Page - 91 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

3.8 Exercises 91 The integrandxx doesnothaveananti-derivative that canbeexpressed in termsof standard functions (visit http://wolframalpha.comand typeintegral(x**x,x) to convince yourself that our claim is right. Note thatWolframalpha does give you an answer, but that answer is an approximation, it is not exact. This is because Wolframalpha toousesnumericalmethods to arriveat the answer, just asyouwill in this exercise). Therefore, we are forced to compute the integral by numerical methods.Computea result that is right to fourdigits. Hint Use ideas fromExercise3.9. Filename:integrate_x2x.py. Exercise3.11: Integrateproductsof sine functions In this exerciseweshall integrate Ij;k D Z sin.jx/sin.kx/dx; wherej andk are integers. a) Plot sin.x/sin.2x/ and sin.2x/sin.3x/ forx 2 ; in separate plots. Ex- plainwhyyouexpect R sinxsin2xdxD0and R sin2xsin3xdxD0. b) Use the trapezoidal rule tocomputeIj;k forj D1;:: :;10andkD1;:: :;10. Filename:products_sines.py. Exercise3.12:Revisitfitof sines toa function This is acontinuationofExercise2.18. The task is toapproximateagivenfunction f.t/on Œ ; byasumofsines, SN.t/D NX nD1 bnsin.nt/: (3.27) We are now interested in computing the unknowncoefficients bn such thatSN.t/ is in some sense the best approximation tof.t/. One commonway of doing this is to first set up a general expression for the approximation error, measured by “summingup” thesquareddeviationofSN fromf : E D Z .SN.t/ f.t//2dt : We may view E as a function of b1;:: :;bN. Minimizing E with respect to b1;:: :;bN willgiveusabestapproximation, in thesense thatweadjustb1;:: :;bN such thatSN deviatesas little aspossible fromf .