(41C) Method of Successive Substitutions

[Albert Chan]

(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

see https://www.hpmuseum.org/forum/thread-12...#pid112232