Small Solver Program

[Albert Chan]

(02-14-2019 07:06 AM)Thomas Klemm Wrote:  \(x=\frac{3}{LOG(x)}\)

4
R/S
4.982892143
R/S
4.301189432...

It would be nice if we can temper the oscillation, or slow convergence.
Let x0 = 4, x1, x2 = the first two iterated values.

Rate = (x2-x1)/(x1-x0) ≈ (4.30 - 4.98) / (4.98 - 4) = -0.694

If the same trend continued, we expect final % = 1/(1-r) ~ 60%

x_∞ ≈ x0/(1-r) = (x1 - (x1-x0))/(1-r) = (x1 - r x0) / (1-r)

Use weighted fixed-point equation x = 0.6 * 3/log10(x) + 0.4 x

4.6
4.555924149
4.555537395
4.555535712
4.555535705 (converged)

Edit: compare with Newton's method, x = (ln(1000) + x) / (ln(x) + 1)
4
4.571001573
4.555546101
4.555535705 (converged)