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Thread: analysis questions ..

  1. #1
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    analysis questions ..

    Let x* be the positive real root of 2sinx = x
    (1) Show that T(x) = 2sinx is a λ-contraction on [π/2,2] ( and find λ)
    (2) Find an n such that │T^n(π/2) x*│<10^(-12) (and prove it)


    in (1), i have to show that there is a constant λ satisfying
    d(2sinx, 2siny)≤λd(x,y) right ?
    here, how do i define d(2sinx, 2siny) ?

    and in (2), does T^n mean composition of the function T of n times?


    help me , please ,,, T_T

    thanks a lot in advance
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  2. #2
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    Start by showing $\displaystyle T(x)=2\sin x\in[\pi/2,2]$ if $\displaystyle x\in[\pi/2,2]$.

    Then use $\displaystyle \sin x-\sin y=2\cos\left(\frac{x+y}2\right) \sin\left(\frac{x-y}2\right)$.

    If $\displaystyle x$ and $\displaystyle y$ lie in the interval $\displaystyle [\pi/2,2]$ then $\displaystyle \left|2\cos\left(\frac{x+y}2\right)\right|\leq2|\c os 2|=\lambda<1.$

    Also $\displaystyle \left|2\sin\left(\frac{x-y}2\right)\right|\leq|x-y|$ for all values of the arguments.

    Therefore $\displaystyle |T(x)-T(y)|=|2\sin x-2\sin y|\leq\lambda|x-y|$ in the given interval.

    Thus $\displaystyle |T^n(x)-T^n(y)|\leq\lambda^n|x-y|$ follows.

    If $\displaystyle \alpha$ is the positive root of $\displaystyle 2\sin x=x$ then $\displaystyle \alpha\in[\pi/2,2]$ and $\displaystyle T^n(\alpha)=\alpha$ for all $\displaystyle n$.

    So $\displaystyle |T^n(\pi/2)-\alpha|=|T^n(\pi/2)-T^n(\alpha)|\leq\lambda^n|\pi/2-\alpha|$.

    A rough calculation shows that the desired inequality can be had for $\displaystyle n\geq150$ or thereabouts.
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