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Keywords = PSLQ algorithm

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Article
High-Precision Arithmetic in Mathematical Physics
by David H. Bailey and Jonathan M. Borwein
Mathematics 2015, 3(2), 337-367; https://doi.org/10.3390/math3020337 - 12 May 2015
Cited by 46 | Viewed by 9495
Abstract
For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This [...] Read more.
For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This article discusses the challenge of high-precision computation, in the context of mathematical physics, and highlights what facilities are required to support future computation, in light of emerging developments in computer architecture. Full article
(This article belongs to the Special Issue Mathematical physics)
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<p>Graph of <math display="inline"> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mstyle displaystyle="true"> <msub> <mo>∑</mo> <mrow> <mn>0</mn> <mo>⩽</mo> <mi>k</mi> <mo>⩽</mo> <mn>10</mn></mrow></msub> <mrow> <msup> <mn>2</mn> <mrow> <mo>−</mo> <mi>k</mi></mrow></msup> <mi>sin</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mi>k</mi></msup> <mi>x</mi> <mo stretchy="false">)</mo></mrow></mstyle></mrow></math>, over (0, <span class="html-italic">π</span>).</p>
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<p>Left: Error vs. CPU time for the numerical solution of the unstable periodic orbit LR for the Lorenz model (in double-double arithmetic) using a Runge-Kutta code (dop853), an extrapolation code (odex) [<a href="#b50-mathematics-03-00337" class="html-bibr">50</a>,<a href="#b51-mathematics-03-00337" class="html-bibr">51</a>] and a Taylor series method (TIDES). Right: Error vs. CPU time for the numerical solution of an unstable periodic orbit for the Lorenz model (in 500-digit arithmetic) using the TIDES code. Taken from [<a href="#b52-mathematics-03-00337" class="html-bibr">52</a>].</p>
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<p>Numerical solution of the L<sup>25</sup>R<sup>25</sup> unstable periodic orbit of the Lorenz model for 16 time periods using the TIDES code with 300 digits, compared with just one time period using 64-bit IEEE arithmetic.</p>
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<p>Fractal property of the Lorenz attractor. The first plot shows a rectangle in the plane. All later plots zoom in on a tiny region (too small to be seen by the unaided eye) at the center of the red rectangle of the preceding plot to show that what appears to be a line is in fact not a line. (Reproduced with permission from [<a href="#b58-mathematics-03-00337" class="html-bibr">58</a>]).</p>
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<p>Plot of log<sub>10</sub>(min |<span class="html-italic">y<sub>i</sub></span>|) versus iteration number in a typical multipair PSLQ run. Note the sudden drop, by nearly 200 orders of magnitude, at iteration 199.</p>
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