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| -rw-r--r-- | README.md | 8 |
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@@ -1,6 +1,8 @@ +# discrete-mf + For functions f on the discrete circle {0,…,N-1} this program computes their uncentered maximal functions and the ratio of the L<sup>p</sup> norms of the kth derivative of the maximal function and the function. -# Building +## Building If `git` is installed you can fetch the repository using @@ -20,7 +22,7 @@ builds three files: - `charf_error` does computations using floating point (`double`) numbers and gives an upper bound for the total rounding error. - `charf_exact` does exact computations using fractions. Nominator and denominator are of type `unsigned long long` and bounded in size accordingly. -# Computation +## Computation By default, the program goes through all characteristic functions on circles from length 2 to 36, considers derivatives from order 0 to 64 and exponents p = 1,2,4,8,∞. Computing the maximal function of a function is the most computationally complex part. This means it is time efficient to consider several orders of derivative and exponents at the same time, in particular since the (k+1)th derivative is computed using the kth derivative. @@ -28,7 +30,7 @@ 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. -# Error bounds +## Error bounds 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: |