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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questions
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About when $\operatorname{Tr}(AB)=0$ can implies $B=0$?
Define
\begin{equation}
\begin{aligned}
F_\theta(t)&=\sum_{1\leq i,j \leq n} \frac{a_{ij}}{t}(\theta_i-\theta_j) t \cdot\sin((\theta_i-\theta_j)t)\\
&=\frac{1}{t}\sum_{1\...
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0
votes
1
answer
263
views
Roots of linear combination of $x \sin x$
Let $\theta=(\theta_1,\theta_2,\cdots \theta_n)$, and $a_{ij}$ are constants. There is no condition on the positiveness of $a_{ij}$.
Under which condition on $\theta$, such that the following function ...
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7
votes
3
answers
499
views
Rigorous estimates on roots of function
We consider the function
$$f(x) = 1- \frac{1}{N} \sum_{i=1}^N \frac{\sin\left(\tfrac{\pi i}{N}\right)^2}{1+\sin\left(\tfrac{\pi i}{2N}\right)^2-x}.$$
The arguments of the two sines differ by a factor ...
3
votes
0
answers
43
views
how to efficiently find level sets (using a modification of a root-finding algorithm)?
I'm trying to find a set of points $\{ a_i | f(a_i) = c_i \}_{i=1}^k$ where $f$ and $\{ c_i \}_{i=1}^k$ are given in sorted order. All $c_i > 0$, $f$ is continuous and monotonically increasing, $f(...
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1
vote
1
answer
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views
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 ...
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2
votes
1
answer
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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 ...
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1
vote
0
answers
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views
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 ...
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votes
0
answers
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views
How to find close form roots or at least good approximation of roots of such function?
I need to solve for $D$:
$$
KD^{N-1} S + P = KD^N + \left(\frac{D}{N}\right)^N,
$$
where
$$
S = \sum_i x_i, \, P = \prod_i x_i,
$$
$$
K_0 = P \left(\frac{N}{D}\right)^N, \, K = AK_0 \frac{\gamma^2}{(\...
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2
votes
1
answer
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views
Square root of prime numbers
I found that the square root of any prime number S can be approximated, at the n-th order, as a rational number represented by the polynomials shown below. $x_0$ is an initial seed, which is a ...
0
votes
1
answer
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views
Calculating derivatives of arbitrary-order at an operator's root
Consider roots $f = 0$ of a nicely-behaved real function $f(x, t)$ of two (real) variables.
Namely, points $(x, t)$ on which $f$ vanishes, $f(x, t) = 0$.
Suppose that $x$ can be written as function of ...
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0
votes
1
answer
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views
Are root finding algorithms stable for bounded polynomials? [closed]
Suppose that we have a bounded polynomial defined on $[0,1]$. I think because it is just polynomial, root finding algorithms would easily and without any instability find all its roots. Am I right?
...
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6
votes
1
answer
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views
Solution to sixth order equation
I'm dealing with the expression $x = \frac{1}{3}y(y+1)(2y+1)^2(2y^2+2y+1)$. What is this approximately, if one is explicitly writing y in terms of x? There's no general formula for sixth powers ...
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Is it possible in principle (but not in practice) to recursively factor away the Riemann zeta zeros as they are computed?
Let: $$f_0(x)=\frac{\zeta (x)}{\sin \left(\frac{\pi x}{2}\right)}$$ and let the seed point be: $$s=\sqrt{-1}$$
which is the input into the limit:
$$\rho_1=s+\lim\limits_{n \rightarrow \infty}\left(1-\...
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votes
1
answer
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How many positive roots can $\sum_{i}\frac{a_i}{x+b_i}$ have where $b_i$'s are all positive? [closed]
What is the maximum number of positive roots $\sum_{i}^N\frac{a_i}{x+b_i}$ can have where $b_i$'s are all positive? (everything here is a real number. To provide context, I encountered this problem ...
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votes
1
answer
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views
Searching the roots of a self-consistent transcendental equation
I have the equation
$$M = c_1 + c_2M - c_3T\ln\left(\left|\frac{e^{(c_4M + c_5)/T}-1}{e^{(c_6M + c_5)/T}-1}\right|\right)$$
where $c_1, \dots, c_6$ are constants.
I am interested in the roots of $$M\...
1
vote
0
answers
22
views
Using Regula-Falsi to determine the solution to a non-linear system [closed]
Apologies, for this isn't a field or subject I know much about.
Regula Falsi (I believe some may know this as "double false position" or something like this) can be used trivially, of course,...
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0
votes
0
answers
40
views
Solving nonlinear equations involving expectations
Let $X$ be a random variable and $g(x,y)$ be a function of two variables. Consider the equation
$$
\mathbb{E}_Xg(X,y) = 0
$$
Are there any specialized techniques for solving such equations (...
