Bounds on remainder term of power series of elementary functions

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: $\Gamma$
• 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:

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.

• Two things: 1. you might be interested in this book: personales.unican.es/segurajj/book ; and 2. don't just confine yourself to implementing things with only power series, they are almost always only effective within a restricted range of arguments (unless your functions of interest satisfy useful reduction formulae). As might be already obvious to you, the easiest bound for the truncation error of a power series is the first neglected term. – J. M. is not a mathematician Aug 17 '10 at 1:27
• You've listed a huge load of references, but not said in detail exactly why these are not enough to solve your problems. I think a detailed question would be helpful; can you give a particular power series for which you cannot determine a good estimate for the remainder, by any of the standard methods you've listed? – Zen Harper Aug 17 '10 at 1:32
• First of all, thanks to all for the comments and answers; all good and all appreciated. – Rhubbarb Aug 17 '10 at 23:13
• @J. Mangaldan (2) are there any particular methods beyond power series and the few others I mentioned above that you would particularly recommend I consider? Do you have a reference for the use of the first neglected term as a bound on the remainder? What are the conditions for that to hold? Thanks again. – Rhubbarb Aug 17 '10 at 23:18
• @Zen Harper: My question was vague partly intentionally (I am very pleased with the responses), and partly as I'm only an amateur mathematician. As a particular case: the terms of the expansion of sin are simple because the sequence of derivatives remains simple. On the other hand, the terms of the expansion of tan seem to become very unwieldy; the closed form is superficially simple, but involves the Bernoulli numbers, which are somewhat erratic. I expect I'm missing something quite simple. In saying that, I wasn't fishing for a specific solution so much as some leads. Thanks again. – Rhubbarb Aug 17 '10 at 23:50

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.

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

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