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Why is it not possible to generate an explicit formula for Newton's method?

First, note that such a formula would violate the "too good to be true" test: it would let you find the minima of any (sufficiently smooth, strictly) convex function in a single step, no matter how ...
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Why is it not possible to generate an explicit formula for Newton's method?

You wouldn't really gain much with a formula for the $n$th iteration. The appeal of the Newton-Raphson method is that a single step: is conceptually easy to understand, is fast to compute, and can ...
• 26.4k
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Convergence of Newton Method for monotonic polynomials

No, not necessarily. In particular, consider the polynomial, $$p(x) = \frac{7}{2}x - \frac{5}{2}x^3 + x^5.$$ Note that $$p'(x) = \frac{7}{2} - \frac{15}{2}x^2 + 5x^4,$$ a positive quadratic in $x^2$ ...
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When does the Newton Raphson method fail?

To visualize geometrically what's going on, I will code an interactive diagram with GNU Dr. Geo (free software of mine) from where I can drag the initial value (the red dot) and see how the method ...
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In practice, what does it mean for the Newton's method to converge quadratically (when it converges)?

A great example is to do what your calculator does when it computes $\sqrt{x}$ for some number. This way, you can see quadratic convergence in practice. Say you want to figure out $\sqrt{17}$, this ...
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Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

It looks closely related to Newton's way to compute the square-root. If you want the square root of $1+x$, you generate the sequence $$a_{n+1}= \frac12\left(a_n + \frac{1+x}{a_n}\right)$$ with $a_0$ ...
• 23.5k
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Faster convergence methods for Lambert W approximation

If you look for higher order formulae, you could consider Householder method which is or order $4$. If you want to go further, have a look at this paper; using it, you can easily build an iteration ...
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• 74.9k
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Is modified Newton's Raphson method redundant?

The convergence for multiplicity $m$ is geometric with factor $1-\frac1m$. This means that you need more than 3 iterations for each digit of the result. Thus you can both detect the slow convergence ...
• 127k

Pattern of Newton-Raphson iteration $x\mapsto\frac{1}{2}(x+\frac{q}{x})$ over finite fields

Whenever $q_1, q_2$ are both quadratic residues modulo $p$, or both quadratic nonresidues modulo $p$, we can find a constant $c \ne 0$ such that $q_2 \equiv c^2 \cdot q_1 \pmod p$. Then multiplication ...
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When to use Newtons's, bisection, fixed-point iteration and the secant methods?

You should never seriously use bisection. If you think that derivatives are hard, use the secant method. If you want to force convergence and can find intervals with opposite signs of the function, ...
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Intuitive Understanding Newton-Raphson method with second derivatives

Read again your new text. With almost certainty you will find that you want to find extremal points of that $f$. Since in general they can be found were $f'(x)=0$, Newton's method for $g(x)=f'(x)$ ...
• 127k
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Can there be a metric space where no contraction has a fixed point?

If $X$ is non-empty then we can choose some $x_0\in X$ and define $f(x)=x_0$ for all $x\in X$. This is a contraction with a fixed point.
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