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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.toolkit.internals1.roots"></a><a class="link" href="roots.html" title="Root Finding With Derivatives"> Root Finding |
| With Derivatives</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.toolkit.internals1.roots.synopsis"></a><h5> |
| <a name="id1200892"></a> |
| <a class="link" href="roots.html#math_toolkit.toolkit.internals1.roots.synopsis">Synopsis</a> |
| </h5> |
| <p> |
| |
| </p> |
| <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> |
| <p> |
| </p> |
| <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="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">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="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">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="identifier">T</span> <span class="identifier">schroeder_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">schroeder_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">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="special">}}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <a name="math_toolkit.toolkit.internals1.roots.description"></a><h5> |
| <a name="id1201759"></a> |
| <a class="link" href="roots.html#math_toolkit.toolkit.internals1.roots.description">Description</a> |
| </h5> |
| <p> |
| These functions all perform iterative root finding: <code class="computeroutput"><span class="identifier">newton_raphson_iterate</span></code> |
| performs second order <a class="link" href="roots.html#newton">Newton Raphson iteration</a>, |
| while <code class="computeroutput"><span class="identifier">halley_iterate</span></code> and |
| <code class="computeroutput"><span class="identifier">schroeder_iterate</span></code> perform |
| third order <a class="link" href="roots.html#halley">Halley</a> and <a class="link" href="roots.html#schroeder">Schroeder</a> |
| iteration respectively. |
| </p> |
| <p> |
| The functions all take the same parameters: |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b>Parameters of the root finding functions</b></p> |
| <dl> |
| <dt><span class="term">F f</span></dt> |
| <dd> |
| <p> |
| Type F must be a callable function object that accepts one parameter |
| and returns a <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a>: |
| </p> |
| <p> |
| For the second order iterative methods (Newton Raphson) the <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a> |
| should have two elements containing the evaluation of the function |
| and it's first derivative. |
| </p> |
| <p> |
| For the third order methods (Halley and Schroeder) the <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a> |
| should have three 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. |
| </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. |
| </p></dd> |
| <dt><span class="term">uintmax_t max_iter</span></dt> |
| <dd><p> |
| An optional maximum number of iterations to perform. |
| </p></dd> |
| </dl> |
| </div> |
| <p> |
| When using these functions you should note that: |
| </p> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| 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 bisection, or in some rare |
| cases even worse than bisection if the initial guess is a long way |
| from the correct value and the derivatives are close to zero. |
| </li> |
| <li> |
| These functions include special cases to handle zero first (and second |
| where appropriate) derivatives, and fall back to bisection in this |
| case. However, it is helpful if F is defined to return an arbitrarily |
| small value <span class="emphasis"><em>of the correct sign</em></span> rather than zero. |
| </li> |
| <li> |
| 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> |
| These functions fall back to bisection 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 bisection. |
| </li> |
| <li> |
| 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 bisection. Remember that the returned value can never |
| be more accurate than f(x) can be evaluated, and that if f(x) 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 f(x) |
| becomes inaccurate. |
| </li> |
| <li> |
| 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> |
| <a name="newton"></a><p> |
| </p> |
| <a name="math_toolkit.toolkit.internals1.roots.newton_raphson_method"></a><h5> |
| <a name="id1202008"></a> |
| <a class="link" href="roots.html#math_toolkit.toolkit.internals1.roots.newton_raphson_method">Newton |
| Raphson Method</a> |
| </h5> |
| <p> |
| Given an initial guess x0 the subsequent values are computed using: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/roots1.png"></span> |
| </p> |
| <p> |
| Out of bounds steps revert to bisection of the current bounds. |
| </p> |
| <p> |
| Under ideal conditions, the number of correct digits doubles with each |
| iteration. |
| </p> |
| <a name="halley"></a><p> |
| </p> |
| <a name="math_toolkit.toolkit.internals1.roots.halley_s_method"></a><h5> |
| <a name="id1202066"></a> |
| <a class="link" href="roots.html#math_toolkit.toolkit.internals1.roots.halley_s_method">Halley's |
| Method</a> |
| </h5> |
| <p> |
| Given an initial guess x0 the subsequent values are computed using: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/roots2.png"></span> |
| </p> |
| <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> |
| <a name="schroeder"></a><p> |
| </p> |
| <a name="math_toolkit.toolkit.internals1.roots.schroeder_s_method"></a><h5> |
| <a name="id1202127"></a> |
