Algebra

Householder's methods

These are a class of root-finding methods; that is, for the numerical solution of a single nonlinear equation, developed by Alston Scott Householder in 1970. They may be considered a generalisation of the well known Newton-Raphson method (also known more simply as Newton’s method) defined by \[ x\leftarrow x-\frac{f(x)}{f’(x)}. \] where the equation to be solved is \(f(x)=0\). From a starting value \(x_0\) a sequence of iterates can be generated by

The trinomial theorem

When I was teaching the binomial theorem (or, to be more accurate, the binomial expansion) to my long-suffering students, one of them asked me if there was a trinomial theorem. Well, of course there is, although in fact expanding sums of greater than two terms is generally not classed as a theorem described by the number of terms. The general result is \[ (x_1+x_2+\cdots+x_k)^n=\sum_{a_1+a_2+\cdots+a_k=n} {n\choose a_1,a_2,\ldots,a_k}x_1^{a_1}x_2^{a_2}\cdots x_k^{a_k} \] so in particular a “trinomial theorem” would be