Dynamical Systems

Mos5180d

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Let h(y) =y21y^2-1 and yny_n be the sequence generated by the Newton-Raphson scheme for approximating the roots of h (which are clearly −1 and 1): yn+1=yn(yn21)/2yny_{n+1}= y_n-(y_n^2-1)/2y_n

Obtain a formula for yny_n in terms of n and y0y_0

Here is my working but I seem to get back to the same position,

1641854407954.png

But when I input numbers for y_0, the results do not follow that pattern
 

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At this point I don't know how to solve it, but your expression for y2y_2, or for yny_n for that matter, does not look right to me.
 
[EDITED] Never mind, you're defining y as the independent variable. Your initial formula is correct.
 
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