Questions tagged [roots]

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Sufficient condition for real roots of a polynomial of order $n>5$ with arbitrary real coefficients

I ask for help in solving the problem. I am developing an optimization program that selects the coefficients of a polynomial of order $n> 5$ so that all its zeros are just real numbers. And I ...
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2 votes
3 answers
116 views

Finding the roots of a function like $3+\cos(x)+0.005/(x-a)$?

I have a blackbox function that (for the purposes of this question) looks like $3+\cos(x) + 0.005/(x-a)$. The location of this $a$ is unknown (but within $0\to2\pi$). (The $3+\cos(x)$ is just a (bad) ...
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1 vote
0 answers
92 views

How can I practice multivariable root-finding?

Recently, I've been reading up on various root-finding / optimization algorithms such as the Levenberg-Marquardt method, Gauss-Newton, Conjugate Gradient, trust-region and trust-region-dogleg. I've ...
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2 votes
1 answer
584 views

Constrained Newton-Raphson root finding

Original Question I have a set of non-linear equations and I need to find the root where a subset of my solution vector is constrained to be greater than or equal to 0. I have implemented the Newton-...
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0 answers
45 views

Fastest way to find roots of second order polynomial upto single decimal point?

What is fasted way to find roots of second order polynomial up to single decimal point using a program.
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  • 101
2 votes
2 answers
393 views

What is an efficient way to calculate zeros of Bessel functions?

One approach is the brute force method of evaluating at all points at fixed intervals and when it nears zero write value, this can be combined with adaptive step size. Another approach is ...
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2 votes
2 answers
106 views

ODE Event detection for calculating multiple roots of continuous sinusoidal equation

I found a paper [1] that has a method for computing rise and set times of a satellite given a closed form solution. It is a complicated sinusoidal function and the paper has a method to calculate a ...
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  • 141
3 votes
1 answer
88 views

Robust ways to find zeros of the Tricomi confluent hypergeometric function as a function of its parameters

I'm solving a quantum mechanical problem, and the quantization condition requires me to solve the equation $$ U\left(\frac12(\ell+1-E), \ell+1, r^2\right) = 0, $$ where $U(a,b,z)$ is the confluent ...
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1 vote
1 answer
115 views

Find the roots of a complicated polynomial

For the polynomial $$p(x) = (x-1)(x-2) \cdots (x-20) - 2^{-23}x^{19}\, ,$$ I tried to use fzero() in MATLAB, and I set the interval to be $[0.5,1.5]\cdots [19.5,...
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  • 29
4 votes
1 answer
327 views

Efficient root finding algorithm for monotonic function

This is my first time asking a question here, so I may not be asking this in the right place. I am trying to find the roots of a monotonic function with as few function evaluations as possible. I ...
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  • 143
6 votes
1 answer
640 views

Defining a condition number and termination criteria for Newton's method

The condition number of function evaluation $$ \mathrm{cond}(f,x) := \left| \frac{x f'(x)}{f(x)} \right| $$ is infinite at a root of $f$. Hence it is useless for rescaling a tolerance which defines an ...
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8 votes
0 answers
114 views

Finding the smallest root of a function on $[0, \infty)$

I would like to find the smallest real root of a 1-D real-valued function $f(x)$ on the domain $x\in [0,\infty)$. In this problem, I can make the following guarantees on $f$: $f$ does have a root at ...
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  • 695
2 votes
1 answer
52 views

Test on a set of high degree polynomials whose coefficients in {-1,0,1}

I'm looking for the best way of implementing the following algorithm: consider the set of all polynomials with a high degree (say, degree 30) whose coefficients ranges from a given set of values (say, ...
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1 vote
1 answer
740 views

Confusion about determining the jacobian in a rootfinding algorithm

I have written some Python code to determine the numerical roots of the following non-linear equation: $$f_m=\tan\lambda_m - \frac{\lambda_m}{1+a}$$ where $\lambda_m\gt0$ and $a\geq0$. The code is: <...
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  • 277
0 votes
1 answer
421 views

