Page - 172 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

172 6 ComputingIntegralsandTestingCode exercise). Therefore, we are forced to compute the integral by numerical methods. Computea result that is right to fourdigits. Hint Use ideas fromExercise6.9. Filename:integrate_x2x.py. Exercise6.11:IntegrateProductsofSine Functions In this exercisewe shall integrate Ij,k = ∫ π −π sin(jx)sin(kx)dx, wherej andk are integers. a) Plotsin(x)sin(2x)andsin(2x)sin(3x)forx∈[−π,π] inseparateplots.Explain whyyouexpect ∫π −π sinxsin2xdx=0and ∫π −π sin2xsin3xdx=0. b) Use the trapezoidal rule tocomputeIj,k forj =1,.. .,10andk=1,.. .,10. Filename:products_sines.py. Exercise6.12:RevisitFitofSines toa Function This is a continuationof Exercise4.13.The task is to approximatea given function f(t)on [−π,π]byasum ofsines, SN(t)= N∑ n=1 bnsin(nt). (6.31) We are nowinterested incomputing theunknowncoefficientsbn such thatSN(t) is in somesense the best approximation tof(t). One commonway of doing this is to firstsetupageneralexpressionfortheapproximationerror,measuredby“summing up” thesquareddeviationofSN fromf : E= ∫ π −π (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 .