How to upgrade the running time of Computer

One of built-in function required in building micro-processor of computer is infinite series of tangent function. Until now, there is one general formula for the infinite series of tangent function that available in mathematical handbook, and also used in all symbolic software-package. The general formula was created by Sir. Bernoulli that is of form

where,is Bernoulli numbers, and n 1,2,3,. Unfortunately, the general formula is not consistent with Maclaurin series that always contains n ! in denominator of each terms. In this paper, we present Rohedi’s reversion for obtaining the infinite series of tangent function without of use the Maclaurin series, but its result is still consistent with the Maclaurin series. Derivation of RohediSmart reversion formula based on solution of the arctangent differential equation

solved ecursively by using short stable modulation technique (S-SMT). Comparison the infinite series of tangent function of Rohedi’s reversion with the result of both Matematica 5.1, Maple 9.5 shows thattime consuming of Rohedi’s reversion is shortest, hence need smallest of computational memory.Finally, we give comparison the time consuming of Rohedi’s reversion for 4.0365 10321 x1635of 1635nd coefficient of infinite series of tangent function calculated using matlab needs2.744 s, while Maple 9.5 soft needs 36.35 s

Discussion and Conclusion

We show that output of Rohedi’s reversion for infinite series of tangent function is still consistent with Maclaurin series. According to the equality of output both of Matematica and Maple software-package, we resume that both software-package have been used equal general formula, that is Bernoulli’s formula. Hence, we take a conclusion that Rohedi’s reversion formula as a new general formula of infinite series of the tangent function that can be used to upgrade the running time of computer.

References :

1. Abranowitz,M., and Stegun, I.A., “Handbook of Mathematical Functions”, New-York, 1972
2. Spiegel,M.R.,”Mathematical Handbook of Formulas and Tables”, Schaum’s Outline Series,McGRAW-Hill Book Company, page 104, 1968.
3. Rohedi,A.Y., “Solving of the homogeneous nonlinear differential equation by using Stable Modulation Technique”, Presented on Conference of Mathematical Analysis and its Applications”, Department of Mathematics, Natural Sciences, ITS, Surabaya, Indonesia, 10-11 August 2006.
4. Rohedi,A.Y., “Analytical Solution Of The Ricatti Differential Equation For High Frequency Derived By Using The Stable Modulation Technique”, Presented on International Conference of Mathematics and Natural Sciences, Poster Edition, Faculty of Natural Sciences, ITB, Bandung, Indonesia, 29-30 Nopember 2006.
5. Shortcut Solution for Bernoulli Equation in AF(A) Formula Based on Stable Modulation Technique (will be submitted for publication

This paper will be submitted for publication.