Numerical analysis

It predates computers by many centuries. [Taylor approximation]? is a product of the seventeenth and eighteenth centuries that is still very important. The logarithms of the sixteenth century are no longer vital to numerical analysis, but the associated and even prehistoric notion of interpolation continues to solve problems for us.

The effect of round-off error is partly quantified in the condition number of an operator. Subtraction of two nearly equal numbers is an ill-conditioned operation, producing [catastrophic loss of significance]?. Using well-conditioned operations helps achieve [numerical stability]?.

An important part of Numerical Analysis is concerned with computing (in an approximate way) the solution of Partial Differential Equations. This is done by first discretizing the equation, bringing it into a finite dimensional subspace, then solving the linear system in this finite dimensional space. The first stage is done by the [Finite Element method]?, [finite difference]? methods, or (particularly in engineering) the method of Finite Volumes.

The Finite element method is the more powerful approach to numerical differential equations, but mathematicians prefer [finite difference]? because the theorems are easier to prove. Shame on them. This is hardly true. Finite elements is based on a vast number of results in functional analysis. FD in contrast is very messy, involving chasing around terms in Taylor series. But I am going to fix this momentarily :)

The linear systems that come form discretized PDEs can then be solved by a variant of Gaussian Elimination, by some Iterative method such as Conjugate Gradients, or by Multigrid?.

[[Newton's method]] (Clifford has promised to make ' work. Looking forward to Huntington's chorea, Goedel's th'm, and Larry's text.) [Gaussian quadrature]?, [least squares]?, Gram-Schmidt?,