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- <div class="titlepage"><div><div><h2 class="title" style="clear: both">
- <a name="math_toolkit.cond"></a><a class="link" href="cond.html" title="Condition Numbers">Condition Numbers</a>
- </h2></div></div></div>
- <h4>
- <a name="math_toolkit.cond.h0"></a>
- <span class="phrase"><a name="math_toolkit.cond.synopsis"></a></span><a class="link" href="cond.html#math_toolkit.cond.synopsis">Synopsis</a>
- </h4>
- <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">condition_numbers</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></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>
- <span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">bool</span> <span class="identifier">kahan</span><span class="special">=</span><span class="keyword">true</span><span class="special">></span>
- <span class="keyword">class</span> <span class="identifier">summation_condition_number</span> <span class="special">{</span>
- <span class="keyword">public</span><span class="special">:</span>
- <span class="identifier">summation_condition_number</span><span class="special">(</span><span class="identifier">Real</span> <span class="keyword">const</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
- <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">+=(</span><span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">x</span><span class="special">);</span>
- <span class="keyword">inline</span> <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">-=(</span><span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">x</span><span class="special">);</span>
- <span class="special">[[</span><span class="identifier">nodiscard</span><span class="special">]]</span> <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()()</span> <span class="keyword">const</span><span class="special">;</span>
- <span class="special">[[</span><span class="identifier">nodiscard</span><span class="special">]]</span> <span class="identifier">Real</span> <span class="identifier">sum</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
- <span class="special">[[</span><span class="identifier">nodiscard</span><span class="special">]]</span> <span class="identifier">Real</span> <span class="identifier">l1_norm</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</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">Real</span><span class="special">></span>
- <span class="keyword">auto</span> <span class="identifier">evaluation_condition_number</span><span class="special">(</span><span class="identifier">F</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">x</span><span class="special">);</span>
- <span class="special">}</span> <span class="comment">// namespaces</span>
- </pre>
- <h4>
- <a name="math_toolkit.cond.h1"></a>
- <span class="phrase"><a name="math_toolkit.cond.summation_condition_number"></a></span><a class="link" href="cond.html#math_toolkit.cond.summation_condition_number">Summation
- Condition Number</a>
- </h4>
- <p>
- Here we compute the condition number of the alternating harmonic sum:
- </p>
- <pre class="programlisting"><span class="keyword">using</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">summation_condition_number</span><span class="special">;</span>
- <span class="keyword">auto</span> <span class="identifier">cond</span> <span class="special">=</span> <span class="identifier">summation_condition_number</span><span class="special"><</span><span class="keyword">float</span><span class="special">,</span> <span class="comment">/* kahan = */</span> <span class="keyword">false</span><span class="special">>();</span>
- <span class="keyword">float</span> <span class="identifier">max_n</span> <span class="special">=</span> <span class="number">10000000</span><span class="special">;</span>
- <span class="keyword">for</span> <span class="special">(</span><span class="keyword">float</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">n</span> <span class="special"><</span> <span class="identifier">max_n</span><span class="special">;</span> <span class="identifier">n</span> <span class="special">+=</span> <span class="number">2</span><span class="special">)</span>
- <span class="special">{</span>
- <span class="identifier">cond</span> <span class="special">+=</span> <span class="number">1</span><span class="special">/</span><span class="identifier">n</span><span class="special">;</span>
- <span class="identifier">cond</span> <span class="special">-=</span> <span class="number">1</span><span class="special">/(</span><span class="identifier">n</span><span class="special">+</span><span class="number">1</span><span class="special">);</span>
- <span class="special">}</span>
- <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</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">float</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">);</span>
- <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"ln(2) = "</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">constants</span><span class="special">::</span><span class="identifier">ln_two</span><span class="special"><</span><span class="keyword">float</span><span class="special">>()</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
- <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Sum = "</span> <span class="special"><<</span> <span class="identifier">cond</span><span class="special">.</span><span class="identifier">sum</span><span class="special">()</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
- <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Condition number = "</span> <span class="special"><<</span> <span class="identifier">cond</span><span class="special">()</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
- </pre>
- <p>
- Output:
- </p>
- <pre class="programlisting"><span class="identifier">ln</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">=</span> <span class="number">0.693147</span>
- <span class="identifier">Sum</span> <span class="special">=</span> <span class="number">0.693137</span>
- <span class="identifier">Condition</span> <span class="identifier">number</span> <span class="special">=</span> <span class="number">22.22282</span>
- </pre>
- <p>
- We see that we have lost roughly two digits of accuracy, consistent with the
- heuristic that if the condition number is 10<sup><span class="emphasis"><em>k</em></span></sup>, then we
- lose <span class="emphasis"><em>k</em></span> significant digits in the sum.
