78 lines
4.9 KiB
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78 lines
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<title>Setting the Termination Condition for Integration</title>
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.double_exponential.de_tol"></a><a class="link" href="de_tol.html" title="Setting the Termination Condition for Integration">Setting the Termination
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Condition for Integration</a>
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</h3></div></div></div>
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<p>
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The integrate method for all three double-exponential quadratures supports
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<span class="emphasis"><em>tolerance</em></span> argument that acts as the termination condition
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for integration.
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<p>
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The tolerance is met when two subsequent estimates of the integral have absolute
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error less than <code class="computeroutput"><span class="identifier">tolerance</span><span class="special">*</span><span class="identifier">L1</span></code>.
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<p>
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It is highly recommended that the tolerance be left at the default value
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of √ε, or something similar. Since double exponential quadrature converges
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exponentially fast for functions in Hardy spaces, then once the routine has
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<span class="bold"><strong>proved</strong></span> that the error is ~√ε, then the error
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should in fact be ~ε.
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</p>
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<p>
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If you request that the error be ~ε, this tolerance might never be achieved
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(as the summation is not stabilized ala Kahan), and the routine will simply
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flounder, dividing the interval in half in order to increase the precision
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of the integrand, only to be thwarted by floating point roundoff.
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</p>
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<p>
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If for some reason, the default value doesn't quite achieve full precision,
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then you could try something a little smaller such as √ε/4 or ε<sup>2/3</sup>. However,
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more likely, you need to check that your function to be integrated is able
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to return accurate values, and that there are no other issues with your integration
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scheme.
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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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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