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lambertw, all branches
01-20-2024, 04:48 PM
Post: #18
RE: lambertw, all branches
Newton's method convergence to prove previous post.

x * e^x = a               // x = Wk(a)
x + ln(x) = lnk(a) = c

If c = ±∞ + θ*I, then (|x| → ∞) ⇒ (slope = 1+1/x ≈ 1)

Iteration formula is practically equivalent to Newton's method.

x = c - ln(x)

x1 = c                      // guess
x2 = c - ln(x1)
x3 = c - ln(x2)          // ... until x's converged

if a = ∞*cis(θ), r = re(c) = ∞

c = r + θ*I ≈ |r| * cis(θ/r)
ln(c) ≈ ln(|r|) + (θ/r)*I
x2 ≈ c - ln(c) ≈ (r-ln(|r|)) + (θ-(θ/r))*I ≈ (r + θ*I) = x1

--> Wk(∞*cis(θ)) = lnk(∞*cis(θ))

if a = 0*cis(θ) and k≠0, r = -∞, sign(θ) = sign(k)

c = r + θ*I ≈ |r| * cis(sign(k)*(pi - |θ/r|))
ln(c)   ≈ ln(|r|) + (sign(k)*(pi - |θ/r|))*I  ≈ ln(|r|) + sign(k)*pi*I
x2 = c - ln(c) ≈ (r-ln(|r|)) + (θ-sign(k)*pi)*I ≈ r + θ2*I

sign(θ2) = sign(θ) = sign(k)      // 1-sided convergence (★)
ln(x2) ≈ ln(|r|) + (sign(k)*(pi - |θ2/r|))*I ≈ ln(|r|) + sign(k)*pi*I

x3 = c - ln(x2) ≈ c - ln(c) = x2

--> Wk≠0(0*cis(θ)) = lnk(0*cis(θ)) - sign(k)*pi*I


(★) Without slope correction, convergence is 1-sided, not crossing discontinuity.

lua> c = I(-99999*log(10), -pi)      -- = ln-1(-10^-99999)
lua> c
(-230256.2067143116-3.141592653589793*I)
lua> c - I.log(_)
(-230268.55366222185-1.3643899976489848e-05*I)
lua> c - I.log(_)
(-230268.55371584298-5.925215873503475e-11*I)
lua> c - I.log(_)
(-230268.5537158432+0*I)          -- W-1(-10^-99999) conjugate

Note that final imaginery part should be -0
This is why I.W(a, k) flip to positive k, iterate, then flip back for negative k

