Chap2 - NUMERICAL ANALYSIS AAOC C341 CHAPTER 2 Root Finding...

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1 NUMERICAL ANALYSIS AAOC C341 CHAPTER 2 Root Finding Methods for Nonlinear Equations
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2 Root Finding Methods for Nonlinear Equations ± Bisection Method ± Secant Method ± Regula-Falsi Method ± Newton’s Method ± Muller’s Method ± Fixed Point Method ± Newton’s Method for Multiple Roots ± System of Non-linear Equations Newton’s Method and Fixed Point Method
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3 Introduction: ± In scientific and engineering studies, a frequently occurring problem is to find the roots or zeros of equations of the form f(x) =0. ± f(x)may be algebraic or transcendental or a combination of both. ± Algebraic functions of the form P(x)= a 0 + a 1 x + … + a n x n , are called polynomials . ± A non-algebraic function is called a transcendental function.
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4 Solution Methods ± Polynomials up to degree 4 can be solved exactly. Since finding the root of f(x)=0 is not always possible by analytical means, we have to go for some other techniques or methods. ± Solution Methods are either Bracketing: Bisection Method, Regula Falsi Method ± or Iterative: Secant Method, Newton's Method, Muller's Method, Fixed-Point Method. ± Rate of Convergence. ± Advantages/Disadvantages.
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5 The Bisection Method The Bisection method is based on the Intermediate Value Theorem and Cantor's Intersection Theorem. Intermediate Value Theorem: Let f(x) be a continuous function in [ a,b ] and let k be any number between f(a) and f(b) . Then, there exists a number c in ( a, b ) such that f(c) = k . Hence, an equation f(x)=0 where f(x) is a real continuous function, has at least one root between a and b if f(a) f(b) < 0 .
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6 Bisection method continued … Cantor's Intersection Theorem: If a sequence of closed intervals [ a n , b n ] is such that [ a n+1 , b n+1 ] [ a n , b n ] for all n , and lim (b n -a n )=0 , then consists of exactly one point. Algorithm for Bisection Method : Suppose f(x) is continuous and we have two numbers a 1 and b 1 such that f(a 1 )f(b 1 )<0 , then by Intermediate Value Theorem we know that there is a root for f(x) between a 1 and b 1 . [] n n n b a , 0 =
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7 Bisection method continued … Then we have to divide the interval into two parts and take the middle point suppose that is c , if f(c ) =0 then we get the root, if not so then by IVT the root lies within a and c or c and b , depending on whether f(a 1 )f(c) <0 or f(c)f(b 1 )<0 . Then we have to take the next interval in which the root lies and continue our process. Since we are dividing the interval by half each time, we are approaching to zero of the function, hence this algorithm gives us the approximate solution. Now we have to put some accuracy criteria to stop the process.
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8 Bisection methodcontinued … Algorithm for Bisection Method: Step 1: Choose a and b as two initial guesses for the root such that f(a) f(b) < 0 . Then by IVT one root is guaranteed for f(x) in [ a,b ].
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This note was uploaded on 02/23/2011 for the course MATH 122 taught by Professor Ritadubey during the Spring '11 term at Amity University.

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Chap2 - NUMERICAL ANALYSIS AAOC C341 CHAPTER 2 Root Finding...

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