New Sum of Powers Log Function
03-31-2021, 09:25 PM
Post: #16
 Albert Chan Senior Member Posts: 1,800 Joined: Jul 2018
RE: New Sum of Powers Log Function
(03-31-2021 01:27 PM)Namir Wrote:  Here is the SopLog implementation in Matlab ...

Code:
    f0 = sumFx(N, S, x);     fp = sumFx(N, S, x + h);     fm = sumFx(N, S, x - h);     diff = 2 * h * f0 / (fp - fm);     x = x - diff;     ...

Without function for the derivative, Newton's method is very inefficient.
We should consider secant's method (or, improved secant)

Also, curve is very close to LambertW, we should use log scale.

>>> n, s = 1000, 5000
>>> x = log(s,n)-1
>>> x1, y1 = x, log(fsum(k**x for k in range(1,n+1))/s)
>>> x1, y1
(mpf('0.23299000144533966'), mpf('-0.20890713759105706')

>>> x = guess3(n,s)
>>> x2, y2 = x, log(fsum(k**x for k in range(1,n+1))/s)
>>> x2, y2
(mpf('0.26718816370743487'), mpf('3.3544082779438775e-6'))

>>> x = x2 - y2 * (x2-x1)/(y2-y1)
>>> x3, y3 = x, log(fsum(k**x for k in range(1,n+1))/s)
>>> x3, y3
(mpf('0.26718761459859175'), mpf('-5.9153427886634265e-9'))

>>> x = x3 - y3 * (x3-x2)/(y3-y2)
>>> x4, y4 = x, log(fsum(k**x for k in range(1,n+1))/s)
>>> x4, y4
(mpf('0.26718761556521503'), mpf('-2.2204460492503136e-16'))

---

Turns out mpmath implemented Euler–Maclaurin Asymptotic expansion of sums
Summation will continue, as long as additional term improve the sum, by at least 1 decimal.

>>> x = solvex(n, s) # my version, without higher derivatives
>>> x, fsum(k**x for k in range(1,n+1))
(mpf('0.26718762849802313'), mpf('5000.0003957177305'))

>>> x = findroot(lambda x: log(sumem(lambda k: k**x, [1,n])/s), log(s,n)-1)
>>> x, fsum(k**x for k in range(1,n+1))
(mpf('0.26718761309749891'), mpf('4999.9999244928886'))
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 Messages In This Thread New Sum of Powers Log Function - Namir - 03-29-2021, 04:53 PM RE: New Sum of Powers Log Function - C.Ret - 03-29-2021, 08:39 PM RE: New Sum of Powers Log Function - Albert Chan - 03-29-2021, 10:47 PM RE: New Sum of Powers Log Function - Albert Chan - 03-30-2021, 01:44 AM RE: New Sum of Powers Log Function - Albert Chan - 03-30-2021, 08:35 PM RE: New Sum of Powers Log Function - Albert Chan - 03-30-2021, 10:26 PM RE: New Sum of Powers Log Function - Namir - 03-30-2021, 11:05 AM RE: New Sum of Powers Log Function - Albert Chan - 03-30-2021, 04:35 PM RE: New Sum of Powers Log Function - Paul Dale - 03-30-2021, 11:41 AM RE: New Sum of Powers Log Function - Gene - 03-30-2021, 01:43 PM RE: New Sum of Powers Log Function - C.Ret - 03-30-2021, 04:01 PM RE: New Sum of Powers Log Function - Namir - 03-30-2021, 05:56 PM RE: New Sum of Powers Log Function - Namir - 03-31-2021, 01:27 PM RE: New Sum of Powers Log Function - Albert Chan - 03-31-2021, 04:06 PM RE: New Sum of Powers Log Function - Albert Chan - 03-31-2021 09:25 PM RE: New Sum of Powers Log Function - Namir - 03-31-2021, 02:19 PM RE: New Sum of Powers Log Function - Albert Chan - 04-01-2021, 02:56 PM RE: New Sum of Powers Log Function - Namir - 04-01-2021, 06:05 PM RE: New Sum of Powers Log Function - Albert Chan - 04-01-2021, 11:05 PM RE: New Sum of Powers Log Function - Namir - 04-01-2021, 11:55 PM RE: New Sum of Powers Log Function - Albert Chan - 04-02-2021, 01:29 AM RE: New Sum of Powers Log Function - Albert Chan - 04-02-2021, 01:18 PM RE: New Sum of Powers Log Function - Namir - 04-04-2021, 03:41 PM

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