1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Root-finding using Boost.Multiprecision</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)"> 9<link rel="prev" href="5th_root_eg.html" title="Computing the Fifth Root"> 10<link rel="next" href="nth_root.html" title="Generalizing to Compute the nth root"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="5th_root_eg.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../root_finding_examples.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="nth_root.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="math_toolkit.root_finding_examples.multiprecision_root"></a><a class="link" href="multiprecision_root.html" title="Root-finding using Boost.Multiprecision">Root-finding 28 using Boost.Multiprecision</a> 29</h3></div></div></div> 30<p> 31 The apocryphally astute reader might, by now, be asking "How do we know 32 if this computes the 'right' answer?". 33 </p> 34<p> 35 For most values, there is, sadly, no 'right' answer. This is because values 36 can only rarely be <span class="emphasis"><em>exactly represented</em></span> by C++ floating-point 37 types. What we do want is the 'best' representation - one that is the nearest 38 <a href="http://en.wikipedia.org/wiki/Floating_point#Representable_numbers.2C_conversion_and_rounding" target="_top">representable</a> 39 value. (For more about how numbers are represented see <a href="http://en.wikipedia.org/wiki/Floating_point" target="_top">Floating 40 point</a>). 41 </p> 42<p> 43 Of course, we might start with finding an external reference source like 44 <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, as above, 45 but this is not always possible. 46 </p> 47<p> 48 Another way to reassure is to compute 'reference' values at higher precision 49 with which to compare the results of our iterative computations using built-in 50 like <code class="computeroutput"><span class="keyword">double</span></code>. They should agree 51 within the tolerance that was set. 52 </p> 53<p> 54 The result of <code class="computeroutput"><span class="keyword">static_cast</span></code>ing 55 to <code class="computeroutput"><span class="keyword">double</span></code> from a higher-precision 56 type like <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code> 57 is guaranteed to be the <span class="bold"><strong>nearest representable</strong></span> 58 <code class="computeroutput"><span class="keyword">double</span></code> value. 59 </p> 60<p> 61 For example, the cube root functions in our example for <code class="computeroutput"><span class="identifier">cbrt</span><span class="special">(</span><span class="number">28.</span><span class="special">)</span></code> 62 compute 63 </p> 64<p> 65 <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cbrt</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(</span><span class="number">28.</span><span class="special">)</span> <span class="special">=</span> 66 <span class="number">3.0365889718756627</span></code> 67 </p> 68<p> 69 WolframAlpha says <code class="computeroutput"><span class="number">3.036588971875662519420809578505669635581453977248111123242141</span><span class="special">...</span></code> 70 </p> 71<p> 72 <code class="computeroutput"><span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(</span><span class="number">3.03658897187566251942080957850</span><span class="special">)</span> 73 <span class="special">=</span> <span class="number">3.0365889718756627</span></code> 74 </p> 75<p> 76 This example <code class="computeroutput"><span class="identifier">cbrt</span><span class="special">(</span><span class="number">28.</span><span class="special">)</span> <span class="special">=</span> 77 <span class="number">3.0365889718756627</span></code> 78 </p> 79<div class="tip"><table border="0" summary="Tip"> 80<tr> 81<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td> 82<th align="left">Tip</th> 83</tr> 84<tr><td align="left" valign="top"> 85<p> 86 To ensure that all potentially significant decimal digits are displayed 87 use <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">max_digits10</span></code> (or if not available on 88 older platforms or compilers use <code class="computeroutput"><span class="number">2</span><span class="special">+</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">*</span><span class="number">3010</span><span class="special">/</span><span class="number">10000</span></code>).<br> 89 </p> 90<p> 91 Ideally, values should agree to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">-</span><span class="identifier">limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits10</span></code> 92 decimal digits. 93 </p> 94<p> 95 This also means that a 'reference' value to be <span class="bold"><strong>input</strong></span> 96 or <code class="computeroutput"><span class="keyword">static_cast</span></code> should have 97 at least <code class="computeroutput"><span class="identifier">max_digits10</span></code> decimal 98 digits (17 for 64-bit <code class="computeroutput"><span class="keyword">double</span></code>). 99 </p> 100</td></tr> 101</table></div> 102<p> 103 If we wish to compute <span class="bold"><strong>higher-precision values</strong></span> 104 then, on some platforms, we may be able to use <code class="computeroutput"><span class="keyword">long</span> 105 <span class="keyword">double</span></code> with a higher precision than 106 <code class="computeroutput"><span class="keyword">double</span></code> to compare with the very 107 common <code class="computeroutput"><span class="keyword">double</span></code> and/or a more 108 efficient built-in quad floating-point type like <code class="computeroutput"><span class="identifier">__float128</span></code>. 