| <a class="link" href="roots.html#math_toolkit.toolkit.internals1.roots.schroeder_s_method">Schroeder's |
| Method</a> |
| </h5> |
| <p> |
| Given an initial guess x0 the subsequent values are computed using: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/roots3.png"></span> |
| </p> |
| <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 bisection of the current bounds. |
| </p> |
| <p> |
| Under ideal conditions, the number of correct digits trebles with each |
| iteration. |
| </p> |
| <a name="math_toolkit.toolkit.internals1.roots.example"></a><h5> |
| <a name="id1204047"></a> |
| <a class="link" href="roots.html#math_toolkit.toolkit.internals1.roots.example">Example</a> |
| </h5> |
| <p> |
| Lets suppose we want to find the cube root of a number, the equation we |
| want to solve along with its derivatives are: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/roots4.png"></span> |
| </p> |
| <p> |
| To begin with lets solve the problem using Newton Raphson iterations, we'll |
| begin be defining a function object that returns the evaluation of the |
| function to solve, along with its first derivative: |
| </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">cbrt_functor</span> |
| <span class="special">{</span> |
| <span class="identifier">cbrt_functor</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">target</span><span class="special">)</span> <span class="special">:</span> <span class="identifier">a</span><span class="special">(</span><span class="identifier">target</span><span class="special">){}</span> |
| <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">z</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">T</span> <span class="identifier">sqr</span> <span class="special">=</span> <span class="identifier">z</span> <span class="special">*</span> <span class="identifier">z</span><span class="special">;</span> |
| <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">tr1</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span><span class="identifier">sqr</span> <span class="special">*</span> <span class="identifier">z</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">,</span> <span class="number">3</span> <span class="special">*</span> <span class="identifier">sqr</span><span class="special">);</span> |
| <span class="special">}</span> |
| <span class="keyword">private</span><span class="special">:</span> |
| <span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| Implementing the cube root is fairly trivial now, the hardest part is finding |
| a good approximation to begin with: in this case we'll just divide the |
| exponent by three: |
| </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="identifier">T</span> <span class="identifier">cbrt</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span> |
| <span class="keyword">int</span> <span class="identifier">exp</span><span class="special">;</span> |
| <span class="identifier">frexp</span><span class="special">(</span><span class="identifier">z</span><span class="special">,</span> <span class="special">&</span><span class="identifier">exp</span><span class="special">);</span> |
| <span class="identifier">T</span> <span class="identifier">min</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">exp</span><span class="special">/</span><span class="number">3</span><span class="special">);</span> |
| <span class="identifier">T</span> <span class="identifier">max</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">2.0</span><span class="special">,</span> <span class="identifier">exp</span><span class="special">/</span><span class="number">3</span><span class="special">);</span> |
| <span class="identifier">T</span> <span class="identifier">guess</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="identifier">exp</span><span class="special">/</span><span class="number">3</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">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">;</span> |
| <span class="keyword">return</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">newton_raphson_iterate</span><span class="special">(</span><span class="identifier">detail</span><span class="special">::</span><span class="identifier">cbrt_functor</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(</span><span class="identifier">z</span><span class="special">),</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">max</span><span class="special">,</span> <span class="identifier">digits</span><span class="special">);</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| Using the test data in libs/math/test/cbrt_test.cpp this found the cube |
| root exact to the last digit in every case, and in no more than 6 iterations |
| at double precision. However, you will note that a high precision was used |
| in this example, exactly what was warned against earlier on in these docs! |
| In this particular case its possible to compute f(x) exactly and without |
| undue cancellation error, so a high limit is not too much of an issue. |
| However, reducing the limit to <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> |
| <span class="special">*</span> <span class="number">2</span> <span class="special">/</span> <span class="number">3</span></code> gave |
| full precision in all but one of the test cases (and that one was out by |
| just one bit). The maximum number of iterations remained 6, but in most |
| cases was reduced by one. |
| </p> |
| <p> |
| Note also that the above code omits error handling, and does not handle |
| negative values of z correctly. That will be left as an exercise for the |
| reader! |
| </p> |
| <p> |
