Questions tagged [roots]
The roots tag has no usage guidance.
41
questions
3
votes
0answers
37 views
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 ...
2
votes
3answers
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) ...
1
vote
0answers
88 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 ...
2
votes
1answer
426 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-...
0
votes
0answers
43 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.
2
votes
2answers
320 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 ...
2
votes
2answers
104 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 ...
3
votes
1answer
74 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 ...
1
vote
1answer
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,...
4
votes
1answer
308 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 ...
6
votes
1answer
549 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 ...
8
votes
0answers
113 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 ...
2
votes
1answer
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, ...
1
vote
1answer
726 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:
<...
0
votes
1answer
399 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 ...
1
vote
0answers
154 views
vectorizing optimization or root finding [closed]
I need to find the roots of a function. I am currently using scipy.optimize.fsolve
...
1
vote
0answers
50 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.
...
1
vote
2answers
498 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 ...
1
vote
2answers
117 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....
2
votes
1answer
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 ...
3
votes
1answer
272 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 ...
6
votes
1answer
184 views
Roots of a function for eigensystem
I want to find the roots for $\kappa$ for the equation
$$\sqrt{\alpha - 1} \cos{\left (\frac{\sqrt{2} \sqrt{\alpha - 1}}{2 \sqrt{\epsilon}} \right )} \cosh{\left (\frac{\sqrt{2} \sqrt{\alpha + 1}}{2 \...
2
votes
1answer
192 views
Difference between Brent's and Alefeld-Potra-Shi for root finding
I need to find the (unique) root of a nonlinear function $f(x)$, $x \in \mathbb{R}$.
For the record, $f(x)$ is the CDF of a probability density minus a constant $0 < p < 1$ (I am inverting the ...
5
votes
1answer
587 views
Efficient and stable computation of inverse CDF
What is the most efficient and numerically stable algorithm for computing the inverse CDF $F^{-1}(y)$ of a probability function, assuming that both the PDF $f(x)$ and the CDF $F(x)$ are known ...
8
votes
2answers
133 views
Bracketing a discontinuity in a step function
I have the function
$f(x) = \begin{cases}
0 \:\: (x < a) \\ 1/2 \:\: (x = a)\\ 1 \:\: (x > a)
\end{cases}$,
where $a$ is unknown. I can compute the function for any value of $x$, and seek to ...
0
votes
0answers
65 views
Finding the root of an equation [closed]
I have given $i_1$, $i_2$ and $\alpha$ which can be real or integer. How can I find the roots of:
$$
(i_1 + i_p )^{\alpha} - i_p^{\alpha} - (i_2 \cdot i_p^{\alpha-1}) = 0
$$
4
votes
1answer
165 views
solving for unknown inside an expectation
I need to find roots for the following function:
$$f(\theta) \equiv E[R(\theta;\eta)]=0$$
for some unknown $\theta$ which is deterministic, while the expectation is taken over a normally ...
4
votes
1answer
138 views
Finding self-kissing points on a plane curve?
I have a curve in the complex plane given by
$$ f(t) = \sum_k r_k\exp(2\pi\mathrm{i}(t+\varphi_k)p_k). $$
Some of the parameters are specially chosen: $r_k>0$, $\sum_k r_k=1$, $p_k\in\mathbb{Z}$, $\...
3
votes
2answers
529 views
Bessel EVP and fzero
I am trying to solve the Eigenvalue problem
$$
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)
$$
The differential equation is the Bessel equation. The solution ...
4
votes
1answer
84 views
Evaluate sine of a polynomial root close to $\pi$
Consider the polynomial
$$ 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^...
2
votes
0answers
66 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
$$E_{t}\left[ b(A_{t+1})^{1-\gamma} *R_{t+1}^{-\...
3
votes
1answer
286 views
Testing 1D root-finding procedures for robustness
How can I test whether a given 1D root-finding procedure is robust? I know that there are data sets and resources online for different kinds of optimization, but I have yet to find anything with ...
4
votes
2answers
185 views
Non-linear root finding with positive definite Jacobian
I am dealing with a system of non-linear equations:
$$
f(\boldsymbol{x}) = \boldsymbol{y}, \;\;\; \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^d.
$$
And I know that the Jacobian $J(\boldsymbol{x})$ ...
5
votes
0answers
724 views
Finding roots of systems of equations with a Jacobian that is singular everywhere
Given $a\in\mathbb{R}^{mn\times n}$, find a $C\in\mathbb{R}^{n}$, $x\in\mathbb{R}^{m\times n}$ such that
$$
0 = f_{k}(\boldsymbol{C}, \boldsymbol{x}):=\sum_{i=1}^{m} C_{i} \left(\prod_{j=1}^{n} a_{kj}^...
6
votes
2answers
242 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.
$$
f(x,y) = \Re\left[\int_0^{x + iy}...
7
votes
4answers
8k views
How to find more than one root of a polynomial?
This program finds the first root of the function f, defined in the code. There are 5 roots of this function. (x=1,2,3,4,5) I wish to find all of the roots in this program and print them to the screen....
10
votes
2answers
3k views
Find all the roots of a function in a given interval
I need to find all the roots of a scalar function in a given interval. The function may have discontinuities. The algorithm can have a precision of ε (e.g. it is ok if the algorithm doesn't find two ...
3
votes
1answer
1k views
Non-linear root finding when the Jacobian is almost singular
I'm trying to solve a system non linear-equations:
$$
\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 ...
11
votes
3answers
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 ...
8
votes
5answers
371 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.
...
6
votes
2answers
209 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 ...