109 </p> 110<p> 111 Almost all platforms can easily use <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>, 112 for example, <a href="../../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html" target="_top">cpp_dec_float</a> 113 or a binary type <a href="../../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_bin_float.html" target="_top">cpp_bin_float</a> 114 types, to compute values at very much higher precision. 115 </p> 116<div class="note"><table border="0" summary="Note"> 117<tr> 118<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td> 119<th align="left">Note</th> 120</tr> 121<tr><td align="left" valign="top"><p> 122 With multiprecision types, it is debatable whether to use the type <code class="computeroutput"><span class="identifier">T</span></code> for computing the initial guesses. 123 Type <code class="computeroutput"><span class="keyword">double</span></code> is like to be 124 accurate enough for the method used in these examples. This would limit 125 the exponent range of possible values to that of <code class="computeroutput"><span class="keyword">double</span></code>. 126 There is also the cost of conversion to and from type <code class="computeroutput"><span class="identifier">T</span></code> 127 to consider. In these examples, <code class="computeroutput"><span class="keyword">double</span></code> 128 is used via <code class="computeroutput"><span class="keyword">typedef</span> <span class="keyword">double</span> 129 <span class="identifier">guess_type</span></code>. 130 </p></td></tr> 131</table></div> 132<p> 133 Since the functors and functions used above are templated on the value type, 134 we can very simply use them with any of the <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a> 135 types. As a reminder, here's our toy cube root function using 2 derivatives 136 and C++11 lambda functions to find the root: 137 </p> 138<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> 139<span class="identifier">T</span> <span class="identifier">cbrt_2deriv_lambda</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span> 140<span class="special">{</span> 141 <span class="comment">// return cube root of x using 1st and 2nd derivatives and Halley.</span> 142 <span class="comment">//using namespace std; // Help ADL of std functions.</span> 143 <span class="keyword">using</span> <span class="keyword">namespace</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> 144 <span class="keyword">int</span> <span class="identifier">exponent</span><span class="special">;</span> 145 <span class="identifier">frexp</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="special">&</span><span class="identifier">exponent</span><span class="special">);</span> <span class="comment">// Get exponent of z (ignore mantissa).</span> 146 <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.</span><span class="special">,</span> <span class="identifier">exponent</span> <span class="special">/</span> <span class="number">3</span><span class="special">);</span> <span class="comment">// Rough guess is to divide the exponent by three.</span> 147 <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">exponent</span> <span class="special">/</span> <span class="number">3</span><span class="special">);</span> <span class="comment">// Minimum possible value is half our guess.</span> 148 <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.</span><span class="special">,</span> <span class="identifier">exponent</span> <span class="special">/</span> <span class="number">3</span><span class="special">);</span> <span class="comment">// Maximum possible value is twice our guess.</span> 149 <span class="keyword">const</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="comment">// Maximum possible binary digits accuracy for type T.</span> 150 <span class="comment">// digits used to control how accurate to try to make the result.</span> 151 <span class="keyword">int</span> <span class="identifier">get_digits</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">int</span><span class="special">>(</span><span class="identifier">digits</span> <span class="special">*</span> <span class="number">0.4</span><span class="special">);</span> <span class="comment">// Accuracy triples with each step, so stop when just</span> 152 <span class="comment">// over one third of the digits are correct.</span> 153 <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">maxit</span> <span class="special">=</span> <span class="number">20</span><span class="special">;</span> 154 <span class="identifier">T</span> <span class="identifier">result</span> <span class="special">=</span> <span class="identifier">halley_iterate</span><span class="special">(</span> 155 <span class="comment">// lambda function:</span> 156 <span class="special">[</span><span class="identifier">x</span><span class="special">](</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">g</span><span class="special">){</span> <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span><span class="identifier">g</span> <span class="special">*</span> <span class="identifier">g</span> <span class="special">*</span> <span class="identifier">g</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">,</span> <span class="number">3</span> <span class="special">*</span> <span class="identifier">g</span> <span class="special">*</span> <span class="identifier">g</span><span class="special">,</span> <span class="number">6</span> <span class="special">*</span> <span