Obtaining extra output argument(s) from the objective function used by fsolve in MATLAB

I have a MATLAB code (see below) that employs 'fsolve' from the optimization toolbox for a root finding problem. The bottleneck is that, within the objective function calculation, there is a ...
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1 vote
0 answers
168 views

vectorizing optimization or root finding [closed]

I need to find the roots of a function. I am currently using scipy.optimize.fsolve ...
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  • 163
1 vote
0 answers
51 views

Calculating fixed point inside limit cycle

So I'm working with a rather complicated dynamical system. Instead of writing it all out. It's probably easier if you just clone my git repository. ...
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  • 111
1 vote
2 answers
504 views

Dekker's method and fixed further border [closed]

Dekker's method and in its evolution Brent's method have as design goal to combine the certainty of a root, certified by function values of opposite sign in an increasingly smaller interval, of ...
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  • 119
1 vote
2 answers
145 views

Iterative root finding of 2-dimensional system of non-linear equations with monotonicity properties

Let $f(x,y)$ and $g(x,y)$ be two $\mathbb{R}^2\rightarrow\mathbb{R}$ functions, both strictly increasing in both arguments. Assume that they are well-behaved functions (continuous, differentiable, etc....
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  • 1,629
2 votes
1 answer
38 views

Finding common roots of two multivariate equations

I am trying to find points of a surface whose normal is parallel with the 3 dimensional axes (i.e. local maxima, minima and saddle points in x, y and z dimensions). The function of the surface is ...
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3 votes
1 answer
278 views

Fastest method forfinding a solution to x*log(x)

more specifically, $C x \log_2(x)- K = 0$, where $C$ and $K$ are constants ($\log_2$ means log base 2). I was solving a topcoder problem, SortEstimate, which requires us to solve the aforementioned ...
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6 votes
1 answer
194 views

Roots of a function for eigensystem

I want to find the roots for $\kappa$ for the equation $$\sqrt{\alpha - 1} \cos{\left (\frac{\sqrt{2} \sqrt{\alpha - 1}}{2 \sqrt{\epsilon}} \right )} \cosh{\left (\frac{\sqrt{2} \sqrt{\alpha + 1}}{2 \...
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  • 8,006
2 votes
1 answer
201 views

Difference between Brent's and Alefeld-Potra-Shi for root finding

I need to find the (unique) root of a nonlinear function $f(x)$, $x \in \mathbb{R}$. For the record, $f(x)$ is the CDF of a probability density minus a constant $0 < p < 1$ (I am inverting the ...
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  • 183
5 votes
1 answer
634 views

Efficient and stable computation of inverse CDF

What is the most efficient and numerically stable algorithm for computing the inverse CDF $F^{-1}(y)$ of a probability function, assuming that both the PDF $f(x)$ and the CDF $F(x)$ are known ...
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  • 183
8 votes
2 answers
136 views

Bracketing a discontinuity in a step function

I have the function $f(x) = \begin{cases} 0 \:\: (x < a) \\ 1/2 \:\: (x = a)\\ 1 \:\: (x > a) \end{cases}$, where $a$ is unknown. I can compute the function for any value of $x$, and seek to ...
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4 votes
1 answer
167 views

solving for unknown inside an expectation

I need to find roots for the following function: $$f(\theta) \equiv E[R(\theta;\eta)]=0$$ for some unknown $\theta$ which is deterministic, while the expectation is taken over a normally ...
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  • 235
4 votes
1 answer
139 views

Finding self-kissing points on a plane curve?

