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Higher order pseudospectral differentiation matrices. (English) Zbl 1086.65016

“We approximate the derivatives of a function $f\left(x\right)$ by interpolating the function with a polynomial at the Chebyshev extrema nodes ${x}_{k}$, differentiating the polynomial, and then evaluating the polynomial at the same nodes.”

The paper investigates the roundoff properties of various ways of setting up the matrix that maps the vector $\left(f\left({x}_{k}\right)\right)$ to the vector of $p$-th derivatives $\left({f}^{\left(p\right)}\left({x}_{k}\right)\right)$ for arbitrary $p$. Simple numerical tests are reported.

Mostly, the paper is clearly written. But it would have been useful to explain the context in which this way of approximating derivatives is valuable. There is a vast difference between using such formulas in explicit approximation of derivatives of a given function (for which very poor results should be expected even in exact arithmetic) and using them as part of an implicit process like solving a differential equation.

##### MSC:
 65D25 Numerical differentiation