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There really isn't a theorem you can use, since most of the applicable theory is for sequences with known asymptotic behavior. For "in the wild" sequences, you can do no better than to check that the …

If your function is badly behaved (e.g. noisy, very oscillatory), no method will perform properly (differentiation is numerically very unstable). That being said, for "nice functions", I have good exp …

You have already seen McNamee's excellent bibliography on polynomial root-finding methods? Personally I have a preference for the "simultaneous iteration" methods (of which Durand-Kerner and Ehrich-A …

I use the technique given by J.C. Nash in the book Compact Numerical Methods for Computers. On page 219, there is the formula $h=\sqrt{\epsilon}\left(|x|+\sqrt{\epsilon}\right)$ where $\epsilon$ is …

Probably still your best bet, after of course reducing your original symmetric matrix to tridiagonal form, would be either bisection (with the help of Gerschgorin bounds) or an appropriate modificatio …

There's always the option of expanding $f$ as a series, integrating that, and performing Lagrange inversion to obtain a series for the inverse function, from which you can determine suitable function …

Well, there are explicit expressions for the numerator $N_{pq}(z)$ and denominator $D_{pq}(z)$ of the $(p,q)$ Padé approximant for $\exp(z)$: $\displaystyle N_{pq}(z)=\sum_{j=0}^p \frac{(p+q-j)!p!}{( …

I haven't gotten around to downloading and reading it (and I'm wondering how I missed this when I was searching for results related to Halley's method), but apparently a multivariate version of the Ha …

I have settled this question here; in brief, a FORTRAN implementation of algorithms for solving this problem have been published in the Collected Algorithms of the ACM. At the risk of being redunda …

There's nothing more straightforward than using the gamma function relationship, I believe (perhaps using Lanczos's approximation to compute the gamma functions). Of course, for integer arguments, the …

As already mentioned by Fedor and Igor, you can ignore the logarithmic term since it zeroes out at $\infty$, and you can just concentrate on the series $$-\frac{i}{2}\sum_{k=1}^\infty \frac1{ik+k^{3/ …

Personally, I'm still a bit torn while giving this prescription. I am of the opinion that for truly reliable computation, one should use the singular value decomposition whenever one can tolerate the …

With credits to J.J. Green, I found this paper on finding the maximum modulus of a polynomial on the disk; it might be of help in this case.

Here's my take on the matter: the difference of philosophy between quadrature routines and ODE solving routines, I believe, is this: Extrapolation is riskier than interpolation. Remember that nu …

This question has been out for a while, and I think it deserves a thorough answer, even tho I came quite late to this party. As both the classical Legendre-Jacobi theory and the Carlson theory have b …