Expand readme on error bounds.

1 files changed, 5 insertions, 2 deletions

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@@ -28,6 +28,9 @@ Computing the maximal function of a function is the most computationally complex
 The program continuously outputs whenever it finds a function that beats the last record for the largest ratio of the L<sup>p</sup> norm of the kth derivative of the maximal function and the function.
 It is also possible to print the results in human readable format and as a latex table.
 
-# Correctness
+# Error bounds
 
-I am relatively confident that the computed error bounds are correct for all operations except exponentiation, which is not used for p ∈ {1,∞}.
+Even for a characteristic function that only take values 0 or 1, its maximal function may not be integer valued. The most widespread way to do non integer arithmetic on a computer is floating point arithmetic. This however is not exact, so results cannot be trusted a priori. This program implements two alternative ways to make results reliable:
+
+1. Interval arithmetic: To each floating point variable we assign an confidence interval, in which we know the true value has to lie. In this model arithmetic operations act on pairs of floating point numbers: the estimated value and the length of the confidence interval. I am relatively confident that the computed confidence intervals are correct for all operations except exponentiation, which is not used for p = 1,∞.
+2. Rational numbers: For integer and even rational valued functions also the maximal function is rational valued. Thus, by storing values as fractions the maximal function and all derivatives can be computed exactly, and so can L<sup>p</sup> norms for p = 1,∞. The downside of this method is that nominator and denominator quickly become too large to be stored in standard integer formats.