Page - 76 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

76 3 LoopsandBranching Exercise3.9:Simple Search:Verification Check your understandingof the search procedure inball_max_height.pyfrom Sect.3.3.3 by comparingwhat you get “by hand” to printouts from the code. Work on a copy ofball_max_height.py. Comment out what you do not need, and use an arrayyof just 4 elements (or so). Fill that array with integers, so that you place a maximum value in a certain location. Then, run through that code by hand for everyiterationof the loop,writingdownthenumbersinlargest_height.Finally, placeaprintcommandintheloop,so thatlargest_heightgetsprintedwithevery iteration.Run theprogramandcompare to whatyoufoundbyhand. Filename:simple_search_verify.py. Exercise3.10:SortArraywithNumbers Write a script that uses theuniform functionfromtherandommodule to generate anarrayof6 randomnumbersbetween0and10. The program should then sort the array so that numbers appear in increasing order.Let theprogrammakeaformattedprintof thearray toscreenbothbeforeand after sorting.Confirmthat the arrayhasbeensortedcorrectly. Filename:sort_numbers.py. Exercise3.11:Computeπ Up through history, great minds have developed different computational schemes for the numberπ. We will here consider two suchschemes, oneby Leibniz (1646– 1716),andonebyEuler (1707–1783). TheschemebyLeibnizmaybe written π =8 ∞∑ k=0 1 (4k+1)(4k+3), whileoneformof the Euler schememayappearas π= √√ √ √6 ∞∑ k=1 1 k2 . If only the first N terms of each sum are used as an approximation to π, each modifiedschemewill havecomputedπ withsomeerror. Write a program that takes N as input from the user, and plots the error development with both schemes as the number of iterations approaches N. Your programshouldalso print out the final errorachievedwith bothschemes, i.e. when the numberof terms isN. Run the programwithN = 100 and explain briefly what thegraphsshow. Filename:compute_pi.py.