(41C) Method of Successive Substitutions

[Thomas Klemm]

We can already see from these examples that the larger the solution, the faster the convergence.

\(
\begin{align}
f^{-1}(x) &= \sin^{-1}\left(\frac{\pi}{x}\right) + 1000 \pi \\
\end{align}
\)

Code:

00 { 9-Byte Prgm }
01 PI
02 X<>Y
03 ÷
04 ASIN
05 PI
06 1000
07 ×
08 +
09 END

If we start with \( 3141 \) and iterate the program we get:

3141.00000000
3141.59365378
3141.59365359
3141.59365359
…

This reminds me of my solution to an older challenge by Valentin: Short & Sweet Math Challenge #19: Surprise ! [LONG]

Quote:Instead of solving tan(x) = x, I solved x = arctan(x) using a fixed-point iteration which converges faster as N grows.

The reason is similar: Larger fixed-points are closer to the poles.
As a result, the derivative of the inverse becomes flatter and tends towards \( 0 \).
This increases the speed of convergence.