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<title>Minimax Approximations and the Remez Algorithm</title>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.internals.minimax"></a><a class="link" href="minimax.html" title="Minimax Approximations and the Remez Algorithm">Minimax Approximations
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and the Remez Algorithm</a>
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</h3></div></div></div>
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<p>
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The directory libs/math/minimax contains a command line driven program for
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the generation of minimax approximations using the Remez algorithm. Both
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polynomial and rational approximations are supported, although the latter
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are tricky to converge: it is not uncommon for convergence of rational forms
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to fail. No such limitations are present for polynomial approximations which
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should always converge smoothly.
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</p>
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<p>
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It's worth stressing that developing rational approximations to functions
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is often not an easy task, and one to which many books have been devoted.
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To use this tool, you will need to have a reasonable grasp of what the Remez
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algorithm is, and the general form of the approximation you want to achieve.
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</p>
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<p>
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Unless you already familar with the Remez method, you should first read the
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<a class="link" href="../remez.html" title="The Remez Method">brief background article explaining the
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principles behind the Remez algorithm</a>.
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</p>
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<p>
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The program consists of two parts:
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</p>
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<div class="variablelist">
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<p class="title"><b></b></p>
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<dl class="variablelist">
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<dt><span class="term">main.cpp</span></dt>
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<dd><p>
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Contains the command line parser, and all the calls to the Remez code.
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</p></dd>
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<dt><span class="term">f.cpp</span></dt>
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<dd><p>
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Contains the function to approximate.
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</p></dd>
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</dl>
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</div>
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<p>
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Therefore to use this tool, you must modify f.cpp to return the function
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to approximate. The tools supports multiple function approximations within
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the same compiled program: each as a separate variant:
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</p>
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<pre class="programlisting"><span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span> <span class="identifier">f</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span><span class="special">&</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">variant</span><span class="special">);</span>
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</pre>
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<p>
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Returns the value of the function <span class="emphasis"><em>variant</em></span> at point
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<span class="emphasis"><em>x</em></span>. So if you wish you can just add the function to approximate
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as a new variant after the existing examples.
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</p>
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<p>
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In addition to those two files, the program needs to be linked to a <a class="link" href="../high_precision/use_ntl.html" title="Using NTL Library">patched NTL library to compile</a>.
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</p>
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<p>
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Note that the function <span class="emphasis"><em>f</em></span> must return the rational part
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of the approximation: for example if you are approximating a function <span class="emphasis"><em>f(x)</em></span>
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then it is quite common to use:
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</p>
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<pre class="programlisting"><span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</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="identifier">Y</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">x</span><span class="special">))</span>
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</pre>
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<p>
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where <span class="emphasis"><em>g(x)</em></span> is the dominant part of <span class="emphasis"><em>f(x)</em></span>,
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<span class="emphasis"><em>Y</em></span> is some constant, and <span class="emphasis"><em>R(x)</em></span> is
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the rational approximation part, usually optimised for a low absolute error
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compared to |Y|.
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</p>
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<p>
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In this case you would define <span class="emphasis"><em>f</em></span> to return <span class="emphasis"><em>f(x)/g(x)</em></span>
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and then set the y-offset of the approximation to <span class="emphasis"><em>Y</em></span>
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(see command line options below).
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</p>
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<p>
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Many other forms are possible, but in all cases the objective is to split
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<span class="emphasis"><em>f(x)</em></span> into a dominant part that you can evaluate easily
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using standard math functions, and a smooth and slowly changing rational
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approximation part. Refer to your favourite textbook for more examples.
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</p>
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<p>
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Command line options for the program are as follows:
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</p>
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<div class="variablelist">
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<p class="title"><b></b></p>
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<dl class="variablelist">
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<dt><span class="term">variant N</span></dt>
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<dd><p>
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Sets the current function variant to N. This allows multiple functions
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that are to be approximated to be compiled into the same executable.
