Using fix point iteration method?

Question

Question: $x\sin x=50$ the root lie between the interval [51,52]

consider, $$x\sin x=50$$ $$x=\arcsin (50/x)=g(x)$$ now, we check $|g'(x)|<1 $ since, $\arcsin (50/x)=g(x)$ $$g'(x)=1/[1-(50/x)^2]^{1/2} (-50/x^2)\\ |g'(x)|=|50/[x(x^2-2500)^{1/2}]|$$ let here $x=52$ $$|g'(x)|=|50/[52(2704-2500)^{1/2}]|\\ |g'(x)|=0.0673<1$$ now we by fix point iteration method, $$x_{n+1}=g(x_n)$$ let $x_{o}=51.5$ $$x_1=\arcsin (50/x_{o})=1.1.3288$$ the value of my $x_{1}$ does not lie in interval $[51,52]$ please help me to point out my mistake in the solution.

Answer 1

The main branch of $\arcsin$ has values between $-\frac\pi2$ and $\frac\pi2$. To get to any other branch you have to shift by multiples of $\pi$ and every other interval has to be mirrored.

Since $51.5/\pi=16.39295913846522$ and $$ [15.5\pi,16.5\pi] \simeq [48.69468613064179,\,51.83627878423159] $$ covers most of the given interval, the fixed point formula that likely will produce a solution is $$ g(x)=16\pi+\arcsin(50/x). $$

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short test in python

In [12]: x=51.5;
In [13]: for k in range(15): x=16*pi+asin(50/x); print "%.20f" % x

51.59433370146560804415
51.58703269445761208090
51.58758931983782503039
51.58754683447086364367
51.58755007695501859644
51.58754982948691036881
51.58754984837379709006
51.58754984693234035831
51.58754984704235369009
51.58754984703395507495
51.58754984703459456341
51.58754984703455193085
51.58754984703455193085
51.58754984703455193085
51.58754984703455193085