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Accurate evaluation of the function is a classic problem in numerical analysis. This function can be seen as an approximation to the first derivative of at , . More generally, is the divided difference of at and and this fact … Continue reading
In the previous post, Complex Magic, Part 1, the use of complex variables in approximating the first derivative of a holomorphic function was discussed. This post delves deeper and introduces formulas suitable for approximating the th derivative of a holomorphic … Continue reading
At the heart of conventional methods of derivative approximation based on divided differences lies a sum of the form in which the coefficients have different signs and vary (usually considerably) in magnitude. As a result, when these coefficients of different … Continue reading
The following function (from L.N. Trefethen, Spectral Methods in MATLAB, with slight modifications) solves the 2nd order wave equation in 2 dimensions () using spectral methods, Fourier for x and Chebyshev for y direction. On its rectangular domain, the equation … Continue reading