Questions tagged [polynomials]

The tag has no usage guidance, but it has a tag wiki.

Filter by
Sorted by
Tagged with
5
votes
2answers
177 views

What are the numerical methods for huge polynomial systems?

Let a system of $n$ polynomial equations of degree $d$ with $m$ variables. I'm interested in a sparse system with $d = 3$, $n \sim 2000000$, $m \sim 50000$ and integer coefficients. What techniques ...
7
votes
1answer
694 views

Accurate Polynomial Evaluation in Floating Point

What are the most accurate algorithms for evaluating a polynomial using floating point arithmetic? The internet seems to suggest that Horner's method is commonly used. In particular I have a cubic ...
2
votes
1answer
1k views

rational functions in Python

I would like use Python to verify the following identities: $ \frac{1}{1-z} = 1 + z + z^2 + z^3 + \dots $ $ \frac{1}{1-z - z^2} = 1 + z + 2z^2 + 3z^3 + \dots $ $ q \prod_{n \geq 1} (1 - q^n)^{24} = ...
1
vote
0answers
84 views

Prove $T_n(x)$ of Chebyshev Polynomial given the recurrence relation [closed]

Using the recursion formula for Chebyshev polynomials, show that $T_n(x)$ can be written as $$T_n(x)=2^{n-1}(x-x_1)(x-x_2)...(x-x_n)$$ where $x_i$ are the $n$ roots of $T_n$ The recurrence relation:...
1
vote
1answer
78 views

Transfrom a Legendre polynomial from $\int_{-1}^{1}\phi_j(x)\phi_k(x)dx $ into $\int_{a}^{b}\phi_j(t)\phi_k(t)dt$ given $t=\dfrac{1}{2}[(b-a)x+(a+b)]$

The Legendre polynomials satisfy $$\int_{-1}^{1}\phi_j(x)\phi_k(x)dx = \begin{cases} 0 &j\neq k\\\\ \dfrac{2}{2j+1} &j=k \end{cases}$$ Suppose that the best fit problem is given on the ...
2
votes
1answer
1k views

Multivariate Orthogonal Polynomial Generation

I'm trying to apply the stochastic galerkin method to partial differential equation with multiple uniform random coefficients. I'm puzzled as to how to extend the corresponding orthogonal (legendre) ...
1
vote
1answer
68 views

Determine low-order polynomial lower bound

I have a function $f$ I'd like to determine numerically and I have a bunch of $(x, y)$ pairs which approximate the function in the following sense: all of the points satisfy $f(x) \leq y$, most of the ...
3
votes
1answer
155 views

What is the most efficient way to represent a 1D function using $hp$-finite element basis functions

Given a one-dimensional function (let's say infinitely differentiable) and a prescribed accuracy of an L2 (or H1) norm, what is the optimal mesh and (in general arbitrary) polynomial orders on each ...
3
votes
3answers
390 views

Evaluating large determinants with multivariate polynomial entries

I have some large (n~100) square matrices with entries two variable polynomials of bounded degree (roughly <20, but many entries are smaller) and integer coefficients, and I'd like to be able to ...
6
votes
0answers
607 views

Stochastic Galerkin projection approach for using generalized polynomial chaos expansion (GPCE) in solving PDE

I want to know if there is any way to define the test and trial function in the way that I want instead of using the default functions. So if I want define the polynomial and basis and coefficient, ...
3
votes
2answers
406 views

Trajectory Planning: Polynomials as universal, integrable approximation functions?

I'd like to program a trajectory planner, let's say for a robot, and I can pass acceleration commands to the robot, the robot can move in one dimension. The outcome of the planner is now a vector with ...
3
votes
2answers
889 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
70 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 $$\...
2
votes
2answers
298 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?
7
votes
4answers
7k 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....
24
votes
5answers
2k views

Why do equi-spaced points behave badly?

Experiment description: 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. Each ...
5
votes
4answers
3k 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 ...
1
vote
1answer
485 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 ...
7
votes
1answer
765 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 ...
10
votes
3answers
3k 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 ...
19
votes
1answer
605 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 ...
6
votes
2answers
189 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 ...
3
votes
2answers
1k 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
105 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 $e_n(u) = e_{n_2}u^2 + e_{n_1} u + e_{n_0}$ ...
9
votes
1answer
108 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 ...
16
votes
3answers
257 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. ...