| Now lets adapt the functor slightly to return the second derivative as |
| well: |
| </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">cbrt_functor</span> |
| <span class="special">{</span> |
| <span class="identifier">cbrt_functor</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">target</span><span class="special">)</span> <span class="special">:</span> <span class="identifier">a</span><span class="special">(</span><span class="identifier">target</span><span class="special">){}</span> |
| <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a><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">T</span><span class="special">></span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">z</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">T</span> <span class="identifier">sqr</span> <span class="special">=</span> <span class="identifier">z</span> <span class="special">*</span> <span class="identifier">z</span><span class="special">;</span> |
| <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">tr1</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span><span class="identifier">sqr</span> <span class="special">*</span> <span class="identifier">z</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">,</span> <span class="number">3</span> <span class="special">*</span> <span class="identifier">sqr</span><span class="special">,</span> <span class="number">6</span> <span class="special">*</span> <span class="identifier">z</span><span class="special">);</span> |
| <span class="special">}</span> |
| <span class="keyword">private</span><span class="special">:</span> |
| <span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| And then adapt the <code class="computeroutput"><span class="identifier">cbrt</span></code> |
| function to use Halley iterations: |
| </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="identifier">T</span> <span class="identifier">cbrt</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span> |
| <span class="keyword">int</span> <span class="identifier">exp</span><span class="special">;</span> |
| <span class="identifier">frexp</span><span class="special">(</span><span class="identifier">z</span><span class="special">,</span> <span class="special">&</span><span class="identifier">exp</span><span class="special">);</span> |
| <span class="identifier">T</span> <span class="identifier">min</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">exp</span><span class="special">/</span><span class="number">3</span><span class="special">);</span> |
| <span class="identifier">T</span> <span class="identifier">max</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">2.0</span><span class="special">,</span> <span class="identifier">exp</span><span class="special">/</span><span class="number">3</span><span class="special">);</span> |
| <span class="identifier">T</span> <span class="identifier">guess</span> <span class="special">=</span> <span class="identifier">ldexp</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="identifier">exp</span><span class="special">/</span><span class="number">3</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">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span> <span class="special">/</span> <span class="number">2</span><span class="special">;</span> |
| <span class="keyword">return</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">halley_iterate</span><span class="special">(</span><span class="identifier">detail</span><span class="special">::</span><span class="identifier">cbrt_functor</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(</span><span class="identifier">z</span><span class="special">),</span> <span class="identifier">guess</span><span class="special">,</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">max</span><span class="special">,</span> <span class="identifier">digits</span><span class="special">);</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| Note that the iterations are set to stop at just one-half of full precision, |
| and yet even so not one of the test cases had a single bit wrong. What's |
| more, the maximum number of iterations was now just 4. |
| </p> |
| <p> |
| Just to complete the picture, we could have called <code class="computeroutput"><span class="identifier">schroeder_iterate</span></code> |
| in the last example: and in fact it makes no difference to the accuracy |
| or number of iterations in this particular case. However, the relative |
| performance of these two methods may vary depending upon the nature of |
| f(x), and the accuracy to which the initial guess can be computed. There |
| appear to be no generalisations that can be made except "try them |
| and see". |
| </p> |
| <p> |
| Finally, had we called cbrt with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> |
| set to 1000 bit precision, then full precision can be obtained with just |
| 7 iterations. To put that in perspective an increase in precision by a |
| factor of 20, has less than doubled the number of iterations. That just |
| goes to emphasise that most of the iterations are used up getting the first |
| few digits correct: after that these methods can churn out further digits |
| with remarkable efficiency. Or to put it another way: <span class="emphasis"><em>nothing |
| beats a really good initial guess!</em></span> |
| </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 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<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> |
| </tr></table> |
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