Seite - 100 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Band Second Edition

100 4 FunctionsandtheWritingofCode 2. Find a mathematical expression for the straight line that goes through the points (i,yi)and (i+1,yi+1). 3. Compute they value by inserting the user’s time value in the expression for the straight line. a) Implement the linear interpolation technique in a function interpolate that takesas inputanarraywith theyi measurements, the timebetweenthemΔt, and some time t, for which the interpolated value is requested. The function should return the interpolatedy valueat time t. b) Write another functionfind_y that finds and prints an interpolated y value at times requested by the user. Let find_yuse a loop in which the user is asked for a time on the interval [0,N]. The loop can terminate when the user gives a negative time. c) Use the following measurements: 4.4,2.0,11.0,21.5,7.5, corresponding to times 0,1,.. .,4 (min), and compute interpolatedvaluesat t = 2.5 and t = 3.1 min. Perform separate hand calculations to check that the output from the programiscorrect. Filename:linear_interpolation.py. Exercise4.11:Test StraightLine Requirement Assumethestraight linefunctionf(x)=4x+1.Writeascript that tests the“point- slope” form for this line as follows. Within a chosen interval on the x-axis (for example, forx between 0 and 10), randomlypick 100 pointson the line and check if the followingrequirement is fulfilled foreachpoint: f(xi)−f(c) xi−c =a, i=1,2,.. .,100, wherea is theslope of the lineandcdefinesa fixedpoint (c,f(c))on the line. Let c=2here. Filename:test_straight_line.py. Exercise4.12:FitStraightLine toData Assume some measurements yi,i = 1,2,.. .,5 have been collected, once every second.Your task is to writea programthatfits a straight line to thosedata. a) Make a function that, for given measurements and parameter values a and b, computes the error between the straight line f(x) = ax + b and the measurements: e= 5∑ i=1 (axi+b−yi)2 . b) Make a function that, in a loop, asks the user to give a and b for the line. The corresponding value of e should then be computed and printed to screen, and a plotof thestraight linef(x)=ax+b, togetherwith thediscretemeasurements, shouldbeproduced.