keyword">struct</span> <span class="identifier">equal_floor</span><span class="special">;</span> |
| <span class="keyword">struct</span> <span class="identifier">equal_ceil</span><span class="special">;</span> |
| <span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span><span class="special">;</span> |
| |
| <span class="special">}}}</span> <span class="comment">// namespaces</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.internals1.roots2.h1"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.roots2.description"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.description">Description</a> |
| </h5> |
| <p> |
| These functions solve the root of some function <span class="emphasis"><em>f(x)</em></span> |
| <span class="emphasis"><em>without the need for any derivatives of <span class="emphasis"><em>f(x)</em></span></em></span>. |
| </p> |
| <p> |
| The <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code> |
| functions use TOMS Algorithm 748 that is asymptotically the most efficient |
| known, and have been shown to be optimal for a certain classes of smooth |
| functions. Variants with and without __policies are provided. |
| </p> |
| <p> |
| Alternatively, there is a simple bisection routine which can be useful in |
| its own right in some situations, or alternatively for narrowing down the |
| range containing the root, prior to calling a more advanced algorithm. |
| </p> |
| <p> |
| All the algorithms in this section reduce the diameter of the enclosing interval |
| with the same asymptotic efficiency with which they locate the root. This |
| is in contrast to the derivative based methods which may <span class="emphasis"><em>never</em></span> |
| significantly reduce the enclosing interval, even though they rapidly approach |
| the root. This is also in contrast to some other derivative-free methods |
| (for example the methods of <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent |
| or Dekker)</a> which only reduce the enclosing interval on the final |
| step. Therefore these methods return a std::pair containing the enclosing |
| interval found, and accept a function object specifying the termination condition. |
| Three function objects are provided for ready-made termination conditions: |
| <span class="emphasis"><em>eps_tolerance</em></span> causes termination when the relative error |
| in the enclosing interval is below a certain threshold, while <span class="emphasis"><em>equal_floor</em></span> |
| and <span class="emphasis"><em>equal_ceil</em></span> are useful for certain statistical applications |
| where the result is known to be an integer. Other user-defined termination |
| conditions are likely to be used only rarely, but may be useful in some specific |
| circumstances. |
| </p> |
| <h4> |
| <a name="math_toolkit.internals1.roots2.h2"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.roots2.bisection"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.bisection">Bisection</a> |
| </h4> |
| <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> |
| <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> |
| <span class="identifier">bisect</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">min</span><span class="special">,</span> |
| <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| |
| <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="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> |
| <span class="identifier">bisect</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">min</span><span class="special">,</span> |
| <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</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="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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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> |
| <span class="identifier">bisect</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">min</span><span class="special">,</span> |
| <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| These functions locate the root using bisection: function arguments are: |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b></b></p> |
| <dl class="variablelist"> |
| <dt><span class="term">f</span></dt> |
| <dd><p> |
| A unary functor which is the function whose root is to be found. |
| </p></dd> |
| <dt><span class="term">min</span></dt> |
| <dd><p> |
| The left bracket of the interval known to contain the root. |
| </p></dd> |
| <dt><span class="term">max</span></dt> |
| <dd><p> |
| The right bracket of the interval known to contain the root. It is |
| a precondition that <span class="emphasis"><em>min < max</em></span> and <span class="emphasis"><em>f(min)*f(max) |
| <= 0</em></span>, the function signals evaluation error if these |
| preconditions are violated. The action taken is controlled by the evaluation |
| error policy. A best guess may be returned, perhaps significantly wrong. |
| </p></dd> |
| <dt><span class="term">tol</span></dt> |
| <dd><p> |
| A binary functor that specifies the termination condition: the function |
| will return the current brackets enclosing the root when <span class="emphasis"><em>tol(min,max)</em></span> |
