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 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 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 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 , and reduces the error by (½) q+1 ii.
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  • Fall '07
  • Fedkiw
  • Numerical Analysis, Degree of a polynomial, Piecewise linear function, xi xi

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