Questions tagged [rootfinding]

Algorithms to approximate numerically a root of a nonlinear equation or system: for instance, Newton's method, secant method, bisection, etc.

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Closed form roots for polynomial $x^9 + ax^6 + bx^5 + cx^3 + d = 0$

I know about Abel–Ruffini theorem, but I have a polynomial of special form. From "Beyond the Quartic Equation" by R.B. King (a very interesting book, btw) I've learned about Tschirnhaus ...
• 355
158 views

How to solve for $a$ in $\sum_{j=i}^n (a -j) \binom{n}{j} y^j (1-y)^{n-j} = 0$

The Problem Given $i, n, y$, I am trying to find a closed form solution for $a$ in the equation $$\sum_{j=i}^n (a -j) \binom{n}{j} y^j (1-y)^{n-j} = 0$$ Do you have any recommendations on how to ...
• 23
1 vote
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Newton-Raphson and bisection for solving for a system of nonlinear equations?

We know that for a single equation root-finding, we can use the Newton's method, or a combination of Newton with bisection to guarantee convergence. Can we use Newton+bisection for a system of ...
• 11
411 views

Square root of prime numbers

I found that the square root of any prime number S can be approximated, at the n-th order, as a rational number represented by the polynomials shown below. $x_0$ is an initial seed, which is a ...
189 views

Calculating derivatives of arbitrary-order at an operator's root

Consider roots $f = 0$ of a nicely-behaved real function $f(x, t)$ of two (real) variables. Namely, points $(x, t)$ on which $f$ vanishes, $f(x, t) = 0$. Suppose that $x$ can be written as function of ...
• 583
150 views

Are root finding algorithms stable for bounded polynomials? [closed]

Suppose that we have a bounded polynomial defined on $[0,1]$. I think because it is just polynomial, root finding algorithms would easily and without any instability find all its roots. Am I right? ...
• 113
973 views

Solution to sixth order equation

I'm dealing with the expression $x = \frac{1}{3}y(y+1)(2y+1)^2(2y^2+2y+1)$. What is this approximately, if one is explicitly writing y in terms of x? There's no general formula for sixth powers ...
1 vote
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1 vote
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Using Regula-Falsi to determine the solution to a non-linear system [closed]

Apologies, for this isn't a field or subject I know much about. Regula Falsi (I believe some may know this as "double false position" or something like this) can be used trivially, of course,...
• 111
43 views

Solving nonlinear equations involving expectations

Let $X$ be a random variable and $g(x,y)$ be a function of two variables. Consider the equation $$\mathbb{E}_Xg(X,y) = 0$$ Are there any specialized techniques for solving such equations (...
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