ROOTS EQUATIONS

• 1. ROOTS OF EQUATIONS FACULTAD DE INGENIERIAS FISICOQUMICAS ESCUELA INGENIERIA DE PETRLEOS CYNDY ARGOTE SIERRA INGENIERIA DE PETRLEOS

2. INTRODUCTION

• 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.

3. CLOSEDMETHODS 4. BISECTION METHOD

• DESCRIPTION OF THE METHOD

• 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 = (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].

5. ADVANTAGES AND DISADVANTAGES

• ADVANTAGES

• You are guaranteed the convergence of the root lock.

• Easy implementation.

• management has a very clear error.

• DISADVANTAGES

• The convergence can be long.

• No account of the extreme values (dimensions) as

• possible roots.

6. BISECTION METHOD

• Bisection=Split

• Suppose that f is a continuous function defined on the interval

• [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.

• If f(a)=0 --> f(a) is root .

• If f(b)=0 --> f(b) is root

7. EXAMPLE

• Applying the bisection method to the function f (x) = x-x-1 for the values a = 1.3 b = 1.4 with a tolerance of