(41C) Method of Successive Substitutions

[Thomas Klemm]

From Plot[{x,Pi/Sin[x]},{x,0,10}]:



… we can find the next solution with the following branch of the inverse function:

\(
\begin{align}
f^{-1}(x) &= 3 \pi - \sin^{-1}\left(\frac{\pi}{x}\right) \\
\end{align}
\)

Code:

00 { 10-Byte Prgm }
01 PI
02 X<>Y
03 ÷
04 ASIN
05 PI
06 3
07 ×
08 X<>Y
09 -
10 END

If we start with \( 9 \) and iterate the program we get:

9.000000000
9.068203896
9.071004065
9.071118067
9.071122707
9.071122896
9.071122903
9.071122904
9.071122904
…