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Defaults to 0.
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</p></dd>
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<dt><span class="term">range a b</span></dt>
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<dd><p>
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Sets the domain for the approximation to the range [a,b], defaults
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to [0,1].
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</p></dd>
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<dt><span class="term">relative</span></dt>
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<dd><p>
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Sets the Remez code to optimise for relative error. This is the default
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at program startup. Note that relative error can only be used if f(x)
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has no roots over the range being optimised.
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</p></dd>
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<dt><span class="term">absolute</span></dt>
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<dd><p>
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Sets the Remez code to optimise for absolute error.
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</p></dd>
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<dt><span class="term">pin [true|false]</span></dt>
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<dd><p>
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"Pins" the code so that the rational approximation passes
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through the origin. Obviously only set this to <span class="emphasis"><em>true</em></span>
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if R(0) must be zero. This is typically used when trying to preserve
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a root at [0,0] while also optimising for relative error.
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</p></dd>
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<dt><span class="term">order N D</span></dt>
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<dd><p>
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Sets the order of the approximation to <span class="emphasis"><em>N</em></span> in the
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numerator and <span class="emphasis"><em>D</em></span> in the denominator. If <span class="emphasis"><em>D</em></span>
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is zero then the result will be a polynomial approximation. There will
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be N+D+2 coefficients in total, the first coefficient of the numerator
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is zero if <span class="emphasis"><em>pin</em></span> was set to true, and the first
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coefficient of the denominator is always one.
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</p></dd>
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<dt><span class="term">working-precision N</span></dt>
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<dd><p>
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Sets the working precision of NTL::RR to <span class="emphasis"><em>N</em></span> binary
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digits. Defaults to 250.
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</p></dd>
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<dt><span class="term">target-precision N</span></dt>
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<dd><p>
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Sets the precision of printed output to <span class="emphasis"><em>N</em></span> binary
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digits: set to the same number of digits as the type that will be used
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to evaluate the approximation. Defaults to 53 (for double precision).
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</p></dd>
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<dt><span class="term">skew val</span></dt>
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<dd><p>
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"Skews" the initial interpolated control points towards one
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end or the other of the range. Positive values skew the initial control
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points towards the left hand side of the range, and negative values
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towards the right hand side. If an approximation won't converge (a
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common situation) try adjusting the skew parameter until the first
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step yields the smallest possible error. <span class="emphasis"><em>val</em></span> should
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be in the range [-100,+100], the default is zero.
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</p></dd>
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<dt><span class="term">brake val</span></dt>
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<dd><p>
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Sets a brake on each step so that the change in the control points
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is braked by <span class="emphasis"><em>val%</em></span>. Defaults to 50, try a higher
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value if an approximation won't converge, or a lower value to get speedier
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convergence.
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</p></dd>
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<dt><span class="term">x-offset val</span></dt>
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<dd><p>
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Sets the x-offset to <span class="emphasis"><em>val</em></span>: the approximation will
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be generated for <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">S</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">X</span><span class="special">))</span> <span class="special">+</span> <span class="identifier">Y</span></code>
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where <span class="emphasis"><em>X</em></span> is the x-offset, <span class="emphasis"><em>S</em></span>
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is the x-scale and <span class="emphasis"><em>Y</em></span> is the y-offset. Defaults
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to zero. To avoid rounding errors, take care to specify a value that
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can be exactly represented as a floating point number.
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</p></dd>
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<dt><span class="term">x-scale val</span></dt>
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<dd><p>
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Sets the x-scale to <span class="emphasis"><em>val</em></span>: the approximation will
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be generated for <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">S</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">X</span><span class="special">))</span> <span class="special">+</span> <span class="identifier">Y</span></code>
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where <span class="emphasis"><em>S</em></span> is the x-scale, <span class="emphasis"><em>X</em></span>
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is the x-offset and <span class="emphasis"><em>Y</em></span> is the y-offset. Defaults
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to one. To avoid rounding errors, take care to specify a value that
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can be exactly represented as a floating point number.
