All Questions
Tagged with perturbation-theory polynomials
23
questions
1
vote
0
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43
views
Degree with which a polynomial changes with some small change
Soft question: I was curious as to how one could measure the degree with which a polynomial is perturbed. More formally, let $P(x) \in \mathbb{C}$ be a polynomial and $\epsilon$ be a very small number,...
- 166
2
votes
0
answers
55
views
Bound on number of positive roots of deformed polynomial
In the comments for the following linked question Descartes rule of sign with positive real exponents, the following was stated:
" The positive roots depend continuously on the exponents. This ...
- 21
0
votes
0
answers
48
views
Finding regular and singular roots of a cubic perturbed polynomial using rescaling
Question: Find the rescalings for the roots of
$$\epsilon^5 x^3 - (3 - 2\epsilon^2 + 10\epsilon^5 - \epsilon^6)x^2 + (30 - 3\epsilon -20 \epsilon^2 + 2\epsilon^3 + 24\epsilon^5 - 2\epsilon^6 - 2\...
- 3,965
1
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0
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45
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Develop perturbation solutions of a cubic polynomial
Question: Develop perturbation solutions to
$$x^3 + (3+4\epsilon + \epsilon^2)x^2 + (3 + 9\epsilon + 7\epsilon^2 + 2\epsilon^3)x + 1 + 5\epsilon + 8\epsilon^2 + 5\epsilon^3 + \epsilon^4 = 0$$
finding ...
- 3,965
1
vote
0
answers
88
views
Asymptotic behavior of the zeros of a polynomials for large values of a parameter
Consider a polynomial in $r$ of the form
$$
r^4+p_3(\lambda)r^3+p_2(\lambda)r^2+p_1(\lambda)r+p_0(\lambda),
$$
where the $p_i$ are polynomials in the parameter $\lambda$. I use degree four to simplify ...
- 1,139
0
votes
0
answers
110
views
Under what conditions is a polynomial root perturbation problem singular?
I'm interested in what we know in general and rigorously. I've encountered lots of rules of thumb, i.e. when you change order or have a repeated root. But I'm interested in both what we can say ...
- 447
1
vote
1
answer
47
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What is the asymptotic solution to the roots of $x^n+a_1 \epsilon x^{n-1} +\cdots+ a_{n-1}\epsilon^{n-1}x+a_n \epsilon^n$?
The polynomial I'm working with is:
$$
\lambda^n
+\frac{\epsilon}{p \ 1!}\lambda^{n-1}
+\frac{\epsilon^2}{p^2 2!}\lambda^{n-1}
+\dots
+\frac{\epsilon^{n-1}}{p^{n-1} (n-1)!}\lambda
+\frac{\epsilon^{n}}...
1
vote
1
answer
63
views
Reed,Simon Theorem XII.1: Use of recursive substitution in the proof
We have a function $F(\beta,\lambda)$ (polynomial of degree $n$) which is analytic near $\beta_0$ and $\lambda_0$. So we can write
$$F(\beta,\lambda)=\sum_{m=0}^n(\lambda-\lambda_0)^mf_m(\beta)$$
...
- 83
2
votes
1
answer
982
views
Perturbation to the coefficients of a polynomial
I am reading 'Trefethen and Bau: Numerical Linear
Algebra' book and I came across the following problem- Assume that the polynomial $p(x) = \sum_{k=0}^n a_k x^k$ with real coefficients has $n$ ...
- 81
2
votes
1
answer
67
views
How to determine Solution from variables in Power
Today, while studying perturbation method for solving polynomial equations, I seen a problem, here I'm going to write those steps where I have problem
$$\epsilon^{1-3p}x_o^3 = \epsilon^{-p}x_o$$
The ...
7
votes
1
answer
884
views
Perturbing a polynomial with repeated real roots to get distinct real roots
Consider a real polynomial $f$ of degree $d$ which has $d$ real roots not necessarily distinct. In general, can we accomplish the following?
For every $\epsilon>0$, can we perturb each ...
- 215
1
vote
1
answer
182
views
Noise in polynomial systems.
Consider equations of the form
\begin{equation}
n_1 x^i + n_2 y^i = s_i
\end{equation}
where $n_1, n_2 \in \mathbb{N}$ are known coefficients and
$s_i > 0$ is a real constant.
Fix two indices $i&...
- 835
2
votes
1
answer
587
views
root perturbation of a polynomial and the method of dominant balance
Consider the following polynomial equation $f(z)=z^{2m}+a(1+z)^m=0$, with $a\to 0$. A simple argument using Rouche's theorem shows that $z_k = (1+o(1))\zeta_k a^{1/2m}$, where $\{\zeta_k\}$ are roots ...
- 1,289
4
votes
1
answer
457
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(Possible) Misunderstanding of perturbation method for finding solution of polynomial equation?
This is the strange moment that I get when I solve this equation:
$$ \frac{w^4}{4} - \frac{w^3}{3} = \varepsilon, $$
where $\varepsilon$ is a small parameter. If I plot the graph $ w \mapsto \frac{w^...
- 5,805
2
votes
1
answer
110
views
Why is $\varepsilon x^5 \sim -x$?
I'm trying to understand what's going on in this lecture on perturbation (the link brings you to 1h 08m 12s).
The original problem is to find the real root of $$x^5+x=1.$$ We have inserted $\...
- 7,720