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Unformatted text preview: Wednesday, January 18 Lecture 9 : Error estimation for numerical integration . (Refers to section 5.3) After having practiced the problems associated to the concepts of this lecture the student should be able to : Find error's bounds when applying either the midpoint rule, the trapezoidal rule or the Simpson's rule. 9.1 Error Estimate formulas We have error estimate formulas for the Trapezoidal rule , the Midpoint rule and the Simpson's rule . 1) Trapezoidal rule and Midpoint rule : Suppose that the second derivative f '' of f is continuous on [ a, b ] and that | f ''( x ) | K for all x in [ a, b ] then: If we approximate with T n then an upper bound | ET n | for the error is given by the expression If we approximate this definite integral with M n an upper bound | EM n | for the error is given by the expression 2) Simpsons rule : Suppose that the fourth derivative f (4) is continuous on [ a, b ] and that | f (4) ( x ) | K for all x in [ a, b ]. If we approximate with S n an upper bound | ES n | for the error is given by the expression Note...
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This note was uploaded on 04/01/2012 for the course MATH 118 taught by Professor Zhou during the Winter '08 term at Waterloo.
- Winter '08