Tricki

## Use an approximation as if it were exact

### 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.