Roots of equations


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Roots of equations

  1. 1. ROOTS OF EQUATIONS A root or solution of equation f(x)=0 are the values of x for which the equation holds true. The numerical methods are used for finding roots of equations, some of them are: 1. GRAPHICAL METHOD It is a simple method to obtain an approximation to the equation root f(x) =0. It consists of to plot the function and determine where it crosses the x-axis. At this point, which represents the x value where f(x) =0, offer an initial approximation of the root. The graphical method is necessary to use any method to find roots, due to it allows us to have a value or a domain values in which the function will be evaluated, due to these will be next to the root. Likewise, with this method we can indentify if the function has several roots. For instance: 2. CLOSED METHODS These are called closed methods because are necessary two initial values to the root, which should “enclose” or to be to the both root sides. The key feature of these methods is that we evaluate a domain or range in which values are close to the function root; these methods are known as convergent. Within the closed methods are the following methods:
  2. 2. 2.1. BISECTION(Also called Bolzano method) The method feature lie in look for an interval where the function changes its sign when is analyzed. The location of the sign change gets more accurately by dividing the interval in a defined amount of sub-intervals. Each of this sub intervals are evaluated to find the sign change. The approximation to the root improves according to the sub-intervals are getting smaller. The following is the procedure: Step 1: Choose lower, xl, and upper, xu, values, which enclose the root, so that the function changes sign in the interval. This is verified by checking that: f  xl  f  xu   0 Step 2: An approximation of the xr root, is determined by: xl  xu xr  2 Step 3: Realize the following evaluations to determine in what subinterval the root is: a. If f  xl  f  xr   0 , then the root is within the lower or left subinterval, so, do xu=xr and return to step 2. b. If f  xl  f  xr   0 , then the root is within the top or right subinterval, so, do xl=xr and return to step 2. c. If f  xl  f  xr   0 , the root is equal to xr; the calculations ends.
  3. 3. The maximum number of iterations to obtain the root value is given by the following equation: 1 ������ − ������ ������������ á������ = ln⁡ (2) ������������������ TOL = Tolerance 2.2. THE METHOD OF FALSE POSITION Although the bisection method is technically valid to determine roots, its focus is relatively inefficient. Therefore this method is an improved alternative based on an idea for a more efficient approach to the root. This method raises draw a straight line joining the two interval points (x, y) and (x1, y1), the cut generated by the x-axis allows greater approximation to the root. Using similar triangles, the intersection can be calculated as follows: ������(������1 ) ������(������2 ) = ������������ − ������������ ������������ − ������2 The final equation for False position method is: ������ ������2 (������1 − ������2 ) ������������ = ������������ − ������ ������1 − ������(������2 ) The calculation of the root xr requires replacing one of the other two values so that they always have opposite signs, what leads these two points always enclose the root. Sometimes, depending on the function, this method works poorly, while the bisection method leads better approximations.
  4. 4. BIBLIOGRAPHY CHAPRA, Steven C. y CANALE, Raymond P.: Métodos Numéricos para Ingenieros. McGraw Hill 2007. 5ª edition.