Estimate logarithm quickly

11262021, 06:24 PM
Post: #14




RE: Estimate logarithm quickly
(11202021 02:02 PM)Albert Chan Wrote: ln(n) ≥ n1 / ((1 + 4*√n + n)/6) // Simpson's Rule A simpler proof of inequality is convert it to atanh(y) Assume x>1, then y = (x1)/(x+1) > 0 To avoid square mess, we apply Doerfler's formula with squared argument. ln(x) = atanh(y = (x1)/(x+1))*2 → atanh(y) = ln(x = (1+y)/(1y))/2 XCAS> D2(x) := 3*(x*x1)/(1 + 4*x + x*x) XCas> factor(D2((1+y)/(1y)) /2) → 3*y/(3+y^2) This is just pade(atanh(y),y,4,2), which expands to: y/(1y^2/3) = y + y^3/3 + y^5/3² + y^7/3³ + ... For y>0, atanh(y) = y + y^3/3 + y^5/5 + y^7/7 + ... is bigger. For x>1, ln(x), which atanh(y) were derived from, is biggger than D2(x) Because of symmetry, For 0<x<1, D2(x) = D2(1/x), same as ln(x). Thus the proof can be extended from x > 1, to x > 0 XCAS> D2(2), D2(1/2) → (9/13 , 9/13) 

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Messages In This Thread 
Estimate logarithm quickly  Albert Chan  08212021, 03:39 PM
RE: Estimate logarithm quickly  Albert Chan  08212021, 03:58 PM
RE: Estimate logarithm quickly  Albert Chan  08212021, 05:41 PM
RE: Estimate logarithm quickly  trojdor  08212021, 06:31 PM
RE: Estimate logarithm quickly  Albert Chan  08212021, 11:42 PM
RE: Estimate logarithm quickly  EdS2  08232021, 06:44 AM
RE: Estimate logarithm quickly  Albert Chan  10062021, 10:58 PM
RE: Estimate logarithm quickly  Albert Chan  10182021, 01:02 PM
RE: Estimate logarithm quickly  Albert Chan  11202021, 02:02 PM
RE: Estimate logarithm quickly  Albert Chan  11222021, 01:49 AM
RE: Estimate logarithm quickly  Albert Chan  11232021, 09:01 PM
RE: Estimate logarithm quickly  Albert Chan  11262021 06:24 PM
RE: Estimate logarithm quickly  Csaba Tizedes  10072021, 06:25 PM
RE: Estimate logarithm quickly  Albert Chan  10222021, 02:15 PM

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