Central difference limit for derivatives of ill-posed functions: Best approach in supercomputing era

摘要

A polynomial-time algorithm based on a revised method of iterative central difference limit is presented for computing the numerical value of the derivative of a given analytic function. Through numerical experiments, we establish that this algorithm is a best one. This can be used to obtain the derivative to a desired accuracy subject to the precision of the computer for violently fluctuating or rapidly oscillatory functions. The concerned time/computational complexity is so small in practice that in the non-main-frame supercomputing era when over estimated 95% of computing resources is unutilized and hence a waste, the complexity here is not an issue. We have, for the purpose of a comparison, also included Matlab symbolic-cum-numerical computation to obtain the derivative of the foregoing functions numerically. Matlab programs in both Matlab standard precision as well as Matlab variable precision are also included for the central difference limit along with the symbolic-cum-numerical computation. The reader concerned with computing the derivative of an ill-conditioned function - large or small - can use these programs by copying, pasting, and executing and can readily check the quality of the derivative.