MACM 316 Lecture Notes - Lecture 12: Mean Value Theorem, Richardson Extrapolation, Machine Epsilon

• School: Simon Fraser University
• Department: Math & Comp Sci
• Course: MACM 316
• Professor: Sarah Huber

Document Summary

Usually, we do not want to use a newton-cotes rule directly on an interval [a, b], where b a is not small. High accuracy on such an interval would require a high degree rule. But we"ve seen that high degree polynomial interpolants on equally-spaced nodes tend to lead to poor approximations. We can avoid these di culties by subdividing the interval [a, b] into smaller pieces and applying the same low degree quadrature rule to each piece, then summing the results. Set x0 = a, xj = x0 + jh, h = (b a)/n. We apply the trapezoidal rule to each subinterval xj, xj+1. Thus, f (x)dx (f (xj) + f (xj+1) h3. Z b a f (x)dx = xj n 1. Xj=0 max x [a,b] f (x) = max x [a,b] f (x) f ( j) min x [a,b] f (x) The mean value theorem gives f ( j) = f ( )