<title>Lambert W function</title>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lambert_wm1_prime</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span> <span class="comment">// W-1 branch 1st derivative.</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">class</span> <a class="link" href="../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lambert_w0</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> <span class="comment">// W0 with policy.</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">class</span> <a class="link" href="../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lambert_wm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> <span class="comment">// W-1 with policy.</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">class</span> <a class="link" href="../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lambert_w0_prime</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> <span class="comment">// W0 derivative with policy.</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">class</span> <a class="link" href="../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lambert_wm1_prime</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> <span class="comment">// W-1 derivative with policy.</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">class</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+ <a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lambert_w0</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> <span class="comment">// W0 with policy.</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">class</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+ <a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lambert_wm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> <span class="comment">// W-1 with policy.</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">class</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+ <a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lambert_w0_prime</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> <span class="comment">// W0 derivative with policy.</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">class</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+ <a class="link" href="result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lambert_wm1_prime</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> <span class="comment">// W-1 derivative with policy.</span>
<span class="special">}</span> <span class="comment">// namespace boost</span>
<span class="special">}</span> <span class="comment">// namespace math</span>
are labelled <code class="computeroutput"><span class="identifier">lambert_w0_prime</span></code>
and <code class="computeroutput"><span class="identifier">lambert_wm1_prime</span></code>.
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../graphs/lambert_w_graph.svg" align="middle"></span>
- <span class="inlinemediaobject"><img src="../../graphs/lambert_w_graph_big_w.svg" align="middle"></span>
- <span class="inlinemediaobject"><img src="../../graphs/lambert_w0_prime_graph.svg" align="middle"></span>
- <span class="inlinemediaobject"><img src="../../graphs/lambert_wm1_prime_graph.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../graphs/lambert_w_graph.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../graphs/lambert_w_graph_big_w.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../graphs/lambert_w0_prime_graph.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../graphs/lambert_wm1_prime_graph.svg" align="middle"></span>
+
+ </p></blockquote></div>
<p>
There is a singularity where the branches meet at <span class="emphasis"><em>e</em></span><sup>-1</sup> ≅ <code class="literal">-0.367879</code>.
Approaching this point, the condition number of function evaluation tends to
arguments are not supported.
</p>
<p>
- The final <a class="link" href="../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
- be used to control how the function deals with errors. Refer to <a class="link" href="../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policies</a>
+ The final <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+ be used to control how the function deals with errors. Refer to <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policies</a>
for more details and see examples below.
</p>
<h6>
<p>
Banwell and Jakaumar equation 5
</p>
-<p>
-     I(V) = μ V<sub>T</sub>/ R <sub>S</sub> ․ W<sub>0</sub>(I<sub>0</sub> R<sub>S</sub> / (μ V<sub>T</sub>))
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">I(V) = μ V<sub>T</sub>/ R <sub>S</sub> ․ W<sub>0</sub>(I<sub>0</sub> R<sub>S</sub> / (μ V<sub>T</sub>))</span>
+ </p></blockquote></div>
<p>
Using these variables
</p>
<p>
to reproduce their Fig 2:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../graphs/diode_iv_plot.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../graphs/diode_iv_plot.svg" align="middle"></span>
+
+ </p></blockquote></div>
<p>
The plotted points for no external series resistance (derived from their published
plot as the raw data are not publicly available) are used to extrapolate back
reference 'best' z50 evaluations using <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>
type reduced to <code class="computeroutput"><span class="keyword">double</span></code> with <code class="computeroutput"><span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">double</span><span class="special">(</span><span class="identifier">z50</span><span class="special">)</span></code> :
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../graphs/lambert_w0_errors_graph.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../graphs/lambert_wm1_errors_graph.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../graphs/lambert_w0_errors_graph.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../graphs/lambert_wm1_errors_graph.svg" align="middle"></span>
+
+ </p></blockquote></div>
<p>
As noted in the implementation section, the distribution of these differences
is somewhat biased for Lambert <span class="emphasis"><em>W</em></span><sub>-1</sub> and this might be reduced
z = 0.0501 to 703, in 0.001 step factor 1.05 when module 7 == 0
</p>
<div class="table">
-<a name="math_toolkit.lambert_w.lambert_w0_Fukushima"></a><p class="title"><b>Table 7.73. Fukushima Lambert <span class="emphasis"><em>W</em></span><sub>0</sub> and typical improvement from
+<a name="math_toolkit.lambert_w.lambert_w0_Fukushima"></a><p class="title"><b>Table 8.73. Fukushima Lambert <span class="emphasis"><em>W</em></span><sub>0</sub> and typical improvement from
a single Halley step.</b></p>
<div class="table-contents"><table class="table" summary="Fukushima Lambert W0 and typical improvement from
a single Halley step.">
bit different, and only 2 two-bits 'wrong'.
</p>
<div class="table">
-<a name="math_toolkit.lambert_w.lambert_w0_plus_halley"></a><p class="title"><b>Table 7.74. Rational polynomial Lambert <span class="emphasis"><em>W</em></span><sub>0</sub> and typical improvement
+<a name="math_toolkit.lambert_w.lambert_w0_plus_halley"></a><p class="title"><b>Table 8.74. Rational polynomial Lambert <span class="emphasis"><em>W</em></span><sub>0</sub> and typical improvement
from a single Halley step.</b></p>
<div class="table-contents"><table class="table" summary="Rational polynomial Lambert W0 and typical improvement
from a single Halley step.">
used.
</p>
<div class="table">
-<a name="math_toolkit.lambert_w.lambert_wm1_fukushima"></a><p class="title"><b>Table 7.75. Lambert <span class="emphasis"><em>W</em></span><sub>-1</sub> using Fukushima algorithm.</b></p>
+<a name="math_toolkit.lambert_w.lambert_wm1_fukushima"></a><p class="title"><b>Table 8.75. Lambert <span class="emphasis"><em>W</em></span><sub>-1</sub> using Fukushima algorithm.</b></p>
<div class="table-contents"><table class="table" summary="Lambert W-1 using Fukushima algorithm.">
<colgroup>
<col>
value, a minority are within 1 <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">Unit
in the last place (ULP)</a> and only a very few 2 ULP.
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../graphs/lambert_w0_errors_graph.svg" align="middle"></span>
- <span class="inlinemediaobject"><img src="../../graphs/lambert_wm1_errors_graph.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../graphs/lambert_w0_errors_graph.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../graphs/lambert_wm1_errors_graph.svg" align="middle"></span>
+
+ </p></blockquote></div>
<p>
For the range of z arguments over the range -0.35 to 0.5, a different algorithm
is used, but the same technique of evaluating reference values using a <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
projects / platform / upstream / boost.git / blobdiff