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</p></dd>
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<dt><span class="term">y-offset val</span></dt>
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<dd><p>
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Sets the y-offset to <span class="emphasis"><em>val</em></span>: the approximation will
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be generated for <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">S</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">X</span><span class="special">))</span> <span class="special">+</span> <span class="identifier">Y</span></code>
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where <span class="emphasis"><em>X</em></span> is the x-offset, <span class="emphasis"><em>S</em></span>
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is the x-scale and <span class="emphasis"><em>Y</em></span> is the y-offset. Defaults
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to zero. To avoid rounding errors, take care to specify a value that
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can be exactly represented as a floating point number.
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</p></dd>
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<dt><span class="term">y-offset auto</span></dt>
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<dd><p>
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Sets the y-offset to the average value of f(x) evaluated at the two
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endpoints of the range plus the midpoint of the range. The calculated
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value is deliberately truncated to <span class="emphasis"><em>float</em></span> precision
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(and should be stored as a <span class="emphasis"><em>float</em></span> in your code).
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The approximation will be generated for <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">X</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">Y</span></code> where <span class="emphasis"><em>X</em></span> is
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the x-offset and <span class="emphasis"><em>Y</em></span> is the y-offset. Defaults to
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zero.
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</p></dd>
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<dt><span class="term">graph N</span></dt>
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<dd><p>
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Prints N evaluations of f(x) at evenly spaced points over the range
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being optimised. If unspecified then <span class="emphasis"><em>N</em></span> defaults
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to 3. Use to check that f(x) is indeed smooth over the range of interest.
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</p></dd>
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<dt><span class="term">step N</span></dt>
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<dd><p>
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Performs <span class="emphasis"><em>N</em></span> steps, or one step if <span class="emphasis"><em>N</em></span>
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is unspecified. After each step prints: the peek error at the extrema
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of the error function of the approximation, the theoretical error term
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solved for on the last step, and the maximum relative change in the
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location of the Chebyshev control points. The approximation is converged
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on the minimax solution when the two error terms are (approximately)
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equal, and the change in the control points has decreased to a suitably
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small value.
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</p></dd>
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<dt><span class="term">test [float|double|long]</span></dt>
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<dd><p>
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Tests the current approximation at float, double, or long double precision.
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Useful to check for rounding errors in evaluating the approximation
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at fixed precision. Tests are conducted at the extrema of the error
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function of the approximation, and at the zeros of the error function.
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</p></dd>
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<dt><span class="term">test [float|double|long] N</span></dt>
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<dd><p>
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Tests the current approximation at float, double, or long double precision.
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Useful to check for rounding errors in evaluating the approximation
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at fixed precision. Tests are conducted at N evenly spaced points over
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the range of the approximation. If none of [float|double|long] are
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specified then tests using NTL::RR, this can be used to obtain the
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error function of the approximation.
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</p></dd>
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<dt><span class="term">rescale a b</span></dt>
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<dd><p>
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Takes the current Chebeshev control points, and rescales them over
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a new interval [a,b]. Sometimes this can be used to obtain starting
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control points for an approximation that can not otherwise be converged.
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</p></dd>
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<dt><span class="term">rotate</span></dt>
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<dd><p>
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Moves one term from the numerator to the denominator, but keeps the
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Chebyshev control points the same. Sometimes this can be used to obtain
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starting control points for an approximation that can not otherwise
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be converged.
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</p></dd>
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<dt><span class="term">info</span></dt>
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<dd><p>
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Prints out the current approximation: the location of the zeros of
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the error function, the location of the Chebyshev control points, the
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x and y offsets, and of course the coefficients of the polynomials.
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</p></dd>
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</dl>
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</div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
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Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
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Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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</p>
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</div></td>
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</tr></table>
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