Page - 48 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

48 2 BasicConstructions Exercise2.11:Compute Up through history, greatminds have developed different computational schemes for thenumber .Wewillhereconsider twosuchschemes,onebyLeibniz(1646 1716), andonebyEuler (1707 1783). TheschemebyLeibnizmaybewritten D8 1X kD0 1 .4kC1/.4kC3/; whileone formof theEuler schememayappearas D vu u t6 1X kD1 1 k2 : If only the firstN terms of each sum are used as an approximation to , each modifiedschemewill havecomputed with someerror. Writeaprogramthat takesN as input fromtheuser, andplots theerrordevelop- mentwith both schemes as thenumber of iterations approachesN . Your program shouldalsoprintout thefinalerrorachievedwithbothschemes, i.e.when thenum- ber of terms is N. Run the programwithN D 100 and explain brieflywhat the graphsshow. Filename:compute_pi.py. Exercise2.12:Computecombinationsof sets ConsideranIDnumberconsistingof two lettersand threedigits, e.g.,RE198.How many different numbers canwe have, and how can a program generate all these combinations? Ifacollectionofn thingscanhavem1variationsof thefirst thing,m2of thesec- ondandsoon, the total numberofvariationsof the collectionequalsm1m2 mn. Inparticular, theIDnumberexemplifiedabovecanhave26 26 10 10 10D676;000 variations. To generate all the combinations, wemust have five nested for loops. Thefirst two runoverall lettersA,B, andsoon toZ,while thenext three runover all digits0;1;:: :;9. To convinceyourself about this result, start outwith an IDnumberon the form A3where thefirst part canvaryamongA,B, andC, and thedigit canbeamong1, 2, or 3. Wemust startwithAand combine itwith 1, 2, and3, then continuewith B,combinedwith1,2, and3, andfinallycombineCwith1,2, and3.Adouble for loopdoes thework. a) Inadeckofcards, eachcard is acombinationofa rankandasuit. Thereare13 ranks: ace (A), 2, 3, 4, 5, 6, 7, 8, 9, 10, jack (J), queen (Q), king (K), and four suits: clubs (C), diamonds (D), hearts (H), and spades (S).A typical cardmay beD3.Write statements thatgenerateadeckofcards, i.e., all thecombinations CA,C2,C3, and soon toSK. b) Avehicle registrationnumber ison the formDE562,where the lettersvaryfrom AtoZandthedigits from0to9.Writestatements thatcomputeall thepossible registrationnumbersandstores themina list.