Re: Simple question or how Mathematica getting on my nerves.

*To*: mathgroup at smc.vnet.net*Subject*: [mg45796] Re: Simple question or how Mathematica getting on my nerves.*From*: Harold.Noffke at wpafb.af.mil (Harold Noffke)*Date*: Sun, 25 Jan 2004 03:04:50 -0500 (EST)*References*: <butdvt$9se$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

George: I downloaded your Notebook entries, and have examples which may save you some hair. You are getting a non-trivial difference between your two integrals because you are "comparing apples to oranges". The example I use below illustrates this point. However, it will not answer all your questions; nor will it answer all my own. Perhaps a good number theorist can comment on the example's mathematical background. So, here goes. When worried about accuracy in Mathematica calculations, it is better to keep every number in "whole number fractional notation" -- that is, as the ratio of two whole numbers. You went from "apples" to "oranges" when you changed 5242/10000 and 6214/10000 to 0.5242 and 0.6214, respectively. If you read "The Mathematica Book > A Practical Introduction to Mathematica > 1.1.4 Arbitrary Precision Calculations", you will find some examples which explain the difference between apple-fractions and orange-decimals. This is the source material I used here. First, we define your "k-function" in TextForm ... In[1]:= k[f_] := 2687176093959399272413585923303421161600 * (1 - f)^67 * f^61 Then, after reading Mathematica Book Section 1.1.4, we determine the accurate fractional equivalent of 0.5242 and 0.6214. In[2]:= u = SetPrecision[ 0.5242, Infinity] 1180393462333807 Out[2]= ---------------- 2251799813685248 In[3]:= l = SetPrecision[ 0.6214, Infinity] 1399268404224013 Out[3]= ---------------- 2251799813685248 Now, we calculate np (where "p" denotes "precise"), the value obtained from using the precise fractional equivalents above. In[4]:= np = Integrate[ k[f], {f, l, u}] Out[4]= -(163 ... / ... 848) <-- Multiline whole-number fraction abbreviated here. Next, we let u and v be the fractions you specified. In[5]:= u = 5242/10000 2621 Out[5]= ---- 5000 In[6]:= l = 6214/10000 3107 Out[6]= ---- 5000 Now, we calculate nf (where "f" denotes your original fractions). In[7]:= nf = Integrate[ k[f], {f, l, u}] Out[7]= -(321 ... / ... 000) <-- Multiline whole-number fraction abbreviated here. Now that we have derived two "apples" answers, we can determine how close they are. In[8]:= Abs[ np - nf] // N -18 Out[8]= 1.87048 10 So, np and nf differ by about 2 parts in 10^18, which is probably good enough for government work. :>) Harold