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2
votes
0
answers
226
views
Could analytically deriving the next non-trivial zero of $\zeta(s)$ be made rigorous up to a fixed accuracy?
In this question., a very inefficient, yet rigorous analytic approach for finding the next prime was established. I wondered whether a similar approach could exist to find the next non-trivial zero ($\...
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A question in regards to finding the roots of numbers
A simple algorithm for finding the square root of any integer goes as the following:
example $√25$: $25 - 1 -3 - 5 - 7 - 9 = 0$
We have deducted 5 numbers (-1, -3, -5, -7, -9) without resulting in a ...
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votes
0
answers
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views
Are the ordinates of the non-trivial zeros of $\zeta(s)$ uniformly distributed around the mid points of Gram point intervals they can be mapped to?
Let $\rho_n$ be the $n$-th non-trivial zero of $\zeta(s)$ and $z_n = \Im(\rho_n)$ with $z_n > 0$ and $z_{n+1} \ge z_n$.
A well known method to establish that all $\rho$s reside on the critical line ...
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vote
0
answers
257
views
How to solve a system of quadratic equations?
Suppose we have a system of $p$ quadratic equations about $\mathbf{x} \in \mathbb{R}^3$ and $\mathbf{x} > 0$
$$ \left\{
\begin{array}{lr}
\mathbf{x}^\top \mathbf{C}_1 \mathbf{x} = 1, ...
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1
vote
1
answer
380
views
When does this limiting ratio give a real root $x$ to the equation of the form $\sum\limits_{k=0}^d \frac{x^k a_{k+1}}{k!}=0$?
By searching the Inverse Symbolic Calculator, we appear to be able to make the following conjecture about a real root to the equation:
$$\sum\limits_{k=0}^d \frac{x^k a_{k+1}}{k!}=0 \tag{1}$$
Let the ...
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1
vote
0
answers
50
views
On a structural decomposition of polynomials based on integral roots
Given an irreducible polynomial of structure $$f(x,y)=\sum_{\substack{i,j\in\{0,1,2\}\\i+j\leq3}}a_{i, j}x^iy^j\in\mathbb Z[x,y]$$ with $a_{2,1}a_{1,2}a_{1,1}a_{1,0}a_{0,1}a_{0,0}\neq0$ if $f(m,n)=0$ ...
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4
votes
0
answers
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views
Finding roots of this polynomial
I have a polynomial, given for parameters $x$ in $\mathbb{R}_+$ and $\alpha$ and $\beta$ in $\mathbb{R}_+^{n}$ by :
$$P(t) = \sum\limits_{i=1}^n \left\{\left(\beta_i - \frac{\alpha_i}{x} - t\right) ...
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vote
0
answers
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What is the distance of a particular root to the farthest one with respect to it as a function of a compacting factor?
Let $f:\mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function defined as
$$f := 2a_{1}(x - x^{p}) + 2a_{2}(x - x^{p})\sum_{k \in K}||x_{k} - x^{p}_{k}||^{2} + \dfrac{2a_{3}}{\alpha}\sum_{j \in J}\dfrac{...
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answer
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Zeros of partial sums of the Ramanujan's zeta function
In this post we consider the Ramanujan tau function $\tau(n)$, see the Wikipedia Ramanujan tau function, and we consider partial sums of its corresponding Dirichlet series (see for example the article ...
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votes
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answers
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Sign of an integer polynomial at a real algebraic number
Given $p\in\mathbb{Z}[x]$, assumed to have a real root, let $r$ be the largest real root of $p$. Now, given $f\in\mathbb{Z}[x]$ (without loss, of lesser degree than $p$), I would like to find out ...
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answers
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Isolating Real Roots of Polynomial–Exponential Function over Algebraic Numbers
Let $\mathbb{A}$ be the set of algebraic numbers and $\mathbb{A}[t]$ the set of polynomials in $t$ with coefficients in $\mathbb{A}$. Given a finite sum of $F(t)=\sum_{k}f_k(t) e^{\eta_k t}$ for ...
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1
vote
4
answers
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views
how to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$ [closed]
I’d like to solve following equation for $x.$
If it is not possible, why I can’t?
$$\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$$
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votes
1
answer
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views
root solving without analytic derivative
I have two related questions about numerical methods for root solving:
1) $f: R \to R$ is continuous and piece-wise smooth, with $f(a)f(b) < 0$.
$f$ has very high number of knot-points and ...
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votes
3
answers
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views
Fast root finding for strictly decreasing function
What is a fast algorithm to find the root of a strictly decreasing function?
If the root is not exact I want to find a root such that the function value is positive to an error.
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answers
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Multi-variate secant method for solving $F(x)=0$
The secant method for solving an equation $F(x)=0$ in one variable is much older than Newton's one. Recall that given two iterates $x_{k-1}$ and $x_k$, it provides an update $x_k$ by taking the ...
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