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Introduction to Numerical Analysis I

Introduction to Numerical Analysis I. Bis ection Method. MATH/CMPSC 455. Solving Nonlinear Equations. General Mathematical Problem: Given a function , find the values of for which. Existence of root :

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Introduction to Numerical Analysis I

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  1. Introduction to Numerical Analysis I Bisection Method MATH/CMPSC 455

  2. Solving Nonlinear Equations General Mathematical Problem: Given a function , find the values of for which Existence of root: Let be a continuous function on , satisfying . Then has a root between and , that is, there exists a number satisfying and

  3. The Bisection Method Basic Idea: Narrow down the interval by halving Start f(a)f(b)<0 c=(a+b)/2 f(a)f(c)<0 No Yes a=c b=c No |a-b|<tol Yes output (a+b)/2 and stop

  4. Example: Find a root of the function on the interval

  5. Error Analysis Theorem (Error Analysis of Bisection Method): If denote the intervals obtained by the Bisection method, then the limits and exist, equal and represent a zero (root) of . If and , then

  6. Example: How many steps is needed, if we use Bisection Method to find a root of in the interval , and require the solution is correct with 6 decimal places? Example: Suppose that the bisection method is started with the interval , how many steps should be taken to compute a root that the relative error is less than .

  7. Practical Considerations

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