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Questions tagged [perturbation-theory]

Perturbation theory describes a range of tools and techniques to find approximate solutions to problems containing small parameters.

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Boundary for eigenvectors of perturbed tridiagonal matrix

Let $A = \left[\begin{array}{cccc} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{array}\right] \; \; $ and $H $ a ...
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Can we approximate a.e. invertible matrices with everywhere invertible matrices in $L^2$ sense?

Let $\mathbb{D}^n=\{ x \in \mathbb{R}^n \, | \, |x| \le 1\}$ be the closed unit ball, and let $A:\mathbb{D}^n \to \mathbb{R}^{n^2}$ be real-analytic on the interior $(\mathbb{D}^n)^o$ and smooth on ...
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20 views

Matrix perturbation and eigenvector

$Ae=\lambda e$ $e^Te=1$ $A$ is a real matrix. $\lambda, e$ are real. $(A+\Delta A)(e+\Delta e)=(\lambda+\Delta \lambda)(e+\Delta e)$ Neglecting small terms, $\Delta Ae +A \Delta e=\Delta \lambda ...
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Ruling out nonzero roots of a transcendental equation

I'm interested in the function $$ s(u) = e^{zu} - (1 + \beta u + \alpha u^2) $$ for $u \geq 0$. Here $\alpha = g z^4$ and $\beta = z^2 (1 - 2gz)$ where $g,z > 0$ are parameters such that $1 + \...
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2answers
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Finding leading order approximation to $\frac{d^2y}{dx^2}-\epsilon y=x$

The conditions provided were $y(0)=1$ and $y'(0)=1$. Since this equation is regular, I can neglect the term involving $\epsilon$, giving $\frac{d^2y}{dx^2}=x$. I tried to solve this in order to ...
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1answer
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Solving IVP exactly with an epsilon variable

I am unsure of how to interpret the question: Given $y''+(1+\epsilon)y=0, y(0)=0,y'(0)=1$. Solve exactly. The context of the problem is that we are practising solving IVP with regular perturbation ...
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57 views

Variation of geometrical quantities under infinitesimal deformation of surfaces

This question is about 2-d surfaces embedded in $\mathbb{R}^{3}$. It's easy to find information on how the metric $g_{\mu\nu}$ changes when $x_{\mu}\rightarrow x_{\mu}+\varepsilon\xi(x)$. So, ...
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1answer
24 views

Eigenvectors of sum of hermitian matrices

Given two real Hermitian matrices $A$ and $B$, what can one say about the eigenvectors of $A+ \epsilon B$ in relation to $A$? Here $\epsilon \in [0,1]$ and $\epsilon B$ is a slight perturbation.
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What is the change of Laplacian operator under small coordinate transformation? [closed]

Consider the Laplacian operator $\nabla^{2}=\frac{1}{\sqrt{g}}\frac{\partial}{\partial q^{i}}\left(\sqrt{g}g^{ij}\frac{\partial}{\partial q^{j}}\right)$. How does the Laplacian operator transform ...
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Inequality on pairs of projections in Kato's book

I do not understand an argument (p. 58, l.2--3) regarding two "close" projections, in the proof of Theorem I.6.34, pp. 56--58, Kato's book "Perturbation Theory for Linear Operators". The setting is ...
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Finding roots of cubic Algebraic perturbation [closed]

So I am confident with finding roots of regular perturbation problems, e.g. $x^3+(3-2ϵ)x^2+(3+ϵ)x+1-2ϵ=0 $ in this case when $ϵ=0$ we have $(x+1)^3=0$. please help.
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Perturbation for non-symmetric matrices

If we have two matrices $A_1$ and $A_2$, with singular value decompositions $A_1 = U_1 \Sigma_1V_1^\top$ and $A_2 = U_2\Sigma_2V_2^\top$ and let $U_1^k$ denote the first $k$ singular vectors of $U_1$. ...
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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 $$F(\beta,\lambda)=\sum_{m=0}^n(\lambda-\lambda_0)^mf_m(\beta)$$ ...
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Can we perturb a map to have distinct singular values?

Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \text{GL}_n^{-}$ be smooth. ($\text{GL}_n^{-}$ is the set of $n \times n$ real matrices with negative ...
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How to evaluate the accuray of quadratic eigenvalue problem (QEP)?

When solving the QEP, we transform it into a GEP and then use qz algorithm to handle it. But there are several formulations of GEP, how to evaluate the accuracy and stability of the solution? I ...
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Divergence of a series using perturbation method to solve a 2nd order polynomial

Consider the following equation $$x^2-2(1+e)x+1-e=0$$ As part of an H.W assignment I'm requested to attempt and find the roots using perturbation theory, and compare to the exact solution. When I ...
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1answer
31 views

Solving a simple O.D.E using perturbation theory

As a part of an H.W assignment, I'm requested to solve the following O.D.E using perturbation theory and compare to the analytical solution $$\dot{y}=y+\epsilon$$ where the solution of this equation ...
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1answer
22 views

Positive and negative powers of small parameter in perturbation problem

I'd like to perturbatively handle an eigenvalue problem similar to this: $$ \lambda f = (\hat{H} + 1/\epsilon^2 \hat{V} + \epsilon {W}) f, $$ where $f$ is a function, $\lambda$ is an eigenvalue, $\...
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Finding an asymptotic expansion for $\displaystyle \int_0^\infty t \exp(-(t-xt^{-1})^2) \ dt$ as $x\to0$.

