Page - 39 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

2.3 ForLoops 39 will contain the largest number from the arrayy. Whenyou run the program, you get The largest height achieved was 1.274210 m whichcompares favorably to theplot thatpopsup. To implement the traversingofarrayswith loopsandindices, is sometimeschal- lenging toget right. Youneed tounderstand the start, stop and step length choices foranindex,andalsohowtheindexshouldenterexpressionsinsidethe loop.At the same time, however, it is something that programmers dooften, so it is important todevelop the right skills on thesematters. Havingoneloopinsideanother,referredtoasadoubleloop, issometimesuseful, e.g., when doing linear algebra. Say we want to find the maximum among the numbersstored ina4 4matrixA.Thecode fragmentcould look like largest_number = A[0][0] for i in range(4): for j in range(4): if A[i][j] > largest_number: largest_number = A[i][j] Here, all thej indices (0 - 3)will be covered for eachvalue of indexi. First, i staysfixedati = 0,whilej runsoverall its indices. Then,i staysfixedati = 1 whilej runsover all its indices again, and soon. SketchAonapieceofpaper and follow the first few loop iterations by hand, then youwill realize how the double loopconstructionworks.Using two loops is just a special caseofusingmultipleor nested loops, andutilizingmore than two loops is just a straightforward extension ofwhatwasshownhere.Note,however, that the loopindexname inmultiple loops must be unique to each of the nested loops. Note also that each nested loopmay haveasmanycode linesasdesired,bothbeforeandafter thenext inner loop. Thevectorizedcomputationofheights thatwedid inball_plot.py(Sect.1.4) could alternatively havebeendoneby traversing the timearray (t) and, for eacht element, computingtheheightaccording to the formulayDv0t 12gt2. However, it is important to know that vectorization goesmuch quicker. So when speed is important,vectorization isvaluable. Useloopstocomputesums One important useof loops, is to calculate sums. Asa simple example, assume somevariablexgivenby themathematical expression xD NX iD1 2 i; i.e., summing up theN first even numbers. For somegivenN , sayN D 5,x would typicallybecomputed inacomputerprogramas: