(HP15C)(HP67)(HP41C) Bernoulli Polynomials

[Albert Chan]

(08-29-2023 11:47 AM)Namir Wrote:  When I saw the nested summation formula in Wikipedia, I decided that it would be an easier approach to code
(albeit at the expense of some extra CPU time).

That's quite true.
But noted that formula had huge cancellation issue with factor (x+k)^m, when x and/or m is big.



For example, if m=6, n=m (for last sum), it may have huge cancellation errors.

\(\displaystyle \frac{\Delta^6(x^6)}{7} =
\frac{x^6-6\,(x\!+\!1)^6+15\,(x\!+\!2)^6-20\,(x\!+\!3)^6+15\,(x\!+\!4)^6-6\,(x\!+\!5)^6+(x\!+\!6)^6}{7}
= \frac{720}{7} \)

Example, tested on XCas 1.9.0, with 48 bits float:

XCas> B(m,x) := sum(1/(n+1) * sum((-1)^k * comb(n,k) * (x+k)^m, k = 0 .. n), n = 0 .. m)

XCas> float(B(16, pi))      → 1462871.932         /* reference */
XCas> B(16, float(pi))      → -423965.265625    /* massive cancellations */
XCas> B2(16, float(pi))    → 1462871.93177      /* falling factorial form more stable */