Download or view TangentNumbers.frink in plain text format
// Calculation of Tangent Numbers, which can be used to also calculate
// Bernoulli numbers.
//
// These can be used to derive the tangent of a number by:
//
// tan[z] = sum[n>0, Tn * z^(2n-1)/(2n-1)!]
//
// See:
// http://arxiv.org/abs/1108.0286 (Brent)
// http://mathworld.wolfram.com/TangentNumber.html
// http://oeis.org/A000182
// https://www.petervis.com/mathematics/maclaurin_series/maclaurin_series_tanx.html
//
// See:
// Fast Algorithms for High-Precision Computation of Elementary Functions,
// Richard P. Brent, 2006
// https://pdfs.semanticscholar.org/bf5a/ce09214f071251bfae3a09a91100e77d7ff6.pdf
//
// This implements the algorithm in figure 2 of the Brent paper.
//
// The main problem for using this to calculate tangents to arbitrary
// precision is that this algorithm alters numbers in-place in several passes,
// and doesn't allow easy calculation of more tangent numbers if you decide
// you need more.
//
// Another problem with using Tangent Numbers around 90 degrees is that
// this converges very slowly and may require way too many terms.
//
// We could also try using Newton's method to invert arctan[x] which
// has a simple series expansion,
// arctan[x] = sum[(-1)^k x^(2k+1) / (2k + 1), {k, 0, infinity}]
// but this only converges for abs[x] <= 1, x != +/- i
//
tangentNumbers[n] :=
{
t = new array
t@1 = 1
for k = 2 to n
t@k = (k-1) t@(k-1)
for k = 2 to n
for j = k to n
t@j = (j-k) t@(j-1) + (j-k+2) t@j
return t
}
Download or view TangentNumbers.frink in plain text format
This is a program written in the programming language Frink.
For more information, view the Frink
Documentation or see More Sample Frink Programs.
Alan Eliasen was born 20217 days, 23 hours, 35 minutes ago.