Numeric Approximation

Introduction

In numeric analysis, the mistake between the real valor and the valor securer is called: approximation error, to reduce these errors effects, we need to do numeric approximations which have some properties that help us to reduce the approximation error.

Numeric Approximation:

Then are techniques through a mathematical model are solved using arithmetic calculus.
The numeric approximation be systematic techniques whose the results be approximation from the true valor that take on the interest variant; constant repetition of the technique (iteration), enable come nearly to the want valor.
We can understand by numeric approximation X* a quantity that represent a number whose exactly valor is X. When the number X* is nearly to the most exactly X valor, will be a better approximation from this number. Examples:
3.1416 is a numeric approximation from p,
2.7183 is a numeric approximation from e,
1.4142 is a numeric approximation from Ö2, and
0.333333 is a numeric approximation from 1/3.

Significant amount:
The number of significant amounts is the number of digit t, that we can use, con trust, when we measure a variable.
The ceros included in a number not ever are significant amount; for example the numbers:
-0.00002415
-0.002415
-2415
-241500

Exactness and precision:
Precision refer the significant amount numbers whose represent a quantity.
The exactness refers at the number approximation our some measure from numeric valor whose supposes represent.
Example: p is irracional number constituted by infinite amounts numbers; 3.141592653589793... is good approximation of p, that could be considerate the exact valor. If we considered the next p approximation, we can say:
p = 3.15 not have precision and are inexact
p = 3.14 not have precision but are exact
p = 3.151692 have precision but are inexact
p = 3.141593 have precision and are exact

Numeric methods should offer exactly solutions and precise. Mistake term is used to refer the inexactly so measure the bad precision in predictions.

Mistakes
When we make numeric approximations, could generate some mistake type when we work with operations and mathematical quantity:
-Un-continuity mistakes: result from the approximation use.
-Round mistakes: result from use a finite quantity of significant amount.

Them are some numeric approximation methods, we designated, but we won’t study them in this blog:
1 Euler
2 Taylor’s methods
3 Runge-Kutta’ method
4 Matlab simulation

Refers:

-UNIVERSIDAD SIMON BOLIVAR, División de Ciencias Físicas y Matemáticas, Departamento de Computo Científico
-Sergio Ramírez, hellnight39@hotmail.com
-http://www.google.com.co/url?sa=t&source=web&ct=res&cd=1&ved=0CBYQFjAA&url=http%3A%2F%2Fdcb.fi-c.unam.mx%2Fusers%2Fgustavorb%2FMN%2FPresentaciones%2F1.2%2520Aproximacion%2520numerica.pps&rct=j&q=Aproximaci%C3%B3n+Num%C3%A9rica&ei=XEfrS8CuIYbGlQf42KCcBA&usg=AFQjCNF18xIiVt1us5qIJ1OqbfB7WdIMNw&sig2=kn3-mqXu9JNVnmlEVNfdUQ, Gustavo Rocha 2005.