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class_14 - CS205 Class 14 Covered in class 1 3 4 Readings...

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CS205 – Class 14 Covered in class: 1, 3, 4 Readings: 7.4, 8.1 to 8.3 Quadrature 1. Numerical quadrature approximate ( ) b a I f x dx for a given f a. These f ’s might be arbitrarily difficult to compute and only available by running a program. b. General approach : Subdivide [ , ] a b into n intervals 1 [ , ] i i x x with 0 x a and n x b and consider each subinterval separately 2. Newton-Cotes quadrature for each subinterval 1 [ , ] i i x x , choose n equally spaced points and use k-1 degree polynomial interpolation to approximate the integral a. Exact on polynomials of degree n-1 when n is even, as expected b. Exact on polynomials of degree n when n is odd, from symmetric cancellation i. c. local accuracy an exact method on k degree polynomials has a local error that scales like 2 ( ) k O h in each subinterval where h is the length of the subinterval d. global accuracy since there are 1 O h subintervals, the total error scales like 1 ( ) k O h i. doubling the number of subintervals, sends 2 h h
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