| becomes true. |
| </p></dd> |
| <dt><span class="term">max_iter</span></dt> |
| <dd><p> |
| The maximum number of invocations of <span class="emphasis"><em>f(x)</em></span> to make |
| while searching for the root. On exist this is set to actual number |
| of invocations performed. |
| </p></dd> |
| </dl> |
| </div> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can |
| be used to control the behaviour of the function: how it handles errors, |
| what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy |
| documentation for more details</a>. |
| </p> |
| <p> |
| Returns: a pair of values <span class="emphasis"><em>r</em></span> that bracket the root so |
| that: |
| </p> |
| <pre class="programlisting"><span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special"><=</span> <span class="number">0</span> |
| </pre> |
| <p> |
| and either |
| </p> |
| <pre class="programlisting"><span class="identifier">tol</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">,</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special">==</span> <span class="keyword">true</span> |
| </pre> |
| <p> |
| or |
| </p> |
| <pre class="programlisting"><span class="identifier">max_iter</span> <span class="special">>=</span> <span class="identifier">m</span> |
| </pre> |
| <p> |
| where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span> |
| passed to the function. |
| </p> |
| <p> |
| In other words, it's up to the caller to verify whether termination occurred |
| as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations |
| (easily done by checking the updated value of <span class="emphasis"><em>max_iter</em></span> |
| when the function returns), rather than because the termination condition |
| <span class="emphasis"><em>tol</em></span> was satisfied. |
| </p> |
| <h4> |
| <a name="math_toolkit.internals1.roots2.h3"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.roots2.bracket_and_solve"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.bracket_and_solve">Bracket |
| and solve</a> |
| </h4> |
| <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> |
| <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> |
| <span class="identifier">bracket_and_solve_root</span><span class="special">(</span> |
| <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span> |
| <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| |
| <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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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> |
| <span class="identifier">bracket_and_solve_root</span><span class="special">(</span> |
| <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span> |
| <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| This is a convenience function that calls <span class="emphasis"><em>toms748_solve</em></span> |
| internally to find the root of <span class="emphasis"><em>f(x)</em></span>. It's usable only |
| when <span class="emphasis"><em>f(x)</em></span> is a monotonic function, and the location |
| of the root is known approximately, and in particular it is known whether |
| the root is occurs for positive or negative <span class="emphasis"><em>x</em></span>. The parameters |
| are: |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b></b></p> |
| <dl class="variablelist"> |
| <dt><span class="term">f</span></dt> |
| <dd><p> |
| A unary functor that is the function whose root is to be solved. f(x) |
| must be uniformly increasing or decreasing on <span class="emphasis"><em>x</em></span>. |
| </p></dd> |
| <dt><span class="term">guess</span></dt> |
| <dd><p> |
| An initial approximation to the root |
| </p></dd> |
| <dt><span class="term">factor</span></dt> |
| <dd><p> |
| A scaling factor that is used to bracket the root: the value <span class="emphasis"><em>guess</em></span> |
| is multiplied (or divided as appropriate) by <span class="emphasis"><em>factor</em></span> |
| until two values are found that bracket the root. A value such as 2 |
| is a typical choice for <span class="emphasis"><em>factor</em></span>. |
| </p></dd> |
| <dt><span class="term">rising</span></dt> |
| <dd><p> |
| Set to <span class="emphasis"><em>true</em></span> if <span class="emphasis"><em>f(x)</em></span> is rising |
| on <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>false</em></span> if <span class="emphasis"><em>f(x)</em></span> |
| is falling on <span class="emphasis"><em>x</em></span>. This value is used along with |
| the result of <span class="emphasis"><em>f(guess)</em></span> to determine if <span class="emphasis"><em>guess</em></span> |
| is above or below the root. |
| </p></dd> |
| <dt><span class="term">tol</span></dt> |
| <dd><p> |
| A binary functor that determines the termination condition for the |
| search for the root. <span class="emphasis"><em>tol</em></span> is passed the current |
| brackets at each step, when it returns true then the current brackets |
| are returned as the result. |
| </p></dd> |
| <dt><span class="term">max_iter</span></dt> |
| <dd><p> |
| The maximum number of function invocations to perform in the search |
