- Matrix and Vector operations
- In a routine for reducing a symmetric matrix to tridiagonal matrix (believe me, there is a reason why I couldn’t use one of the numerous BLAS packages)
- Comparing real matrices for equality overload operator
- Comparing coefficients of two polynomials for solving ARIMA equations in time-series forecasting
- Calculating approximate correlation coefficients for groups of variables
- Coding Quasi-Newton methods (specially for solving box-constrained problems)

__Absolute and Relative Comparison__

For starters, I have been using absolute comparison with constant epsilon value (1E-15), and it has been working pretty well. The code is simple, and looks like below:

public static bool AreEqual(double a, double b, double EPS)

{

if ((a < 0 && b > 0) || (a > 0 && b < 0))

return false;

else

return (Math.Abs(a - b) < EPS);

}

Peter, however, mentions: “Any fixed epsilon would likely be too small when comparing large numbers, or too large when comparing small numbers.” That is indeed true, and I generally get by with passing separate epsilon value for other situations. The relative version of the method, from Peter’s post, looks like this:

public static bool AlmostEqual(double a, double b)

{

if (a == b)

return true;

else

return Math.Abs(a - b) < Math.Abs(a) / 281474976710656.0;

}

For an explanation about the constant, see the original post. As is obvious, this won’t work well with numbers close to zero. There are other schemes, where one calculates (a-b)/a, or (a-b)/b, or (a-b)/max(a,b) and compares with an epsilon.

__Using Double.Epsilon__

Note, however, that using Double.Epsilon available in .NET as the epsilon value is not a good idea in these comparisons. Double.Epsilon is only useful if one is subtracting values between 0 and 1 as it's the "smallest positive Double value greater than zero". Since double values aren't evenly spaced and their distance increases with their magnitude, one won't ever get something around Double.Epsilon if one is subtracting two arbitrary values.

__Other Methods__

Recently, I came across another interesting post which shows how to compare two floats (and by extension, doubles) using integers. If you are interested, definitely check this out. Another suggestion that has been floated in these discussions, is to use arbitrary precision arithmetic (APA), but in most applications, doing everything in APA would be really slow. An example of the use of APA is Microsoft Windows Calculator (the calc.exe program).

__End Word__

One sentiment that repeatedly seems to appear in posts related to this topic is that “the chosen method would really depend on what one would consider equal”. It is definitely application specific, and with enough spread in test cases, a wrong choice of method would show up in wrong result (or, so we hope !!).

__References__

Following are two really good references I have found on this topic:

[1] Jon Skeet’s coverage for floats and doubles and decimals

[2] Bruce Dawson’s coverage

Update (March 5th, 2017): Updated dead link.

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