Page - 71 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

3.4 Testing 71 Theseare twoequations for twounknownsC and r.Wecaneasily eliminateC by dividing theequationsbyeachother. Thensolving forr gives ri 1 D ln.Ei=Ei 1/ ln.ni=ni 1/ : (3.24) We have introduced a subscript i 1 in r since the estimated value for r varies with i. Hopefully, ri 1 approaches the correct convergence rate as the number of intervals increasesand i !q. 3.4.3 FinitePrecisionofFloating-PointNumbers The test procedures above lead to comparisonof numbers for checking that calcu- lationswere correct. Suchcomparison ismorecomplicated thanwhat anewcomer might think. Supposewehaveacalculationa + bandwant tocheckthat theresult iswhatweexpect.Westartwith1+2: >>> a = 1; b = 2; expected = 3 >>> a + b == expected True Thenweproceedwith0.1+0.2: >>> a = 0.1; b = 0.2; expected = 0.3 >>> a + b == expected False Sowhyis0:1C0:2¤0:3?Thereason is that realnumberscannot ingeneralbe exactly representedon a computer. Theymust insteadbe approximatedbyafloat- ing-pointnumber3 that canonlystoreafinite amountof information,usuallyabout 17digitsofa realnumber. Letusprint0.1,0.2,0.1+0.2,and0.3with17decimals: >>> print ’%.17f\n%.17f\n%.17f\n%.17f’ % (0.1, 0.2, 0.1 + 0.2, 0.3) 0.10000000000000001 0.20000000000000001 0.30000000000000004 0.29999999999999999 Wesee that all of the numbers have an inaccurate digit in the 17th decimal place. Because 0.1C 0.2 evaluates to 0.30000000000000004 and 0.3 is represented as 0.29999999999999999, these twonumbersarenot equal. Ingeneral, real numbers inPythonhave (atmost)16correctdecimals. When we compute with real numbers, these numbers are inaccurately repre- sented on the computer, and arithmetic operations with inaccurate numbers lead to small rounding errors in the final results. Depending on the type of numerical algorithm, the roundingerrorsmayormaynotaccumulate. 3https://en.wikipedia.org/wiki/Floating_point