Bounds on remainder term of power series of elementary functions - MathOverflow

Bounds on remainder term of power series of elementary functions

2010-08-17T00:00:54Z

This is mainly a question about the remainder term of power series for elementary functions.

I'm very interested in aspects of calculating or computing elementary operations and functions, by which I mean:

trigonometric: sin, cos, tan
inverse trig.: sin⁻¹, cos⁻¹, tan⁻¹
log and exponential: ln, exp
hyperbolic: sinh, cosh, tanh
inverse hyp.: sinh⁻¹, cosh⁻¹, tanh⁻¹
powers, reciprocation, $\sqrt{\ \ \ }$

perhaps also:

gamma function: Γ
and a few other important functions

There are many contexts (of calculation). For example:

real versus complex arguments
known, fixed precision versus variable precision
numerical versus symbolic

There are many approaches and techniques available too. For example:

power series expansions and polynomial approximations
use of relationships between the functions
use of periodic or similar properties to shrink the domain
lookup tables and interpolation
CORDIC (used within some hand calculators I believe)
exact methods
interval or other error-tracking methods

Some good references to certain aspects include:

Digital Library of Mathematical Functions: Elementary Functions
Chee-Keng Yap, Fundamental problems of algorithmic algebra
Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Designs

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.

Any pointers on this matter would be much appreciated.

Likewise for any further references on any other aspects of or techniques for calculating elementary functions.

Answer by Jacques Carette for Bounds on remainder term of power series of elementary functions

2010-08-17T01:05:28Z

These bounds you are looking for can be obtained from majorant series. What you seek is implemented in the Dynamic Dictionary of Mathematical Functions; Bruno Salvy gave a very nice talk on this topic at CICM 2010 in early July.

For the guaranteed numerics aspect, the expert is Marc Mezzarobba, a PhD student of Bruno's. On that page, see the links to NumGfun (software, presentation and paper) for all the details you would ever want on the topic.

Answer by SandeepJ for Bounds on remainder term of power series of elementary functions

2010-08-17T01:39:46Z

finding bounds for the error or remainder term in partial power series expansions

I think you want the Euler-Maclaurin Summation formula. That bounds the remainder terms, although it would require knowing the closed form of the integral representation of the function you are calculating.

$ \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) $

The paper by Apostol "Elementary view of Euler-Maclaurin" AMM vol 106 (1999) pp. 409-418 is very accessible.

The following papers/books may also be helpful

R.P. Boas "Estimating Remainders." Math. Mag. 51, pp 83-89, (1978)
http://www.tricki.org/article/Estimating_sums
Bridger and Frampton Bounding Power Series Remainders Math. Mag. 71 (1998), pp. 204-207
Sofo. Computational Techniques for the Summation of Series
Ross. Methods of Summation
Davis. Summation of Series.