The document discusses three methods for finding the roots of equations: 1) The graphic method involves plotting the equations on a graph to find the point where they intersect, representing the root. It can determine if a system is compatible, incompatible, or has infinite solutions. 2) The bisection method iteratively narrows down the interval containing the root by dividing it in half. 3) The false position method uses the slopes of lines through two initial points to estimate a new test point, converging faster than bisection by incorporating information about the function.

2. BISECTIONMETHOD

3. FALSE POSITION METHOD

4. BY:ÓSCAR FABIÁN CARPIO PATIÑO

6. Is constructed for each of the two functions of first degree obtained, the value table corresponding.

7. Both lines are plotted on the coordinate axes.

8. In this last step there are three possibilities :

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

10. 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.

13. BISECTIONMETHOD In 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.

14. BISECTIONMETHOD ALGORITHM To apply the method consider three sequences an≤ pn≤ bndefined 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 :

15. METHOD OF FALSE POSITION The 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.

16. METHOD OF FALSE POSITION The 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.

17. METHOD OF FALSE POSITION