Andreas Lauschke Consulting

Convergence Acceleration with Canonical Contractions of Continued Fractions

Tangent Function

The following compares two function approximations to the tangent function. One is a continued fraction expansion that was developed as an even contraction of a continued fraction derived from a power series using Euler's method to convert a series into an S-fraction (gray curve). This continued fraction expansion is already the fastest-converging one from among the ones presented in: Cuyt, Petersen, et al.: "Handbook of Continued Fractions for Special Functions" (Springer, 2008), which is the reason I chose it as the starting point for my own contraction method. The second (blue-dashed curve) is a canonical even contraction I have developed from the second.

In this example the Taylor series is not even considered. It is quite complicated and converges only between the two poles at +/- Pi/2 whereas the continued fraction approximations converge everywhere on the complex plane EXCEPT at the poles! The Taylor series is entirely unsuitable for high-precision approximations to the tangent function.

Comparison: Continued Fraction Expansion, and Contracted Continued Fraction Expansion of tan(z)

The original continued fraction expansion for tan(z) is (gray curve above)


The contracted continued fraction expansion for tan(z) I have developed from this is (blue-dashed curve above)


partial numerators

partial denominators

Using the backwards-recursion algorithm (see General page) this can be implemented in a procedural manner fairly straight-forward using a simple loop. Here in Java:

Java code

Notice that all polynomials use the Horner form (see General page) for much better efficiency.

It is now up to the individual computing platform and tweaking of compiler and runtime options to find out which one of the two continued fraction expansions is faster for a prescribed accuracy. In the case of the contracted continued fraction the terms to be computed are more complicated, which costs computing time, but fewer steps are required to attain a prescribed accuracy. This is a question that cannot be answered a-priori as compiled computing platforms are too different and the values chosen for compiler and runtime settings are too diverse to make a definitive statement here. However, already for n=0 or n=1 the convergence acceleration of the contracted continued fraction is so significant that the number of additional computations required in every step by the contracted continued fraction is not of importance. On today's computing platforms the standard arithmetic operations +,-,*, and / are "cheap" and fast, which ties in with another advantage of continued fraction expansions: they simply don't need any further arithmetic operations than these four!

For example, picking n=0, the comparison is between




Both yield the same accuracy, the latter is the contraction of the former, and which one is calculated faster depends entirely on the computing platform and compiler and runtime settings. Using simpler terms means having to go "deeper", and going "shallow" means using more complicated terms.

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