Estimate the number of iterations of Newton's method needed to find a root of $f(x)=\mathrm{cos}(x)-x$ to within ${10}^{-100}$.

Izabelle Lowery
2022-10-12
Answered

Estimate the number of iterations of Newton's method needed to find a root of $f(x)=\mathrm{cos}(x)-x$ to within ${10}^{-100}$.

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Adalyn Pitts

Answered 2022-10-13
Author has **15** answers

The idea behind the reasoning is the quadratic convergence of Newton's algorithm (if the zero is simple).

When you are near the zero $\alpha $, an iteration takes you from $\alpha +\delta $ to

$(\alpha +\delta )-\frac{f(\alpha +\delta )}{{f}^{\prime}(\alpha +\delta )}\approx \alpha +\delta -\frac{{f}^{\prime}(\alpha )\delta +{f}^{\u2033}(\alpha )\frac{{\delta}^{2}}{2}}{{f}^{\prime}(\alpha )+{f}^{\u2033}(\alpha )\delta}\approx \alpha +\frac{{f}^{\u2033}(\alpha )}{2{f}^{\prime}(\alpha )}{\delta}^{2},$

so each step roughly doubles the number of correct digits.

If you start with approximately one correct digit, after seven steps, you have roughly ${2}^{7}=128$ correct digits.

When you are near the zero $\alpha $, an iteration takes you from $\alpha +\delta $ to

$(\alpha +\delta )-\frac{f(\alpha +\delta )}{{f}^{\prime}(\alpha +\delta )}\approx \alpha +\delta -\frac{{f}^{\prime}(\alpha )\delta +{f}^{\u2033}(\alpha )\frac{{\delta}^{2}}{2}}{{f}^{\prime}(\alpha )+{f}^{\u2033}(\alpha )\delta}\approx \alpha +\frac{{f}^{\u2033}(\alpha )}{2{f}^{\prime}(\alpha )}{\delta}^{2},$

so each step roughly doubles the number of correct digits.

If you start with approximately one correct digit, after seven steps, you have roughly ${2}^{7}=128$ correct digits.

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To find approximate $\sqrt{a}$ we can use Newton's method to approximately solve the equation ${x}^{2}-a=0$ for $x$, starting from some rational ${x}_{0}$.

Newton's method in general is only locally convergent, so we have to be careful with initialization.

Show that in this case, the method always converges to something if ${x}_{0}\ne 0$.

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Show that in this case, the method always converges to something if ${x}_{0}\ne 0$.

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2.Using the answer from problem I (hopefully it's correct), determine your weight in Newton's if you were on the moon. Yes, you have to look up something to complete this problem.

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3. You measured the mass of a rock to be 355g. What is its weight?

4.You are now holding the rock in your hand from problem 3. What force is the rock on your hand? How much force do you need to exert on the rock to hold in stationary.

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Using Newton's method below:

${x}_{n+1}={x}_{n}-\frac{f({x}_{n})}{{f}^{\prime}({x}_{0})}$

using this chord formula where the chord length $c$ is $1$ cm:

$c=2r\mathrm{sin}\frac{\theta}{2}$

supposing the radius is $1.1$ cm and the angle $\theta $ is unknown, show the iterative Newton's Method equation you would use to find an approximate value for $\theta $ in the context of this problem (using the appropriate function and derivative).

${x}_{n+1}={x}_{n}-\frac{f({x}_{n})}{{f}^{\prime}({x}_{0})}$

using this chord formula where the chord length $c$ is $1$ cm:

$c=2r\mathrm{sin}\frac{\theta}{2}$

supposing the radius is $1.1$ cm and the angle $\theta $ is unknown, show the iterative Newton's Method equation you would use to find an approximate value for $\theta $ in the context of this problem (using the appropriate function and derivative).