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ROOTS EQUATIONS

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ROOTS EQUATIONS

1. 1. ROOTS OF EQUATIONS FACULTAD DE INGENIERIAS FISICOQUÍMICAS ESCUELA INGENIERIA DE PETRÓLEOS CYNDY ARGOTE SIERRA INGENIERIA DE PETRÒLEOS
2. 2. INTRODUCTION <ul><li>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. </li></ul>
3. 3. CLOSED METHODS
4. 4. BISECTION METHOD <ul><li>DESCRIPTION OF THE METHOD </li></ul><ul><li>The method is to divide several times by half the sub-intervals [a, b], and in each step, find the half that contains p. To begin suppose that a1 = a and b1 = b, and p1 is the midpoint of [a, b] is: p1 = ½ (a1 + b1). If f (p) = 0, then p = p1; if not so, then f (p1) has the same sign as f (a1) of (b1). If f (p1) f (a1) have the same sign, then p exists between (p1, b1), and we took a2 = p1 and b2 = b1. If f (p1) f (a1) have opposite signs, then p exists in the inte5rvalo (a1, p1) and take a1 and a2 = b2 = p1. then reapply the process to the interval [a2, b2]. </li></ul>
5. 5. ADVANTAGES AND DISADVANTAGES <ul><li>ADVANTAGES </li></ul><ul><li>• You are guaranteed the convergence of the root lock. </li></ul><ul><li>• Easy implementation. </li></ul><ul><li>• management has a very clear error. </li></ul><ul><li>DISADVANTAGES </li></ul><ul><li>• The convergence can be long. </li></ul><ul><li>• No account of the extreme values (dimensions) as </li></ul><ul><li>possible roots. </li></ul>
6. 6. BISECTION METHOD <ul><li>Bisection=Split </li></ul><ul><li>Suppose that f is a continuous function defined on the interval </li></ul><ul><li>[a, b] with f (a) f (b) of different signs. According to the intermediate value theorem, there exists a number p in (a, b) such that f (p) = 0. </li></ul><ul><li>If f(a)=0 --> f(a) is root . </li></ul><ul><li>If f(b)=0 --> f(b) is root </li></ul>
7. 7. EXAMPLE <ul><li>Applying the bisection method to the function f (x) = x³-x-1 for the values a = 1.3 b = 1.4 with a tolerance of <= 0.1 </li></ul>First compute f (a) and f (b): f (1.3) = -0103, f (1.4) = 0.344 1st iteration p1 = 1.3 + 1.4 / 2 = 1.35 f (1.35) = 0.1104. and is larger than tolerance. 2nd iteration We analyze what value will be a2 and b2. As no change sign are opposite ie a1 and a2 = b2 = p1. a2 = 1.3 and b2 = 1.35 p2 = 1.3 + 1.35 / 2 = 1325 F (1325) = 0.00120312, which is less than the tolerance, then we can say that a root for f (x) is 1,325.
8. 8. FALSE POSITION <ul><li>DESCRIPTION OF THE METHOD </li></ul><ul><li>The function is approximated through a line straight where it is assumed his court with the the shaft x corresponds to the value approximate root. </li></ul>
9. 9. EXAMPLE <ul><li>Using the False position method to approximate the root, starting at the interval and until. Solution This is the same example a bisection method. So, we know that is continuous in the given interval and takes opposite signs at the ends of this range. </li></ul><ul><li> Therefore we can apply the wrong rule method. </li></ul>
10. 10. <ul><li>Since we have only an approximation, we must continue with the process. </li></ul><ul><li>Therefore assess </li></ul><ul><li>We now sign our table </li></ul><ul><li>we see that the root is in the range </li></ul>
11. 11. <ul><li>With this new range, we calculate the new approach: </li></ul><ul><li>since the objective is not satisfied with the process we. </li></ul><ul><li>We evaluate and table signs </li></ul>
12. 12. <ul><li>Hence we see that the root lies in the interval, with which we can calculate the new approach: </li></ul>
13. 13. <ul><li>And the approximate error: </li></ul><ul><li>As the objective is fulfilled, we conclude that the approach sought: </li></ul><ul><li>Observe the speed with which the method converges False position to the root, unlike the slow method of bisection. </li></ul>
14. 14. <ul><li>BIBLIOGRAFIA </li></ul><ul><li>Material del Ingeniero Elkin Rodolfo Santafe CAPITULO III: Raices de Ecuaciones. </li></ul><ul><li>http://noosfera.indivia.net/metodos/posicionFalsa.html </li></ul>