Algorithm 579: CPSC: Complex Power Series Coefficients [D4]
Bengt Fornberg · ACM Transactions on Mathematical Software · 1981
An alg0rithm CPSC is presented here.It evaluates numerically the leading coefficients in a power ~eries expansion of an analytic function (or, equivalently, a number of leading derivatives of an analytic function).A detailed description of the theoretical background of the code is given in [1], together with some warnings about cases in which full accuracy may not be reached. Such cases are(1) very low-order polynomials (for example, f(z) ffi 1 + z);(2) functiofis whose Taylor expansions contain very large isolated terms (for example, f(z) = 106 + (1/(1 -z))); (3) certain entire functions (for example, f(z) = eZ);(4) functions whose radius of convergence is limited, by a branch point at which the function remains many times differentiable (for example, f(z) = (1 + z) 16 log(1 + z) expanded around z ffi 0).In the case 1, the routine will normally fail to even approximate the correct answer.In cases 2, 3, and 4, problems are normally encountered only if large numbers' of coefficients are wanted (e.g., more than 30).The supplied error estimate will, in most of these cases, give correct information about the lowered accuracy.Inevitably there is a risk that the routine will attempt to evaluate the given function exactly at a singularity.This will happen, for example, for f(z) ffi 1/