Bernoulli Number generator

function fB(nB) // recursive function for Bernoulli Number variable nB // number index argument variable v if(nB==0) return 1 else make/O/D/FREE/N=(nB) wB wB = binomial(nB+1,p)*fB(p) v = -sum(wB)/(nB+1) return v // Bernoulli number value endif end

s.r.chinn

After further research, I have found a new function in IP7 that simplifies the generation of Bernoulli numbers:


DisplayHelpTopic "zeta"


function fBernoulli(n)
	variable n
	
	if(n==0)
		return 1
	elseif(n==1)
		return -0.5
	elseif(mod(n,2)!=0) // n>1 is odd here
		return 0
	else // n>1 is even here
		return (-1)^(n/2-1)*factorial(n)*2*zeta(n,1,400)/( ((2*pi)^(n)) )
	endif
end

The second argument, 1, in

zeta(n,1,400)

makes it the Riemann zeta function; the third argument improves the precision for small 'n'.

October 2, 2017 at 08:18 am - Permalink

jjweimer

A more compact implementation (perhaps faster)?


function fBernoulli(n)
	variable n
 
	variable rtn
	make/N=2/FREE rtw = {1,-0.5}
	if (n<=1)
		return rtw[n]
	endif
	rtn = mod(n,2) != 0 ? 0 : (-1)^(n/2-1)*factorial(n)*2*zeta(n,1,400)/( ((2*pi)^(n)) )
	return rtn
end

--
J. J. Weimer
Chemistry / Chemical & Materials Engineering, UAH

October 5, 2017 at 07:06 am - Permalink

s.r.chinn

Jeffrey,
I think the importance of computation speed depends on what the user needs. For computing many Bn at the same time a recursive approach is widely used. As Brent notes (https://arxiv.org/abs/1108.0286v3): "However, this is unsatisfactory if floating-point numbers are used, because the recurrence in numerically unstable." Harvey (http://web.maths.unsw.edu.au/~davidharvey/talks/bernoulli.pdf) gives an interesting approach wherein ONE division operation provides all necessary coefficients for the Bn up to a given order. The hitch is that huge numerators and denominators are required, and the division is best done in base 2. I tested the concept using

APmath

in base 10, and it works!

October 5, 2017 at 08:21 am - Permalink

jjweimer

> I think the importance of computation speed depends on what the user needs.

I agree. I was posting more to explore other options.

--
J. J. Weimer
Chemistry / Chemical & Materials Engineering, UAH

October 9, 2017 at 06:19 pm - Permalink

Bernoulli Number generator

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