A fast algorithm for spectral differentiation (Q1184647)

If \(u\) is a function and \(u'=(u'(x_ 0),\dots,u'(x_ n))\) is the vector of approximate derivatives, then \(u'\) can be obtained by matrix multiplication, i.e. \(u'=Du\), where \(D\) is called a derivative matrix. A matrix vector multiplication takes \(O(n^ 2)\) operations. In this paper an algorithm is proposed to reduce twice the necessary number of operations. This algorithm uses some regularity properties of the matrix \(D\).