# mathematics

## 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 Joukowsky Transform

The Joukowksy Transform is an elegant and simple way to create an airfoil shape. Let $$C$$ be a circle in the complex plane that passes through the point $$z=1$$ and encompasses the point $$z=-1$$. The transform is defined as $\zeta=z+\frac{1}{z}.$ We can explore the transform by looking at the circles centred at $$(-r,0)$$ with $$r<0$$ and with radius $$1+r$$: $\|z-r\|=1+r$ or in cartesian coordinates with parameter $$t$$:

## 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