Quick description
Often we want to apply a method to a general class of functions, but we only know how to do this for a certain class of functions (e.g., polynomials). Then use an approximation to our general function from our certain class, and act as if our approximation were exact.
A more sophisticated version, is to use this to update a guess by means of this approach, obtaining successively (we hope) approximations to the unknown we wish to compute.
Prerequisites
Linear algebra, calculus.
Example 1
Numerical integration is mostly done this way.
We take a polynomial interpolant of a given function
at certain chosen points
and then use
as the approximation to
.
The error in the result is usually best estimated by the error in the interpolant, or by noting that the method is exact for all polynomials of up to a certain degree (at least ), and then using the error in the interpolant to estimate the error in the integral. (See also Interpolation and approximation.)
Example 2
Solving linear or nonlinear equations iteratively.
For a linear system of equations, where we wish to find
, if we have an approximate matrix
for which we can readily solve systems
, we can compute
iteratively via:

This is known as iterative refinement.
For nonlinear systems we use the Taylor series approximation to 1st order: , and then set the linearization to zero.
If we update the value of
to be
, this gives the Newton–Raphson method.