numerical methods: calculation of roots of equations

Calculation of 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 ... The determination of the solutions of equation can be a very difficult problem.

if f (x) is a polynomial function of grade 1 or 2, know simple expressions that allow us to determine its roots. For polynomials of degree 3 or 4 is necessary to use complex and laborious methods. However, if f (x) is of degree greater than four is either not polynomial, there is no formula known to help identify the zeros of the equation (except in very special cases).

There are a number of rules that can help determine the roots of an equation: Bolzano's theorem, which states that if a continuous function, f (x) takes on the ends of the interval [a, b] values of opposite sign, then the function accepts at least one root in that interval. In the case where f (x) is an algebraic function (polynomial) of degree n real coefficients, we can say that will have n real roots or complex.

The most important property to verify the rational roots of an algebraic equation states that if p / q is a rational root of the equation with integer coefficient

Example: We intend to calculate the rational roots of the equation

3x3 + 3x2 - x - 1 = 0

First, you make a change of variable x = y / 3:

and then multiply by 32:

y3 + 3y2-3y = -9

0with candidates as a result of the polynomial are:

Substituting into the equation, we obtain that the only real root is y = -3,

(Which is also the only rational root of the equation). Logically, this method is very powerful, so we can serve only as guidelines.

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.

GRAPHIC METHOD
This method is used primarily to locate an interval where the function
any root.
Example 1
Locate an interval where the function f (x) = e ^ (- x) - ln x
has a root.

Solution
To calculate the root of f (x) do f (x) = 0, where e ^ (- x) = ln x.
Therefore, the problem is finding the intersection point of the functions
g (x) = e ^ (- x) and h (x) = ln x.
We know these graphs:

From which, we conclude that an interval where the only root is [1,1.5].
In fact, we are not interested in being the finest in the search interval as then apply systematic methods to approximate better root. Let's say that the usefulness of the graphic method is in providing a range with which we start work

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