I have a curve in the complex plane given by $$ f(t) = \sum_k r_k\exp(2\pi\mathrm{i}(t+\varphi_k)p_k). $$ Some of the parameters are specially chosen: $r_k>0$, $\sum_k r_k=1$, $p_k\in\mathbb{Z}$, $\...
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  • 11.4k
3 votes
2 answers
543 views

Bessel EVP and fzero

I am trying to solve the Eigenvalue problem $$ x^2 y''+ x y' + x^2 y = \lambda^2 y\,,\quad x\in(0,1)\,,\quad y(0)=0\,,\quad y'(1)=y(1) $$ The differential equation is the Bessel equation. The solution ...
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4 votes
1 answer
84 views

Evaluate sine of a polynomial root close to $\pi$

Consider the polynomial $$ p(x) = -514-462 x+359 x^2+1129 x^3+165 x^4+490 x^5-418 x^6+497 x^7-227 x^8+60 x^9-10 x^{10}, $$ whose root $A\approx 3.14$ is very close to $\pi$: $$|A-\pi|=2.0746\times 10^...
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  • 11.4k
2 votes
0 answers
67 views

Solving a nonlinear equation with a Markov process and RVs

Assume that we have the following equation and the following assumption. The scope is to solve for some particular variables expressed later. Update $$E_{t}\left[ b(A_{t+1})^{1-\gamma} *R_{t+1}^{-\...
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  • 235
3 votes
1 answer
310 views

Testing 1D root-finding procedures for robustness

How can I test whether a given 1D root-finding procedure is robust? I know that there are data sets and resources online for different kinds of optimization, but I have yet to find anything with ...
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4 votes
2 answers
190 views

Non-linear root finding with positive definite Jacobian

I am dealing with a system of non-linear equations: $$ f(\boldsymbol{x}) = \boldsymbol{y}, \;\;\; \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^d. $$ And I know that the Jacobian $J(\boldsymbol{x})$ ...
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  • 697
5 votes
0 answers
733 views

Finding roots of systems of equations with a Jacobian that is singular everywhere

Given $a\in\mathbb{R}^{mn\times n}$, find a $C\in\mathbb{R}^{n}$, $x\in\mathbb{R}^{m\times n}$ such that $$ 0 = f_{k}(\boldsymbol{C}, \boldsymbol{x}):=\sum_{i=1}^{m} C_{i} \left(\prod_{j=1}^{n} a_{kj}^...
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6 votes
2 answers
252 views

Continuation procedure to solve for a 2D curve that satisfies f(x,y) = 0

I have some function of $R^2$, that must be numerically computed. For instance, I might be interested in a real-valued contour integral that begins from (x,y) = 0. $$ f(x,y) = \Re\left[\int_0^{x + iy}...
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  • 303
7 votes
4 answers
8k views

How to find more than one root of a polynomial?

This program finds the first root of the function f, defined in the code. There are 5 roots of this function. (x=1,2,3,4,5) I wish to find all of the roots in this program and print them to the screen....
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10 votes
2 answers
3k views

Find all the roots of a function in a given interval

I need to find all the roots of a scalar function in a given interval. The function may have discontinuities. The algorithm can have a precision of ε (e.g. it is ok if the algorithm doesn't find two ...
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3 votes
1 answer
1k views

Non-linear root finding when the Jacobian is almost singular

I'm trying to solve a system non linear-equations: $$ \frac{\partial K(\mathbf{\lambda})}{\partial \lambda_i} - c_i = 0 $$ for $i = 1, \dots, 15$, using Newton's method: $$ \lambda^{k + 1} = \lambda^k ...
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  • 697
11 votes
3 answers
4k views

Solution of quartic equation

Is there a open C-implementation for the solution of quartic equations: $$ax⁴+bx³+cx²+dx+e=0$$ I am thinking of an implementation of Ferrari's solution. On Wikipedia I read that the solution is ...
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  • 1,109
8 votes
5 answers
372 views

Iterative solution to a nonlinear equation

I appologize in advance if this question is silly. I need to compute the root of \begin{equation} u -f(u) =0 \end{equation} Where $u$ is a real vector and $f(u)$ is a real-vector valued function. ...
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6 votes
2 answers
213 views

Is there a backward stable $\tilde{O}(n \log(1/\epsilon))$ algorithm to factor a complex polynomial?

Finding the roots of a complex polynomial is in general extremely numerically unstable, as discussed in (1). According to Pan ((2), (3)), this produces a cubic complexity lower bound, and he presents ...
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