Page - 50 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

50 2 BasicConstructions b) Write another functionwith a loopwhere the user is asked for a time on the interval Œ0;N and the corresponding (interpolated) y value is written to the screen. The loop is terminatedwhen theusergivesanegative time. c) Use the following measurements: 4:4;2:0;11:0;21:5;7:5, corresponding to times0;1;:: :;4 (min),andcompute interpolatedvaluesat t D2:5and t D3:1 min. Perform separate hand calculations to check that the output from the programis correct. Filename:linear_interpolation.py. Exercise2.16:Test straight linerequirement Assume the straight line function f.x/ D 4xC 1. Write a script that tests the “point-slope” form for this line as follows. Within a chosen interval on thex-axis (for example, forx between 0 and 10), randomlypick 100 points on the line and check if the followingrequirement is fulfilled foreachpoint: f.xi/ f.c/ xi c Da; i D1;2;:: :;100; wherea is theslopeof the lineandcdefinesafixedpoint .c;f.c//onthe line. Let cD2here. Filename:test_straight_line.py. Exercise2.17:Fit straight line todata Assume somemeasurements yi;i D 1;2;:: :;5 have been collected, once every second.Your task is towrite aprogramthatfits a straight line to thosedata. a) Makeafunctionthatcomputestheerrorbetweenthestraight linef.x/DaxCb and themeasurements: eD 5X iD1 .axi Cb yi/2 : b) Makeafunctionwitha loopwhereyougiveaandb, thecorrespondingvalueof e iswritten to thescreen,andaplotof thestraight linef.x/DaxCb together with thediscretemeasurements is shown. Hint Tomake the plotting from the loop to work, youmay have to insert from matplotlib.pylab import * at the topof the script and also addshow()after theplot command in the loop. c) Giventhemeasurements0:5;2:0;1:0;1:5;7:5,at times0;1;2;3;4,use thefunc- tion inb) to interactivelysearch fora andb such thate isminimized. Filename:fit_straight_line.py. Remarks Fittingastraight line tomeasureddatapoints isaverycommontask. The manual search procedure in c) can be automated by using amathematicalmethod called themethod of least squares.