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Results tagged with na.numerical-analysis
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user 14830
Numerical algorithms for problems in analysis and algebra, scientific computation
27
votes
Accepted
show that $ \frac{\Gamma(\frac{1}{24})\Gamma(\frac{11}{24})}{\Gamma(\frac{5}{24})\Gamma(\fra...
This formula can actually be proved using only properties of the Gamma
function already known to Gauss, with no need to invoke special values of
Dirichlet series. The relevant identities are
$$
\Gamm …
- 79.9k
9
votes
Accepted
Alternating binomial Dirichlet series
Does this reduce to values of a known special function for arbitrary real (or complex) $s$?
Answered by Johannes Trost in a comment: it's also known as a
"Roman harmonic number". But this …
- 79.9k
11
votes
Accepted
Sparse approximation of the inverse of a sparse matrix
Not in general. An explicit and elementary counterexample
is the sparse triangular matrix with $1$'s on the diagonal
and $-1$'s just above it: the inverse is the triangular matrix with
every entry …
- 79.9k
5
votes
Accepted
Approximating high-dimensional integrals by low-dimensional ones
Here's how to average $e^{sx}$ over $A_t$. Let $r = t/2$, so at the $N$-th step of the construction of $A_t$ we have the disjoint union $A_t^{(N)}$ of $2^N$ intervals of length $r^N$ whose left endpo …
- 79.9k
5
votes
Well-balanced covering of transpositions in $n$ elements
$R_n \geq n/16$ can be obtained by starting from an arbitrary $f$ and then switching pairs of transpositions to get rid of any overlapping pairs whose images are too close to each other.
Suppose $r(f …
- 79.9k