The polynomials tag has no wiki summary.
2
votes
2answers
40 views
How to properly use polynomial projection to get values at visualization nodes?
I am trying to implementing a nodal discontinuous Galerkin spectral element method for linear and non-linear systems of equations. The solution at each time step is given at ...
4
votes
1answer
48 views
Fastest method for evaluating the limit of the sign of a polynomial
Consider a multivariate polynomial $f(x) = f(x_1, \ldots, x_n)$ with maximum degree $d$. Following the linear symbolic perturbation scheme described in Seidel 1998, I want to evaluate the limit
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1
vote
2answers
64 views
Stable formula for a specific root of a depressed cubic
I have a cubic of the form
$$x^3 - a x - b = 0$$
where $a, b > 0$. Thus, I know there is exactly one positive root.
Is there a nice stable formula for this unique positive root?
2
votes
4answers
152 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 ...
13
votes
2answers
159 views
Why do equi-spaced points behave badly?
Description of experiment:
In Lagrange interpolation, the exact equation is sampled at $N$ points (polynomial order $N - 1$) and it is interpolated at 101 points. Here $N$ is varied from 2 to 64. ...
4
votes
3answers
156 views
Fastest polynomial root finder for a given accuracy
I am looking for a very fast polynomial root finder, hopefully with a matlab code. I don't need very accurate results, 2-3 decimal places would be fine. Also the method should be able to optionally ...
2
votes
1answer
149 views
How to apply a Galerkin finite element method to a linear, one-dimensional boundary value problem
I have the following boundary value problem:
$$-(\alpha u')' + \gamma u = f $$
in $\Omega = (a,b)$ with b.c. $u(a) = u(b) = 0$
and $\alpha > 0, \gamma ≥ 0$ and $f:(a,b) \to \Re $
The weak ...
6
votes
1answer
126 views
Polynomial Fitting from Chebyshev Coefficients
I have been reading Numerical recipes about how to create a power series approximation to a function once you have a Chebyshev approximation to the function. However it is still very unclear to me how ...
5
votes
2answers
292 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 ...
9
votes
0answers
185 views
Difficulty with Spectral Method using Chebyshev Polynomials
I am having a bit of difficulty in trying to understand a paper. The paper uses spectral method to solve for an eigenvalue that comes from a system of coupled ODEs. I will write out only one equation ...
5
votes
2answers
122 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 ...
2
votes
2answers
267 views
Evaluation of multivariate polynomials
I am seeking for an efficient algorithm to compute a multivariate polynomial of a fixed structure, but different coefficients and evaluation points. The question is the same as this one, which ...
3
votes
1answer
80 views
Numerically stable real solution(s) to a system of bivariate quadratics
I have a a system of bivariate polynomials as follows:
$ E(u,v): e_2(u) v^2 + e_1(u) v + e_0(v) = 0 \\
F(u,v): f_2(u) v^2 + f_1(u) v + f_0(v) = 0$
where $e_n(u) = e_{n_2}u^2 + e_{n_1} u + e_{n_0}$ ...
7
votes
1answer
71 views
Numerically stable algorithms for computing remainder of polynomials
Let $f, g \in \mathbb{R}[x]$ and $\deg f > \deg g$. I am looking for asymptotically fast and numerically stable algorithms for computing $f \bmod g$. In the applications intended, both $f, g$ are ...
12
votes
3answers
200 views
Uses of power series maps
I'm from the field of accelerator physics, specifically related to circular storage rings for synchrotron light sources. High energy electrons circulate around the ring, guided by magnetic fields. ...
