Approximations, Round-off errors y Taylor

APROXIMATIONS AND ROUND-OFF ERRORS

Numerical method involves an approximation, for this reason an exact result is almost impossible. We need criteria to specify how confident a number is ; this is the origin of round-off errors.

For example e (euler) and are quantities, but we express them with a limited number of digits.

The only way to minimize round-off errors is to increase the numbe of significant digits.

ACCURACY AND PRECISION

Accuracy says how closely a value agree with the real one, in the other hand precision refers how closely some values agree each others.

THE TAYLOR SERIES

Taylor gives a serie to predict an answer, a fuction value at one point. We can predict the value at the point based on another point, in this way we assume that the value of the function at the new point (x+1) is the same as the value at the old point (x).

f(x_i+1)≈f(x_i)

in this way Taylor’s Theorem is express as:

Source: "Numerical Methods for Engineers" Steven C. Chapra

Where,

Source: "Numerical Methods for Engineers" Steven C. Chapra

We can approximate the serie according of the order: the higher order the better approximation.

Source: "Numerical Methods for Engineers" Steven C. Chapra