ROOTS OF EQUATIONS
The purpose of calculating the roots of an equation to determine the values of x for which holds:
f (x) = 0 (28)
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 (28) 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).
In general, the methods for finding the real roots of algebraic equations and transcendental methods are divided into intervals and open methods.
The determination of the solutions of equation (28) 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).
In general, the methods for finding the real roots of algebraic equations and transcendental methods are divided into intervals and open methods.
- INTERVAL METHODS: exploit the fact that typically a function changes sign in the vicinity of a root. They get this name because it needs two initial values to be "encapsulated" to the root. Through such methods will gradually reduce the size of the interval so that the repeated application of the methods always produce increasingly close approximations to the actual value of the root, so methods are said to be convergen.
In Figure 2.1 is seen as the function changes + f (x) a - f (x), as it passes through the root c. This is because f (c) = 0 and necessarily pass function of positive to negative quadrant x. In some cases, to be seen later this does not happen, for now it will be assumed as shown. The methods they use open sign changes in order to place the root (point c), but it must then establish a range (such as [a, b]).
Similarly happens when the function passes through the point e, the change occurs-f (x) + f (x), to find the root of the method requires an interval as [d, f].
The main methods are Interval:
a. Graphical Method
b. Bisection Method
c. Linear Interpolation Method
d. methods of false position
- OPEN METHODS: in contrast, are based on formulas that require a single initial value x (initial approach to the root). Sometimes these methods away from the real value of the root grows the number of iterations.
Open the main methods are:
a. Newton Raphson method.
b. Secant method
c. Multiple roots
.
fuente:
· Burden Richard L. & Faires J. Douglas, Análisis numérico. 2ª. ed., México, Grupo Editorial Iberoamérica, 1993.
· Chapra Steven C. & Canale Raymond P., Métodos numéricos para ingenieros. 4ª. ed., México, McGraw-Hill, 2003
