CS 205A class_14 notes

Scientific Computing

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CS205 – Class 14 Covered in class: 1, 2 Readings: 8.1 to 8.3 1. Numerical quadrature approximate () b a I fxd x = 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 [,] ab into n intervals 1 [, ] ii x x + with 0 x a = and n x b = and consider each subinterval separately 2. Newton-Cotes quadrature for each subinterval 1 x x + , choose n equally spaced points and use k-1 degree polynomial interpolation to approximate the integral a. Exact on polynomials of degree q=k-1 when k is even, as expected b. Exact on polynomials of degree q=k when k is odd, from symmetric cancellation i. c. local accuracy an exact method on q degree polynomials has a local error that scales like 2 k Oh + in each subinterval where h is the length of the subinterval d. global accuracy since there are ( ) 1 subintervals, the total error scales like 1 k + i. doubling the number of subintervals, sends 2 hh , and reduces the error by (½) q+1 ii. order of accuracy is q+1 e.
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CS 205A class_14 notes - CS205 Class 14 Covered in class:...

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