Bounds on remainder term of power series of elementary functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T08:05:54Z http://mathoverflow.net/feeds/question/35812 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35812/bounds-on-remainder-term-of-power-series-of-elementary-functions Bounds on remainder term of power series of elementary functions Rhubbarb 2010-08-17T00:00:54Z 2010-08-17T23:11:37Z <p>This is mainly a question about the remainder term of power series for elementary functions.</p> <p>I'm very interested in aspects of calculating or computing elementary operations and functions, by which I mean:</p> <ul> <li>trigonometric: sin, cos, tan</li> <li>inverse trig.: sin<sup>&minus;1</sup>, cos<sup>&minus;1</sup>, tan<sup>&minus;1</sup></li> <li>log and exponential: ln, exp</li> <li>hyperbolic: sinh, cosh, tanh</li> <li>inverse hyp.: sinh<sup>&minus;1</sup>, cosh<sup>&minus;1</sup>, tanh<sup>&minus;1</sup></li> <li>powers, reciprocation, $\sqrt{\ \ \ }$</li> </ul> <p>perhaps also:</p> <ul> <li>gamma function: &Gamma;</li> <li>and a few other important functions</li> </ul> <p>There are many contexts (of calculation). For example:</p> <ul> <li>real <em>versus</em> complex arguments</li> <li>known, fixed precision <em>versus</em> variable precision</li> <li>numerical <em>versus</em> symbolic</li> </ul> <p>There are many approaches and techniques available too. For example:</p> <ul> <li>power series expansions and polynomial approximations</li> <li>use of relationships between the functions</li> <li>use of periodic or similar properties to shrink the domain</li> <li>lookup tables and interpolation</li> <li>CORDIC (used within some hand calculators I believe)</li> <li><em>exact</em> methods</li> <li>interval or other error-tracking methods</li> </ul> <p>Some good references to certain aspects include:</p> <ul> <li>Digital Library of Mathematical Functions: <a href="http://dlmf.nist.gov/4" rel="nofollow">Elementary Functions</a></li> <li>Chee-Keng Yap, <a href="http://books.google.co.uk/books?id=YzjvQ-TSQ4gC&amp;dq=polynomials+inauthor%3ayap&amp;hl=en&amp;ei=1cxpTPK3Kti4jAeOyqTUBA&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=10&amp;ved=0CGIQ6AEwCQ" rel="nofollow">Fundamental problems of algorithmic algebra</a></li> <li>Behrooz Parhami, <a href="http://www.amazon.co.uk/Computer-Arithmetic-Algorithms-Hardware-Designs/dp/0195125835/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1282002469&amp;sr=1-2" rel="nofollow">Computer Arithmetic: Algorithms and Hardware Designs</a></li> </ul> <p><strong>The main gap in my knowledge is in finding bounds for the error or remainder term in partial power series expansions of certain of the above functions. Some are fairly simple to determine, whilst others seem to be awkward.</strong></p> <p><strong>Any pointers on this matter would be much appreciated.</strong></p> <p>Likewise for any further references on any other aspects of or techniques for calculating elementary functions.</p> http://mathoverflow.net/questions/35812/bounds-on-remainder-term-of-power-series-of-elementary-functions/35821#35821 Answer by Jacques Carette for Bounds on remainder term of power series of elementary functions Jacques Carette 2010-08-17T01:05:28Z 2010-08-17T01:05:28Z <p>These bounds you are looking for can be obtained from <em>majorant series</em>. What you seek is implemented in the <a href="http://ddmf.msr-inria.inria.fr/ddmf" rel="nofollow">Dynamic Dictionary of Mathematical Functions</a>; <a href="http://algo.inria.fr/salvy/" rel="nofollow">Bruno Salvy</a> gave a very nice <a href="http://algo.inria.fr/salvy/Talks/aisc.pdf" rel="nofollow">talk</a> on this topic at <a href="http://cicm2010.cnam.fr/index.html" rel="nofollow">CICM 2010</a> in early July.</p> <p>For the <em>guaranteed numerics</em> aspect, the expert is <a href="http://www.normalesup.org/~mezzarobba/" rel="nofollow">Marc Mezzarobba</a>, a PhD student of Bruno's. On that page, see the links to <strong>NumGfun</strong> (software, presentation and paper) for all the details you would ever want on the topic.</p> http://mathoverflow.net/questions/35812/bounds-on-remainder-term-of-power-series-of-elementary-functions/35824#35824 Answer by SandeepJ for Bounds on remainder term of power series of elementary functions SandeepJ 2010-08-17T01:39:46Z 2010-08-17T01:39:46Z <blockquote> <p>finding bounds for the error or remainder term in partial power series expansions</p> </blockquote> <p>I think you want the <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula#Asymptotic_expansion_of_sums" rel="nofollow">Euler-Maclaurin Summation formula</a>. That bounds the remainder terms, although it would require knowing the closed form of the integral representation of the function you are calculating. </p> <p>$\sum_{n=a}^b f(n) \sim \int_a^b f(x)\,dx + \frac{f(a)+f(b)}{2} + \sum_{k=1}^\infty \,\frac{B_{2k}}{(2k)!}\left(f^{(2k-1)}(b)-f^{(2k-1)}(a)\right)$</p> <p>The paper by <a href="http://www.jstor.org/stable/2589145" rel="nofollow">Apostol "Elementary view of Euler-Maclaurin" AMM vol 106 (1999) pp. 409-418 </a> is very accessible. </p> <p>The following papers/books may also be helpful </p> <ol> <li>R.P. Boas "<a href="http://www.jstor.org/stable/2690143" rel="nofollow">Estimating Remainders</a>." Math. Mag. 51, pp 83-89, (1978)</li> <li><a href="http://www.tricki.org/article/Estimating_sums" rel="nofollow">http://www.tricki.org/article/Estimating_sums</a></li> <li>Bridger and Frampton <a href="http://www.jstor.org/stable/2691205" rel="nofollow">Bounding Power Series Remainders</a> Math. Mag. 71 (1998), pp. 204-207</li> <li>Sofo. <a href="http://www.springer.com/mathematics/analysis/book/978-0-306-47805-5" rel="nofollow">Computational Techniques for the Summation of Series</a></li> <li>Ross. <a href="http://www.amazon.com/Methods-summation-Bertram-Ross/dp/B0007BHA3C/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1282008695&amp;sr=8-1" rel="nofollow">Methods of Summation</a></li> <li>Davis. <a href="http://www.amazon.com/summation-Harold-Thayer-Davis/dp/B0007FQCIW/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1282008645&amp;sr=1-1" rel="nofollow">Summation of Series.</a></li> </ol>