Page - 101 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

4.3 Exercises 101 c) Given the measurements 0.5,2.0,1.0,1.5,7.5, at times 0,1,2,3,4, use the function inb) to interactivelysearchfora andb such thate isminimized. Filename:fit_straight_line.py. Remarks Fittingastraightlinetomeasureddatapointsisaverycommontask.The manual search procedure in c) can be automated by using a mathematical method called themethodof least squares. Exercise4.13:FitSines toStraightLine A lot of technology, especially most types of digital audio devices for processing sound, is based on representinga signal of time as a sum of sine functions.Say the signal is some functionf(t)on the interval [−π,π] (a moregeneral interval [a,b] can easily be treated, but leads to slightly more complicated formulas). Instead of workingwithf(t)directly,we approximatef by thesum SN(t)= N∑ n=1 bnsin(nt), (4.1) where thecoefficientsbn mustbeadjustedsuch thatSN(t) is agoodapproximation tof(t). We shall in this exerciseadjustbn bya trial-and-errorprocess. a) Makeafunctionsinesum(t, b) that returnsSN(t), giventhecoefficientsbn in anarraybandtimecoordinatesinanarrayt.Note that ift is anarray, thereturn value is alsoanarray. b) Write a functiontest_sinesum() that calls sinesum(t, b) in a) and deter- minesif thefunctioncomputesa testcasecorrectly.Astestcase, lettbeanarray with values−π/2 andπ/4, chooseN = 2, andb1 = 4 andb2 =−3. Compute SN(t)byhand toget referencevalues. c) Makeafunctionplot_compare(f, N, M) thatplots theoriginalfunctionf(t) together with the sum of sines SN(t), so that the quality of the approximation SN(t) can be examined visually. The argument f is a Python function imple- mentingf(t),N is thenumberof termsin thesumSN(t), andM is thenumberof uniformlydistributed t coordinatesused toplotf andSN. d) Write a functionerror(b, f, M) that returns a mathematical measure of the error inSN(t)asanapproximationtof(t): E= √∑ i (f(ti)−SN(ti))2, where the ti values are M uniformly distributed coordinates on [−π,π]. The arraybholds thecoefficients inSN andf is aPythonfunction implementing the mathematical functionf(t). e) Make a function trial(f, N) for interactively giving bn values and getting a plot on the screen where the resulting SN(t) is plotted together with f(t). The error in the approximationshould also be computedas indicated in d). The