Page - 153 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

6.6 TestingCode 153 These are two equations for two unknownsC and r. We can easily eliminateC by dividing,e.g., (6.26)by(6.27).Doingso,andproceedingtosolveforr, followedby also introducingasubscript i−1for r, gives ri−1 =−ln(Ei/Ei−1) ln(ni/ni−1) . (6.28) The subscript is introduced, since the estimated value for r will vary with i. Hopefully,ri−1 approaches the correct convergencerate as the number of intervals increasesand i→q. 6.6.3 FinitePrecisionofFloating-PointNumbers The test procedures above lead to comparison of numbers for checking that calculations were correct. Such comparison is more complicated than what a newcomer might think. Suppose we have a calculation a + b and want to check that the result iswhat weexpect. AddingIntegers We start with1+ 2: In [1]: a = 1; b = 2; expected = 3 In [2]: a + b == expected Out[2]: True AddingRealNumbers Thenwe proceedwith 0.1+ 0.2: In [3]: a = 0.1; b = 0.2; expected = 0.3 In [4]: a + b == expected Out[4]: False ApproximateRepresentationofRealNumbersonaComputer Sowhyis0.1+ 0.2 =0.3?Thereasonis that,generally,realnumberscannotberepresentedexactly on a computer. They must instead be approximated by a floating-point number5 that can only store a finite amount of information,usually about 17 digits of a real number.Let usprint0.1,0.2,0.1+0.2, and0.3with17decimals: In [5]: print(’{:.17f}\n{:.17f}\n{:.17f}\n{:.17f}’\ .format(0.1, 0.2, 0.1 + 0.2, 0.3)) 0.10000000000000001 0.20000000000000001 0.30000000000000004 0.29999999999999999 We see that all of the numbers have an inaccurate digit in the 17th decimal place. Because 0.1 + 0.2 evaluates to 0.30000000000000004 and 0.3 is represented as 0.29999999999999999, these two numbers are not equal. In general, real numbers inPythonhave (atmost)16correctdecimals. 5https://en.wikipedia.org/wiki/Floating_point.