Skip to main content

numerics - Machine Epsilon


I'm trying to evaluate the machine epsilon of my computer (see below). I wrote this:


eps = 1.0;
p = 0;
While[(1.0 + eps) > 1.0, eps = eps/2.0; p += 1];
eps
p

eps*2

and got:


1.42109*10^-14
46
2.84217*10^-14

which is not what I expected (IEEE754). So I did this


EngineeringForm[1.0 + 2 * eps, 20]
EngineeringForm[1.0 + eps, 20]

EngineeringForm[1.0 + (eps/2.0), 20]
EngineeringForm[1.0 + (eps/4.0), 20]
EngineeringForm[1.0 + (eps/16.0), 20]
EngineeringForm[1.0 + (eps/32.0), 20]

and got that:


1.000000000000028
1.000000000000014
1.000000000000007
1.000000000000004

1.000000000000001
1.

which seems to contradict the termination condition of the 'while' loop. So I would like to understand what's going on under Mathematica's hood. I tried to look for some information on Mathematica's internal float representation without success. Any help to explain these results is welcome. I'm sorry if my question is a bit vague.


Eric


Context:


SystemInformationData[{"Kernel" ->
{"Version" -> "9.0 for Mac OS X x86 (64-bit) (January 24, 2013)",
"ReleaseID" -> "9.0.1.0 (4055646, 4055073)", ...}]


EDIT


It was suggested the following program that works directly for some reason:


eps = 1.0;
n = 0;
While[(1.0 + eps) - 1.0 > 0.0,
eps = eps/2;
n++;
];
eps*2
n

Log[2, eps*2]

Answer



There is a tolerance Internal`$EqualTolerance that is applied when testing Real numbers. If the numbers agree up to the last Internal`$EqualTolerance digits, then they are treated as equal. Try this:


eps = 1.0;
p = 0;
Block[{Internal`$EqualTolerance = 0.},
While[(1.0 + eps) > 1.0, eps = eps/2.0; p += 1];
]
eps
p

eps*2
(*
1.11022*10^-16
53
2.22045*10^-16
*)

See How to make the computer consider two numbers equal up to a certain precision and also this answer by Alexey Popkov.


Comments

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...