class="identifier">g</span><span class="special">);</span> <span class="special">},</span> 157 <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">get_digits</span><span class="special">,</span> <span class="identifier">maxit</span><span class="special">);</span> 158 <span class="keyword">return</span> <span class="identifier">result</span><span class="special">;</span> 159<span class="special">}</span> 160</pre> 161<p> 162 Some examples below are 50 decimal digit decimal and binary types (and on 163 some platforms a much faster <code class="computeroutput"><span class="identifier">float128</span></code> 164 or <code class="computeroutput"><span class="identifier">quad_float</span></code> type ) that 165 we can use with these includes: 166 </p> 167<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_bin_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> <span class="comment">// For cpp_bin_float_50.</span> 168<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_dec_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> <span class="comment">// For cpp_dec_float_50.</span> 169<span class="preprocessor">#ifndef</span> <span class="identifier">_MSC_VER</span> <span class="comment">// float128 is not yet supported by Microsoft compiler at 2013.</span> 170<span class="preprocessor"># include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">float128</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> <span class="comment">// Requires libquadmath.</span> 171<span class="preprocessor">#endif</span> 172</pre> 173<p> 174 Some using statements simplify their use: 175 </p> 176<pre class="programlisting"> <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span><span class="special">;</span> <span class="comment">// decimal.</span> 177 <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_bin_float_50</span><span class="special">;</span> <span class="comment">// binary.</span> 178<span class="preprocessor">#ifndef</span> <span class="identifier">_MSC_VER</span> <span class="comment">// Not supported by Microsoft compiler.</span> 179 <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">float128</span><span class="special">;</span> 180<span class="preprocessor">#endif</span> 181</pre> 182<p> 183 They can be used thus: 184 </p> 185<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">);</span> 186 187<span class="identifier">cpp_dec_float_50</span> <span class="identifier">two</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> <span class="comment">// </span> 188<span class="identifier">cpp_dec_float_50</span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">cbrt_2deriv</span><span class="special">(</span><span class="identifier">two</span><span class="special">);</span> 189<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cbrt("</span> <span class="special"><<</span> <span class="identifier">two</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 190 191<span class="identifier">r</span> <span class="special">=</span> <span class="identifier">cbrt_2deriv</span><span class="special">(</span><span class="number">2.</span><span class="special">);</span> <span class="comment">// Passing a double, so ADL will compute a double precision result.</span> 192<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cbrt("</span> <span class="special"><<</span> <span class="identifier">two</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 193<span class="comment">// cbrt(2) = 1.2599210498948731906665443602832965552806854248047 'wrong' from digits 17 onwards!</span> 194<span class="identifier">r</span> <span class="special">=</span> <span class="identifier">cbrt_2deriv</span><span class="special">(</span><span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>(</span><span class="number">2.</span><span class="special">));</span> <span class="comment">// Passing a cpp_dec_float_50, </span> 195<span class="comment">// so will compute a cpp_dec_float_50 precision result.</span> 196<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cbrt("</span> <span class="special"><<</span> <span class="identifier">two</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 197<span class="identifier">r</span> <span class="special">=</span> <span class="identifier">cbrt_2deriv</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>(</span><span class="number">2.</span><span class="special">);</span> <span class="comment">// Explicitly a cpp_dec_float_50, so will compute a cpp_dec_float_50 precision result.</span> 198<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cbrt("</span> <span class="special"><<</span> <span class="identifier">two</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 199<span class="comment">// cpp_dec_float_50 1.2599210498948731647672106072782283505702514647015</span> 200</pre> 201<p> 202 A reference value computed by <a href="http://www.wolframalpha.com/" target="_top">Wolfram 203 Alpha</a> is 204 </p> 205<pre class="programlisting"><span class="identifier">N</span><span class="special">[</span><span class="number">2</span><span class="special">^(</span><span class="number">1</span><span class="special">/</span><span class="number">3</span><span class="special">),</span> <span class="number">50</span><span class="special">]</span> <span class="number">1.2599210498948731647672106072782283505702514647015</span> 206</pre> 207<p> 208 which agrees exactly. 