(41C) Method of Successive Substitutions
10-06-2020, 12:57 AM (This post was last modified: 10-06-2020 02:05 AM by Albert Chan.)
Post: #2
 Albert Chan Senior Member Posts: 1,658 Joined: Jul 2018
RE: (41C) Method of Successive Substitutions
(10-04-2020 04:21 PM)Eddie W. Shore Wrote:  Take the equation ln(3*x) = x with initial guess x0 = 1.512

Sometimes, convergence may be slow, or not at all.
We can place a weight on it.

With my Casio FX-115MS

1.512 =
ln(3 Ans ﻿
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.512045566
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.512075703
﻿= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.512095633

r = (95633-75703) / (75703-45566) = 19930 / 30317

Convegence is slow (we wanted small |r|)
Assume same trend continued (constant r), estimated converged to:

1.512045566 + 0.000030317/(1-r) = 1.512134548

Let's check if assumption is good. Continued on ...
w = 1/(1-r) ≈ 3 ﻿ ﻿ ﻿ ﻿ ﻿ ﻿
x = (1-w)*x + w*ln(3*x) = -2 x + 3 ln(3*x)

-2 Ans + 3 ln( 3 Ans ﻿
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.512135175
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.512134542
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.512134552, converged