Video Mistakes II: The Sequel

Chapters
0:00 Intro
1:58 Outro

#NumericalAnalysis

[alexandrevachon]

Thank you for rectifying those. I understand better now.

[nenadilic]

Regarding the Wegstein's method, it doesn't seem to be a mistake: according to the formula, you need exactly two previous root approximations. So what you first did was finding first two approximations using the fixed-point method (starting value + one iteration), followed by 5 Wegstein's steps.

In your correction, you first found first three approximations (starting value + 2 iterations) by fixed-point method in order to compute initial q, restarted and did 5 Wegstein's steps. The total number of iterations is thus smaller in the first case (6 vs. 7).

Why it worked? I think because, in deriving the formula, the approximated derivative at the point on the interval (for computing a_n) is anyway calculated using the fixed-point, not the Wegstein's conditions (e.g. x_n is replaced simply by g(x_n-1). So, if that formula works (and we know it does) then it also works when switching from fixed-point to Wegstein's iterations.