Page - 173 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

6.8 Exercises 173 Minimization of a function of N variables, E(b1,.. .,bN) is mathematically performedbyrequiringall thepartialderivatives to bezero: ∂E ∂b1 =0, ∂E ∂b2 =0, ... ∂E ∂bN =0 . a) Compute the partial derivative ∂E/∂b1 and generalize to the arbitrary case ∂E/∂bn, 1≤n≤N. b) Showthat bn= 1 π ∫ π −π f(t)sin(nt)dt . c) Write a functionintegrate_coeffs(f, N, M) that computesb1,.. .,bN by numerical integration,usingM intervals in the trapezoidal rule. d) A remarkable property of the trapezoidal rule is that it is exact for inte- grals ∫π −π sinntdt (when subintervals are of equal size). Use this property to create a function test_integrate_coeff to verify the implementation of integrate_coeffs. e) Implement the choice f(t) = 1 π t as a Python function f(t) and call integrate_coeffs(f, 3, 100) to see what the optimal choice ofb1,b2,b3 is. f) Make a function plot_approx(f, N, M, filename)where you plot f(t) together with the best approximationSN as computed above, usingM intervals fornumerical integration.Save theplot toa file withnamefilename. g) Runplot_approx(f, N, M, filename)forf(t)= 1 π t forN =3,6,12,24. Observehowtheapproximationimproves. h) Run plot_approx forf(t) = e−(t−π) andN = 100. Observe a fundamental problem: regardless ofN,SN(−π)= 0, note2π ≈ 535. (There are ways to fix this issue.) Filename:autofit_sines.py. Exercise6.13:Derive the TrapezoidalRule foraDouble Integral Use ideas in Sect.6.7.1 to derive a formula for computing a double integral∫b a ∫d c f(x,y)dydxby the trapezoidal rule. Implementand test this rule. Filename:trapezoidal_double.py.