Page - 52 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

52 2 BasicConstructions f) Choosef.t/ to be a straight line f.t/ D 1 t on Œ ; . Call trial(f, 3) and try tofind throughexperimentationsomevaluesb1,b2, andb3 such that the sumofsinesSN.t/ is agoodapproximation to the straight line. g) Nowwe shall try to automate the procedure in f). Write a function that has threenested loopsovervaluesofb1,b2, andb3. Leteach loopcover the interval Œ 1;1 in steps of 0.1. For each combinationofb1, b2, andb3, the error in the approximationSN shouldbecomputed.Use this tofind, andprint, the smallest error and the correspondingvaluesofb1,b2, andb3. Let the programalso plot f and theapproximationSN corresponding to thesmallest error. Filename:fit_sines.py. Remarks 1. The function SN.x/ is a special case of what is called a Fourier series. At thebeginningof the19th century, JosephFourier (1768-1830)showed that any function can be approximated analytically by a sumof cosines and sines. The approximationimprovesas thenumberof terms(N) is increased. Fourier series arevery important throughoutscienceandengineering today. (a)Finding thecoefficientsbn is solvedmuchmoreaccurately inExercise3.12, byaprocedure that also requiresmuch lesshumanandcomputerwork! (b)In real applications, f.t/ is not known as a continuous function, but func- tionvaluesoff.t/areprovided. Forexample, indigital soundapplications, music in aCD-qualityWAVfile is a signalwith 44,100 samples of the cor- respondinganalogsignalf.t/per second. Exercise2.19:Countoccurrencesofa string inastring In theanalysisofgenesoneencountersmanyproblemsettings involvingsearching for certain combinations of letters in a long string. For example, we may have astring like gene = ’AGTCAATGGAATAGGCCAAGCGAATATTTGGGCTACCA’ Wemaytraverse this string, letterby letter, by the for loopfor letter in gene. The lengthof the string is givenbylen(gene), so an alternative traversal over an index i is for i in range(len(gene)). Letter number i is reached through gene[i], and a substring from index i up to, but not including j, is created by gene[i:j]. a) Write a functionfreq(letter, text) that returns the frequencyof the letter letter in the string text, i.e., the number of occurrences ofletterdivided by the lengthoftext. Call the function todetermine the frequencyofC andG in thegenestringabove.Compute the frequencybyhand too. b) Write a function pairs(letter, text) that counts howmany times a pair of the letterletter (e.g.,GG) occurswithin the stringtext. Use the function to determine howmany times the pair AA appears in the string gene above. Performamanualcounting too tocheck theanswer.