NUMERICAL METHODS

ITERATIVE METHODS

An iterative method is a method that is progressively calculating approximations to the solution of a problem. In mathematics, in an iterative method is repeated the same process of improvement on an approximate solution: is expected to be obtained is a more approximate than the first. The process is repeated on this new solution until the most recent result meets certain requirements. Unlike direct methods, in which they must complete the process to have the answer, the iterative methods can suspend the process at the end of an iteration and gives an approximation to the solution.

Jacobi method:

The Jacobi method is the iterative method for solving systems of linear equations as simple and applies only to square systems, ie systems with as many unknowns as equations.

1. First, we determine the equation of recurrence. For this order the equations and unknowns. From the equation i cleared the unknown i. In matrix notation is written as:

x = c + Bx (1)
where x is the vector of unknowns.

2. It takes an approach to the solutions and this is designated by xo
3.Is iterated in the cycle that changes the approach

xi +1 = c + Bxi


The Gauss-Seidel Method:

The Gauss-Seidel method is very similar to the method of Jacobi. While in the Jacobi using the value of the unknowns to determine a new approach, the Gauss-Seidel is using the values of the unknowns recently calculated in the same iteration, and not the next. For example, the Jacobi method is obtained in the first calculation xi +1, but the value of x is not used until the next iteration. In the Gauss-Seidel method instead is used instead of xi xi +1 immediately to calculate the value of yi +1 equally applicable to the following variables are always used the newly calculated variables.

REFERENCES:
http://www.uv.es/~diaz/mn/node32.html
http://www.mty.itesm.mx/etie/deptos/m/ma95-843/lecturas/l843-13.pdf
http://es.wikipedia.org/wiki/M%C3%A9todo_iterativo

ROOTS OF EQUATIONS

The purpose of calculating the roots of an equation to determine the values of x for which holds:

f (x) = 0

The determination of the roots of an equation is one of the oldest problems in mathematics and there have been many efforts in this regard. Its importance is that if we can determine the roots of an equation we can also determine the maximum and minimum eigenvalues of matrices, solving systems of linear differential equations, etc ...

Most of the methods used to calculate the roots of an equation are iterative and are based on models of successive approximations. These methods work as follows: from a first approximation to the value of the root, we determine a better approximation by applying a particular rule of calculation and so on until it is determined the value of the root with the desired degree of approximation.



• GRAPHICS METHODS: is to plot the function and see where it crosses the x-axis This item, which represents the value of x for which f (x) = 0, provides an initial approximation of the root.

• OPEN METHOD: based on formulas that require only one value of x for a couple of them but do not necessarily contain the root. As such sometimes diverge or move away from the root as it grows the number of iterations. However, when the open methods in general do converge faster than methods that use ranges.

Bisection method

Is to consider an interval (xi, xs) which ensures that the function root. Its aim is to divide the interval always half in successive iterations

Procedure:

1. Find an interval (xi, xs) which ensures the root

F(xi)*F(xs)<0>

2. Find the midpoint

Xr=(xi+xs)/2

3. It identifies which of the two intervals is the root

F(xi)*F(xr)<0 Xs=xr

F(xs)*F(xr)<0 style=""> Xi=xr

4. Therm is the method with the maximum number of iterations or

when reaches the% E

ž Tolerance Range

False position

Based on the graphical display, where relations of triangles
As in the bisection of comparing the functions of f (xi), f (xs) and f (xr)
Compared with the bisection method, this method is the long since you have to generate new lines to reach the line crossing the root.


CLOSED METHOD

Fixed point method

The method is to leave the left side of the equation equal to X where f (x) = 0 and looking for a value of X that is equal to the function clear

X--> F(X)=0

F(X)=X +G (X)

X =G (X)

Newton raphson method

It involves taking an initial value and from the same draw tangents to approach the value of the root.
The disadvantage of this method is that we must calculate the derivative of the function

Procedure:

1. f (x) = 0

2. Choose one Xi

3. Calculate Xi +1

4.Calculate the% Ea

Secant method

Is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite difference approximation of Newton's method.

The secant method is defined by the recurrence relation

As can be seen from the recurrence relation, the secant method requires two initial values, x₀ and x₁, which should ideally be chosen to lie close to the root.

CALCULATION OF MULTIPLE ROOTS.

A multiple root corresponds to a point where a function is tangential to the x axis For example, two resulting from repeated roots

(1) f(x)=(x-3)(x-1)(x-1)

or multiplying terms

(2)f(x)=(x^3)-(5x^2)+7x-3

The equation has a double root because a value of x cancels two terms of equation (1).

FUENTE:NUMERICAL METHODS FOR ENGINEERS WITH PERSONAL COMPUTER APPLICATIONS.STEVEN C. CHAPRA/ RAYMOND P. CANALE