| for the root. On exit is set to the actual number of invocations performed. |
| </p></dd> |
| </dl> |
| </div> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can |
| be used to control the behaviour of the function: how it handles errors, |
| what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy |
| documentation for more details</a>. |
| </p> |
| <p> |
| Returns: a pair of values <span class="emphasis"><em>r</em></span> that bracket the root so |
| that: |
| </p> |
| <pre class="programlisting"><span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special"><=</span> <span class="number">0</span> |
| </pre> |
| <p> |
| and either |
| </p> |
| <pre class="programlisting"><span class="identifier">tol</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">,</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special">==</span> <span class="keyword">true</span> |
| </pre> |
| <p> |
| or |
| </p> |
| <pre class="programlisting"><span class="identifier">max_iter</span> <span class="special">>=</span> <span class="identifier">m</span> |
| </pre> |
| <p> |
| where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span> |
| passed to the function. |
| </p> |
| <p> |
| In other words, it's up to the caller to verify whether termination occurred |
| as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations |
| (easily done by checking the value of <span class="emphasis"><em>max_iter</em></span> when |
| the function returns), rather than because the termination condition <span class="emphasis"><em>tol</em></span> |
| was satisfied. |
| </p> |
| <h4> |
| <a name="math_toolkit.internals1.roots2.h4"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.roots2.algorithm_toms_748_alefeld_potra"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.algorithm_toms_748_alefeld_potra">Algorithm |
| TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a> |
| </h4> |
| <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> |
| <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> |
| <span class="identifier">toms748_solve</span><span class="special">(</span> |
| <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| |
| <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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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> |
| <span class="identifier">toms748_solve</span><span class="special">(</span> |
| <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><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="keyword">class</span> <span class="identifier">Tol</span><span class="special">></span> |
| <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> |
| <span class="identifier">toms748_solve</span><span class="special">(</span> |
| <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| |
| <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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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> |
| <span class="identifier">toms748_solve</span><span class="special">(</span> |
| <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span> |
| <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span> |
| <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> |
| <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> |
| <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| These two functions implement TOMS Algorithm 748: it uses a mixture of cubic, |
| quadratic and linear (secant) interpolation to locate the root of <span class="emphasis"><em>f(x)</em></span>. |
| The two functions differ only by whether values for <span class="emphasis"><em>f(a)</em></span> |
| and <span class="emphasis"><em>f(b)</em></span> are already available. The toms748_solve parameters |
| are: |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b></b></p> |
| <dl class="variablelist"> |
| <dt><span class="term">f</span></dt> |
| <dd><p> |
| A unary functor that is the function whose root is to be solved. f(x) |
| need not be uniformly increasing or decreasing on <span class="emphasis"><em>x</em></span> |
| and may have multiple roots. |
| </p></dd> |
| <dt><span class="term">a</span></dt> |
| <dd><p> |
| The lower bound for the initial bracket of the root. |
| </p></dd> |
| <dt><span class="term">b</span></dt> |
| <dd><p> |
| The upper bound for the initial bracket of the root. It is a precondition |
| that <span class="emphasis"><em>a < b</em></span> and that <span class="emphasis"><em>a</em></span> |
| and <span class="emphasis"><em>b</em></span> bracket the root to find so that <span class="emphasis"><em>f(a)*f(b) |
| < 0</em></span>. |
| </p></dd> |
| <dt><span class="term">fa</span></dt> |
| <dd><p> |
| Optional: the value of <span class="emphasis"><em>f(a)</em></span>. |
| </p></dd> |
| <dt><span class="term">fb</span></dt> |
| <dd><p> |
| Optional: the value of <span class="emphasis"><em>f(b)</em></span>. |
| </p></dd> |
| <dt><span class="term">tol</span></dt> |
| <dd><p> |
| A binary functor that determines the termination condition for the |
| search for the root. <span class="emphasis"><em>tol</em></span> is passed the current |