lua> c = c:conj()
lua> c
(-230256.2067143116+3.141592653589793*I)
lua> c - I.log(_)
(-230268.55366222185+1.3643899976489848e-05*I)
lua> c - I.log(_)
(-230268.55371584298+5.925215873503475e-11*I)
lua> c - I.log(_)
(-230268.5537158432+0*I)
lua> _:conj()
(-230268.5537158432-0*I)          -- W-1(-10^-99999)
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Messages In This Thread
lambertw, all branches - Albert Chan - 04-07-2023, 01:24 PM
RE: lambertw, all branches - Albert Chan - 04-07-2023, 02:47 PM
RE: lambertw, all branches - Albert Chan - 04-19-2023, 01:30 AM
RE: lambertw, all branches - pier4r - 04-07-2023, 06:04 PM
RE: lambertw, all branches - Albert Chan - 04-07-2023, 07:54 PM
RE: lambertw, all branches - Albert Chan - 04-08-2023, 03:21 PM
RE: lambertw, all branches - Albert Chan - 04-08-2023, 05:54 PM
RE: lambertw, all branches - Albert Chan - 04-07-2023, 08:40 PM
RE: lambertw, all branches - Albert Chan - 04-09-2023, 03:59 AM
RE: lambertw, all branches - Albert Chan - 04-09-2023, 04:36 PM
RE: lambertw, all branches - Albert Chan - 04-10-2023, 04:44 PM
RE: lambertw, all branches - Albert Chan - 04-10-2023, 06:47 PM
RE: lambertw, all branches - Albert Chan - 04-13-2023, 03:03 PM
RE: lambertw, all branches - floppy - 04-13-2023, 04:14 PM
RE: lambertw, all branches - Albert Chan - 04-23-2023, 02:49 PM
RE: lambertw, all branches - Albert Chan - 04-23-2023, 04:40 PM
RE: lambertw, all branches - Albert Chan - 01-19-2024, 04:14 PM
RE: lambertw, all branches - Albert Chan - 01-20-2024 04:48 PM
RE: lambertw, all branches - Gil - 01-20-2024, 10:52 PM
RE: lambertw, all branches - Albert Chan - 01-21-2024, 01:14 AM
RE: lambertw, all branches - Albert Chan - 01-21-2024, 01:54 AM
RE: lambertw, all branches - Gil - 01-21-2024, 01:53 PM
RE: lambertw, all branches - Albert Chan - 01-21-2024, 04:19 PM
RE: lambertw, all branches - Gil - 01-21-2024, 04:35 PM
RE: lambertw, all branches - Albert Chan - 01-21-2024, 06:03 PM
RE: lambertw, all branches - Albert Chan - 01-21-2024, 07:01 PM
RE: lambertw, all branches - Gil - 01-21-2024, 07:30 PM
RE: lambertw, all branches - Gil - 01-21-2024, 08:39 PM
RE: lambertw, all branches - Albert Chan - 01-21-2024, 10:06 PM
RE: lambertw, all branches - Gil - 01-21-2024, 09:51 PM
RE: lambertw, all branches - Gil - 01-21-2024, 10:56 PM
RE: lambertw, all branches - Albert Chan - 01-22-2024, 01:34 AM
RE: lambertw, all branches - Gil - 01-21-2024, 11:15 PM
RE: lambertw, all branches - Gil - 01-22-2024, 06:09 PM
RE: lambertw, all branches - Albert Chan - 01-22-2024, 07:29 PM
RE: lambertw, all branches - Gil - 01-22-2024, 11:33 PM
RE: lambertw, all branches - Albert Chan - 01-23-2024, 02:32 AM
RE: lambertw, all branches - Gil - 01-23-2024, 02:35 PM
RE: lambertw, all branches - Albert Chan - 01-23-2024, 03:54 PM
RE: lambertw, all branches - Gil - 01-23-2024, 04:57 PM
RE: lambertw, all branches - Albert Chan - 01-23-2024, 06:17 PM
RE: lambertw, all branches - Gil - 01-23-2024, 06:44 PM
RE: lambertw, all branches - Gil - 01-23-2024, 11:00 PM
RE: lambertw, all branches - Gil - 01-24-2024, 03:18 PM
RE: lambertw, all branches - Albert Chan - 01-24-2024, 08:53 PM
RE: lambertw, all branches - Gil - 01-25-2024, 12:37 AM
RE: lambertw, all branches - Gil - 01-25-2024, 01:10 AM
RE: lambertw, all branches - Gil - 01-25-2024, 03:04 AM
RE: lambertw, all branches - Albert Chan - 01-25-2024, 07:02 AM
RE: lambertw, all branches - Gil - 01-25-2024, 10:09 AM
RE: lambertw, all branches - Albert Chan - 01-25-2024, 04:13 PM
RE: lambertw, all branches - Gil - 01-25-2024, 05:14 PM
RE: lambertw, all branches - Albert Chan - 01-25-2024, 05:57 PM
RE: lambertw, all branches - Gil - 01-25-2024, 06:19 PM
RE: lambertw, all branches - Albert Chan - 01-28-2024, 11:18 PM
RE: lambertw, all branches - Albert Chan - 02-01-2024, 02:17 AM
RE: lambertw, all branches - Albert Chan - 02-01-2024, 04:16 PM
RE: lambertw, all branches - Albert Chan - 02-02-2024, 11:49 AM



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