CALCULATIONS OF ROOTS•GRAPHIC METHOD

•BISECTION METHOD•FALSE POSITION METHOD

•BY:ÓSCAR FABIÁN CARPIO PATIÑO

GRAPHIC METHOD•The graphical method for solving such systems is therefore in representing Cartesian axes or coordinate system, both straight and see if they are cut and, if so, where. Keep in mind that, in the plane, two lines can only have three relative positions (each other): intersect at a point, are parallel or coincide (the same line).

GRAPHIC METHOD

•If the two lines intersect at one point, the coordinates of this are the pair (x, y) that form the unique solution of the system, since they are the only values of the two unknowns that satisfy the two equations of the system, thus , it is compatible determined.

GRAPHIC METHODThe process of solving a system of equations by the graphical method is summarized in the following phases:•It clears the mystery and in both equations.•Is constructed for each of the two functions of first degree obtained, the value table corresponding.•Both lines are plotted on the coordinate axes.•In this last step there are three possibilities :

If two lines intersect, the coordinates of the court are the only values of the unknowns x and y given support system.

If both lines are coincident, the system has infinite solutions that are the respective coordinates of all points of that line in that match both. Unspecified compatible system.If both lines are parallel, the system has no solution. Incompatible system.

GRAPHIC METHOD•The graphical method for solving such systems is therefore in representing Cartesian axes or coordinate system, both straight and see if they are cut and, if so, where. Keep in mind that, in the plane, two lines can only have three relative positions (each other): intersect at a point, are parallel or coincide (the same line).

GRAPHIC METHOD

SOLVE THE FOLLOWING SYSTEM OF EQUATIONS x + y = 600

2x - y = 0 To solve the system by the graphical method and solve for the unknown in both equations and we have: y = -x + 600

y = 2x

EXAMPLE

GRAPHIC METHOD•Come on now, to represent both straight, to calculate their tables of values:

y = -x + 600 y = 2x

X Y X Y

200 400 100 200

600 0 200 400

GRAPHIC METHOD•With these tables of values for the two lines and choosing appropriate scales on the axes OX and OY, we can now plot :

Looking at the graph, we see clearly that the two lines intersect at the point (200, 400), then the solution of the system is : 

x = 200 y = 400

BISECTION METHODIn mathematics, the bisection method is a root finding algorithm that works by dividing the interval in half and then selecting the subinterval which is the root.

Suppose we want to solve the equation f (x) = 0, given two points a and b such that f (a) and f (b) have opposite signs, we know from Bolzano's theorem that f must have at least one root in the interval [a, b]. The bisection method divides the interval into two, using a third point c = (a + b) / 2. At this time, there are two possibilities: f (a) and f (c), or f (c) and f (b) have opposite signs. The bisection algorithm is applied to the subinterval where the sign change occurs.

BISECTION METHODALGORITHMTo apply the method consider three sequences an ≤ pn ≤ bn defined by the following relationships:

Where initial values are given by :a0 :=a ; b0 :=b

There is evidence that the three sequences converge to the value of the only root of the interval :

METHOD OF FALSE POSITIONThe poor convergence of the bisection method as well as its poor adaptability to higher dimensions motivate the use of better techniques. One such method is the Method of False Position. Here, we start with an initial interval [x1,x2], and we assume that the function changes sign only once in this interval. Now we find an x3 in this interval, which is given by the intersection of the x axis and the straight line passing through (x1,f(x1)) and (x2,f(x2)). It is easy to verify that x3 is given by

Now, we choose the new interval from the two choices [x1,x3] or [x3,x2] depending on in which interval the function changes sign.

METHOD OF FALSE POSITIONThe false position method differs from the bisection method only in the choice it makes for subdividing the interval at each iteration. It converges faster to the root because it is an algorithm which uses appropriate weighting of the intial end points x1 and x2 using the information about the function, or the data of the problem. In other words, finding x3 is a static procedure in the case of the bisection method since for a givenx1 and x2, it gives identical x3, no matter what the function we wish to solve. On the other hand, the false position method uses the information about the function to arrive at x3.

METHOD OF FALSE POSITION