Page - 127 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

5.7 Exercises 127 larger theq, thequicker theerrorgoes tozeroas thenumberof iterations (n)grows (whenen < 1). With the given error model, we may compute the convergencerate from q= ln(en+1/en) ln(en/en−1) . This is derived by considering the error model for three consecutive iterations, dividingoneequationby the otherandsolving forq. If then a seriesof iterations is run,wecancomputeasequenceofvalues forq as the iterationcountern increases. As n increases, the computed q values are expected to approach the convergence rate that characterizes the particular iterative method. For the ratio we are looking athere, theconvergenceratio is1. Extend your module with a functioncompute_rates, which takes an array (or a list) F with (e.g., 20) Fibonacci numbers as input and computes (and prints) the corresponding values for q. Call the function from the test block and run the program.Do theconvergenceratesapproach theexpectedvalue? Later, in Sect.6.6.2, you will learn that convergence rates are very useful when testing (verifying)software. Filename:Fibonacci_numbers.py. Exercise5.5:ReadFile:TotalVolume ofBoxes A file box_data.dat contains volume data for a collection of rectangular boxes. These boxes all have the same bottom surface area, but (typically) differ in height. Thefile could, forexample, read: Volume data for rectangular boxes 10.0 3.0 4.0 2.0 3.0 5.0 Apart from the header, each line represents one box. However, since they all have the same bottom surface area, that area (10.0) is only given for the first box. For that first box, also the height (3.0) is given, as it is for each of the following boxes. a) Writedownaformulaforcomputingthe totalvolumeofallboxesrepresentedin the file. That formula should be written such that a minimum of multiplications andadditions isused. b) Write a program that reads the filebox_data.dat, computes the total volume of all boxes represented in the file, and prints that volume to the screen. In the calculations,apply the formula just derived. (Note that, as afirst step,youmayread thefile and just print (to screen)what is read.Comparingthisprintoutwithfilecontent(usesomeeditor) is thenagood idea.) c) In the filebox_data.dat, after the last line (containing the height of the “last” box), insert a coupleof empty lines, i.e. just press enter a few times. Then, save thefile and run theprogramanew.Whathappens?Explainbriefly.