ROOTS OF EQUATIONS

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

f (x) = 0 (1)

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:
As 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 METHODS: 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.

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

f (x) = (x-3) (x-1) (x-1) or (2)

Multiplying terms f (x) = x3-5x2 +7 x-3 (3)

The equation has a double root because a value of x cancels two terms of equation (2).

REFERENCES: NUMERICAL METHODS FOR ENGINEERS WITH PERSONAL COMPUTER APPLICATIONS. STEVEN C. Chapra / Raymond P. CANALE