- </p>
- <p>
- Our guess it that if you're worried about whether your sum is ill-conditioned,
- the <span class="emphasis"><em>last</em></span> thing you want is for your condition number estimate
- to be inaccurate as well. Since the condition number estimate relies on computing
- the (perhaps ill-conditioned) sum, we have defaulted the accumulation to use
- Kahan summation:
- </p>
- <pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">cond</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">summation_condition_number</span><span class="special"><</span><span class="keyword">float</span><span class="special">>();</span> <span class="comment">// will use Kahan summation.</span>
- <span class="comment">// ...</span>
- </pre>
- <p>
- Output:
- </p>
- <pre class="programlisting"><span class="identifier">ln</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">=</span> <span class="number">0.693147</span>
- <span class="identifier">Kahan</span> <span class="identifier">sum</span> <span class="special">=</span> <span class="number">0.693147</span>
- <span class="identifier">Condition</span> <span class="identifier">number</span> <span class="special">=</span> <span class="number">22.2228</span>
- </pre>
- <p>
- If you are interested, the L<sub>1</sub> norm is also generated by this computation, so
- you may query it if you like:
- </p>
- <pre class="programlisting"><span class="keyword">float</span> <span class="identifier">l1</span> <span class="special">=</span> <span class="identifier">cond</span><span class="special">.</span><span class="identifier">l1_norm</span><span class="special">();</span>
- <span class="comment">// l1 = 15.4</span>
- </pre>
- <h4>
- <a name="math_toolkit.cond.h2"></a>
- <span class="phrase"><a name="math_toolkit.cond.condition_number_of_function_eva"></a></span><a class="link" href="cond.html#math_toolkit.cond.condition_number_of_function_eva">Condition
- Number of Function Evaluation</a>
- </h4>
- <p>
- The <a href="https://en.wikipedia.org/wiki/Condition_number" target="_top">condition number</a>
- of function evaluation is defined as the absolute value of <span class="emphasis"><em>xf</em></span>'(<span class="emphasis"><em>x</em></span>)<span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>)<sup>-1</sup>.
- It is computed as follows:
- </p>
- <pre class="programlisting"><span class="keyword">using</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">evaluation_condition_number</span><span class="special">;</span>
- <span class="keyword">auto</span> <span class="identifier">f</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">)-></span><span class="keyword">double</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">log</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span> <span class="special">};</span>
- <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">1.125</span><span class="special">;</span>
- <span class="keyword">double</span> <span class="identifier">cond</span> <span class="special">=</span> <span class="identifier">evaluation_condition_number</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="number">1.125</span><span class="special">);</span>
- <span class="comment">// cond = 1/log(x)</span>
- </pre>
- <h4>
- <a name="math_toolkit.cond.h3"></a>
- <span class="phrase"><a name="math_toolkit.cond.caveats"></a></span><a class="link" href="cond.html#math_toolkit.cond.caveats">Caveats</a>
- </h4>
- <p>
- The condition number of function evaluation gives us a measure of how sensitive
- our function is to roundoff error. Unfortunately, evaluating the condition
- number requires evaluating the function and its derivative, and this calculation
- is itself inaccurate whenever the condition number of function evaluation is
- large. Sadly, this is also the regime when you are most interested in evaluating
- a condition number!
- </p>
- <p>
- This seems to be a fundamental problem. However, it should not be necessary
- to evaluate the condition number to high accuracy, valuable insights can be
- obtained simply by looking at the change in condition number as the function
- evolves over its domain.
- </p>
- <h4>
- <a name="math_toolkit.cond.h4"></a>
- <span class="phrase"><a name="math_toolkit.cond.references"></a></span><a class="link" href="cond.html#math_toolkit.cond.references">References</a>
- </h4>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
- <li class="listitem">
- Gautschi, Walter. <span class="emphasis"><em>Orthogonal polynomials: computation and approximation</em></span>
- Oxford University Press on Demand, 2004.
- </li>
- <li class="listitem">
- Higham, Nicholas J. <span class="emphasis"><em>The accuracy of floating point summation.</em></span>
- SIAM Journal on Scientific Computing 14.4 (1993): 783-799.
- </li>
- <li class="listitem">
- Higham, Nicholas J. <span class="emphasis"><em>Accuracy and stability of numerical algorithms.</em></span>
- Vol. 80. Siam, 2002.
- </li>
- </ul></div>
- </div>
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- Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
- Daryle Walker and Xiaogang Zhang<p>
- Distributed under the Boost Software License, Version 1.0. (See accompanying
- file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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