NUMERICAL METHODS

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