MATH 405 Study Guide - Midterm Guide: Von Neumann Stability Analysis, Spline Interpolation, Leapfrog Integration
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Find a natural cubic spline interpolant, s(x), for a function sin(x) over the interval x (0, ). You are given three data points xk = {0, /2, } and corresponding three function values sin(xk) = {0, 1, 0}. Once you nd the spline function, evaluate it at x = /4, and compare the answer to sin( /4) = 1/ 2. Estimate the accuracy of the spline approximation by calculating m in |s( /4) 1/ 2| = o(10 m). Hint: a2 b2 = (a b)(a+b), 2 1. 4: numerical integration. Use the fact that gauss-legendre quadrature with n points allows to calculate the integrals of polynomials of degree 2n 1 exactly to nd the locations and weights for n = 2. Find the formula for calculating the n dimensional integral over n dimensional hypercube (i. e. n integrals with identical -1 to 1 limits) using n gauss points. Hint: use 1d integration recursively: initial value ode.