| brackets at each step, when it returns true, then the current brackets |
| are returned as the result. |
| </p></dd> |
| <dt><span class="term">max_iter</span></dt> |
| <dd><p> |
| The maximum number of function invocations to perform in the search |
| for the root. On exit <span class="emphasis"><em>max_iter</em></span> is set to actual |
| number of function invocations used. |
| </p></dd> |
| </dl> |
| </div> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can |
| be used to control the behaviour of the function: how it handles errors, |
| what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy |
| documentation for more details</a>. |
| </p> |
| <p> |
| toms748_solve returns: a pair of values <span class="emphasis"><em>r</em></span> that bracket |
| the root so that: f(r.first) * f(r.second) <= 0 and either tol(r.first, |
| r.second) == true or max_iter >= m where <span class="emphasis"><em>m</em></span> is the |
| initial value of <span class="emphasis"><em>max_iter</em></span> passed to the function. |
| </p> |
| <p> |
| In other words, it's up to the caller to verify whether termination occurred |
| as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations |
| (easily done by checking the updated value of <span class="emphasis"><em>max_iter</em></span> |
| against its previous value passed as parameter), rather than because the |
| termination condition <span class="emphasis"><em>tol</em></span> was satisfied. |
| </p> |
| <pre class="programlisting"><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">struct</span> <span class="identifier">eps_tolerance</span> |
| <span class="special">{</span> |
| <span class="identifier">eps_tolerance</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span> |
| <span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| <code class="computeroutput"><span class="identifier">eps_tolerance</span></code> is the usual |
| termination condition used with these root finding functions. Its operator() |
| will return true when the relative distance between <span class="emphasis"><em>a</em></span> |
| and <span class="emphasis"><em>b</em></span> is less than twice the machine epsilon for T, |
| or 2<sup>1-bits</sup>, whichever is the larger. In other words, you set <span class="emphasis"><em>bits</em></span> |
| to the number of bits of precision you want in the result. The minimal tolerance |
| of twice the machine epsilon of T is required to ensure that we get back |
| a bracketing interval: since this must clearly be at least 1 epsilon in size. |
| </p> |
| <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_floor</span> |
| <span class="special">{</span> |
| <span class="identifier">equal_floor</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">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| This termination condition is used when you want to find an integer result |
| that is the <span class="emphasis"><em>floor</em></span> of the true root. It will terminate |
| as soon as both ends of the interval have the same <span class="emphasis"><em>floor</em></span>. |
| </p> |
| <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_ceil</span> |
| <span class="special">{</span> |
| <span class="identifier">equal_ceil</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">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| This termination condition is used when you want to find an integer result |
| that is the <span class="emphasis"><em>ceil</em></span> of the true root. It will terminate |
| as soon as both ends of the interval have the same <span class="emphasis"><em>ceil</em></span>. |
| </p> |
| <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span> |
| <span class="special">{</span> |
| <span class="identifier">equal_nearest_integer</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">bool</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| This termination condition is used when you want to find an integer result |
| that is the <span class="emphasis"><em>closest</em></span> to the true root. It will terminate |
| as soon as both ends of the interval round to the same nearest integer. |
| </p> |
| <h5> |
| <a name="math_toolkit.internals1.roots2.h5"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.roots2.implementation"></a></span><a class="link" href="roots2.html#math_toolkit.internals1.roots2.implementation">Implementation</a> |
| </h5> |
| <p> |
| The implementation of the bisection algorithm is extremely straightforward |
| and not detailed here. <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS |
| Algorithm 748: enclosing zeros of continuous functions</a> is described |
| in detail in: |
| </p> |
| <p> |
| <span class="emphasis"><em>Algorithm 748: Enclosing Zeros of Continuous Functions, G. E. Alefeld, |
| F. A. Potra and Yixun Shi, ACM Transactions on Mathematica1 Software, Vol. |
| 21. No. 3. September 1995. Pages 327-344.</em></span> |
| </p> |
| <p> |
| The implementation here is a faithful translation of this paper into C++. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| 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></td> |
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