# Taylor or Fourier applications in computer science

I'm an undergrad computer science student. As part of my calculus course I need to write a 'paper' (of course not as serious as a PhD paper) telling about a cool (for me) application of taylor/fourier series in software development or general computer science.

The only rule is 'not PI approximation'.

Any good ideas, I've made a research in EBSCO, ACM library and even IEEE but I can't find something useful to me.

Best regards!

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One of the main original applications of Fourier series were to find solutions of differential equations. Maybe you can write a program that uses truncated Fourier series to approximate the solution to an appropriately interesting differential equation? – treble May 8 '12 at 22:29
There's lot of Fourier stuff going on in signal and image processing. For example (randomly chosen among myriad applications) you could look into pitch-correcting software. – Henning Makholm May 8 '12 at 22:32
Just to stave off any probable future comments: Taylor series are not that commonly used for the numerical evaluation of transcendental functions; since Taylor expansions are only really useful for evaluation near the expansion point, additional machinery is often needed to make them practical for a wider argument range, and even then, there are often better methods for the numerics. Nevertheless, they are a useful starting point. – J. M. May 8 '12 at 22:33

Here are some ideas:

• The Schönhage-Strassen fast multiplication algorithm is based on FFT.
• Media compression methods are based on variants of FFT. For example, JPEG is based on a two-dimensional version of a variant of FFT known as DCT.
• In general, FFT is a basic tool in digital signal processing. It's use for example in speech recognition to extract the formants.
• Algorithms for computing transcendental functions are sometimes based on Padé approximants, which are a generalization of Taylor series.
• Algorithms used in CAS to calculate the asymptotics of general expressions use Taylor series.
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DCT is not intrinsically two-dimensional; rather it is a simplification of the usual discrete Fourier transform that exploits inherent symmetry of the data. See this for instance. – J. M. May 8 '12 at 22:36
@J.M. not exactly a simplification, but a variant, I'd say. – leonbloy May 8 '12 at 23:23
That's correct, reworded. – Yuval Filmus May 8 '12 at 23:30
@leon: I don't know about you, but anything that cuts effort in half is a simplification for me... – J. M. May 9 '12 at 1:39
I dont see how DCT cuts effort in half. Instead of computing a N complex DFT (N complex=2N scalars, half of them redundant because signal is real), DCT computes the 2N DFT of an mirrored (even) signal (which results in a 2N scalars -DFT is real- half of which are again redundant). There is no simplification, the cosine transform is just another unitary transform, they're basically equal in performance, and just equal in storage and precision (both have an exact inverse except for rounding noise), it's just that the DCT performs better than Fourier for compacting energy for typical images. – leonbloy May 9 '12 at 2:51