I want a fixed point scheme that will converge for all values of ' $a$ ' . $$x = \sin(x) - ax + 30$$
I have tried a couple of options but none works for all values of '$a$'
2026-03-25 21:30:56.1774474256
Fixed Point Scheme
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$x=(\sin x + 30)/(1+a)$ (note the parentheses) is the natural try and will work well when $a \not \in [-2,0]$ When $a$ is in that interval the derivative can be greater than $1$ in absolute value. In that case you want $x=\arcsin ((1+a)x-30)$ but I don't see a way to get one expression that works for all $a$.