209 </p> 210<p> 211 To <span class="bold"><strong>show</strong></span> values to their full precision, 212 it is necessary to adjust the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span></code> 213 <code class="computeroutput"><span class="identifier">precision</span></code> to suit the type, 214 for example: 215 </p> 216<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span> 217<span class="identifier">T</span> <span class="identifier">show_cube_root</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">value</span><span class="special">)</span> 218<span class="special">{</span> <span class="comment">// Demonstrate by printing the root using all definitely significant digits.</span> 219 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">);</span> 220 <span class="identifier">T</span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">cbrt_2deriv</span><span class="special">(</span><span class="identifier">value</span><span class="special">);</span> 221 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"value = "</span> <span class="special"><<</span> <span class="identifier">value</span> <span class="special"><<</span> <span class="string">", cube root ="</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 222 <span class="keyword">return</span> <span class="identifier">r</span><span class="special">;</span> 223<span class="special">}</span> 224</pre> 225<pre class="programlisting"><span class="identifier">show_cube_root</span><span class="special">(</span><span class="number">2.</span><span class="special">);</span> 226<span class="identifier">show_cube_root</span><span class="special">(</span><span class="number">2.L</span><span class="special">);</span> 227<span class="identifier">show_cube_root</span><span class="special">(</span><span class="identifier">two</span><span class="special">);</span> 228</pre> 229<p> 230 which outputs: 231 </p> 232<pre class="programlisting">cbrt(2) = 1.2599210498948731647672106072782283505702514647015 233 234value = 2, cube root =1.25992104989487 235value = 2, cube root =1.25992104989487 236value = 2, cube root =1.2599210498948731647672106072782283505702514647015 237</pre> 238<div class="tip"><table border="0" summary="Tip"> 239<tr> 240<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td> 241<th align="left">Tip</th> 242</tr> 243<tr><td align="left" valign="top"> 244<p> 245 Be <span class="bold"><strong>very careful</strong></span> about the floating-point 246 type <code class="computeroutput"><span class="identifier">T</span></code> that is passed to 247 the root-finding function. Carelessly passing a integer by writing <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span> <span class="identifier">r</span> 248 <span class="special">=</span> <span class="identifier">cbrt_2deriv</span><span class="special">(</span><span class="number">2</span><span class="special">);</span></code> 249 or <code class="computeroutput"><span class="identifier">show_cube_root</span><span class="special">(</span><span class="number">2</span><span class="special">);</span></code> will 250 provoke many warnings and compile errors. 251 </p> 252<p> 253 Even <code class="computeroutput"><span class="identifier">show_cube_root</span><span class="special">(</span><span class="number">2.F</span><span class="special">);</span></code> will 254 produce warnings because <code class="computeroutput"><span class="keyword">typedef</span> 255 <span class="keyword">double</span> <span class="identifier">guess_type</span></code> 256 defines the type used to compute the guess and bracket values as <code class="computeroutput"><span class="keyword">double</span></code>. 257 </p> 258<p> 259 Even more treacherous is passing a <code class="computeroutput"><span class="keyword">double</span></code> 260 as in <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">cbrt_2deriv</span><span class="special">(</span><span class="number">2.</span><span class="special">);</span></code> which 261 silently gives the 'wrong' result, computing a <code class="computeroutput"><span class="keyword">double</span></code> 262 result and <span class="bold"><strong>then</strong></span> converting to <code class="computeroutput"><span class="identifier">cpp_dec_float_50</span></code>! All digits beyond 263 <code class="computeroutput"><span class="identifier">max_digits10</span></code> will be incorrect. 264 Making the <code class="computeroutput"><span class="identifier">cbrt</span></code> type explicit 265 with <code class="computeroutput"><span class="identifier">cbrt_2deriv</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>(</span><span class="number">2.</span><span class="special">);</span></code> will 266 give you the desired 50 decimal digit precision result. 267 </p> 268</td></tr> 269</table></div> 270<p> 271 Full code of this example is at <a href="../../../../example/root_finding_multiprecision_example.cpp" target="_top">root_finding_multiprecision_example.cpp</a>. 272 </p> 273</div> 274<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 275<td align="left"></td> 276<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 277 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 278 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 279 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 280 Daryle Walker and Xiaogang Zhang<p> 281 Distributed under the Boost Software License, Version 1.0. (See accompanying 282 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>) 283 </p> 284</div></td> 285</tr></table> 286<hr> 287<div class="spirit-nav"> 288<a accesskey="p" href="5th_root_eg.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../root_finding_examples.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="nth_root.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 289</div> 290</body> 291</html> 292