I am looking to find the first few terms in an asymptotic expansion for the integral $$ \int_{0}^{\infty}t \exp\left(-\left[t-xt^{-1}\right]^{2}\right)\,\mathrm{d}t \quad\mbox{as}\quad x \to 0 $$ I am ...
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Big-O / smal-o for perturbed matrix-inversion: What is $(A_n + \mathcal O_n(n^{-\alpha}))^{-1}$?

Let $\alpha > 0$ and for each natural number, let $B_n$ be a square matrix $B_n = \mathcal O(n^{-\alpha})$ as $n \rightarrow \infty$. Suppose $A_n$ is ineverible for every $n$ and $A_n \rightarrow ...
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Expression for the remainder when approximating $f(x)$ as $x\to \infty$ in $xf^\prime + f = x^{-3} + \frac{1}{2}f^{\prime\prime}$

I am working through the exercises in the textbook http://inis.jinr.ru/sl/vol2/Mathematics/Hinch,_Perturbation_Methods,1995.pdf and I am struggling to complete exercise 2.2. For the first part, I have:...
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67 views

Change in eigenvalues if row and column added to highly symmetric matrix

I have a symmetric matrix like the following:$$\begin{bmatrix}a&a&a&a\\a&b&b&b\\a&b&b&b\\a&b&b&b\end{bmatrix}$$It's a symmetric real matrix with only 3 ...
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Verifying perturbation theory proof mentioned in an online article

I am reading an article mentioned here http://www.bcamath.org/documentos_public/archivos/personal/comites/course.pdf. I am stuck at page 14 where it says "Thus from (2.1.7) it follows that the ...
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How do eigenvalues change if we perturbe the diagonal entries of a matrix?

Suppose $A \in M_n(\mathbb R)$ is a stable matrix, i.e., all eigenvalues are on the left open half plane of $\mathbb C$. If in particular, all the Gershgorin disks $\Gamma_j$ corresponding to rows $j=...
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1answer
32 views

How the perturbation of a Markov chain affect the stationary distribution

I wonder whether there is such kind of relation between the scale of the perturbation and the stationary distribution of a Markov chain. Suppose $\hat{P}=P+F\in\mathbb{R}^{n\times n}$, where $F$ is ...
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139 views

How do the eigenvalues change if we change the diagonal entries of the matrix?

Suppose $A \in M_n(\mathbb R)$ is stable. By stable, we mean the eigenvalues are all on the left open half plane of $\mathbb C$. Now if we decrease the value of $A_{11}$, does the matrix remain stable?...
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Singular perturbation theory in non-standard form

Singular perturbation theory in ODE's is a well treated and highly studied subject. Most of the references I can find take the form, \begin{align} \dot{x} &=f\left( x,z,\varepsilon \right) \\...
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109 views

Can we choose smoothly the singular vectors of a matrix?

Let $X$ be the space of all real $n \times n$ matrices, with strictly negative determinant, and pairwise distinct singular values. $X$ is an open subset of the space of all real square matrices. Is ...
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1answer
50 views

Find a two term asymptotic expansion of the following problem

I want to find a two term asymptotic expansion, for small $\epsilon$, of the solution of the following problem: $$ y'' - \epsilon y' - y = 1\tag{1}, \, y(0) = 0, \,y(1) = 1 $$ My approach: I assume ...
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1answer
60 views

Matching expansion of an ODE: $\epsilon y'' + xy' + y = 0$

I am trying to solve a boundary layer problem using matched expansion $$\epsilon y'' + xy' + y = 0$$ where the boundary condition is $$y(0) = 1, y(1) = 1$$ and $x\in (0,1)$. So far, I have the outer ...
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1answer
45 views

Perturbation Methods ODE Example

I am currently doing a Masters-level class in Perturbation Methods and I am stuck on a question. I have done similar questions but the middle term is confusing me. It could be that I'm rescaling wrong ...
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Expected value of eigenvalues of perturbed matrix

Let $A$ be an arbitrary $n \times n$ real matrix and $P$ be a random perturbation matrix with zero-mean, i.i.d. entries. Can we say anything about the eigenvalues of $A+P$? In particular, are there ...
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Definition of perturbation using norms

Can someone give me the definition of a perturbation vector? I know that it is a vector composed of $x_i$ that have a small value and are independent, but I know I can define it by using norms because ...
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1answer
44 views

Solution Poisson Equation by Homotopy Perturbation Method

I need to solve the following Boundary Value problem $$\frac{\partial^2w }{\partial x^2}+\frac{\partial^2w }{\partial y^2}=c$$ Boundary conditions are $$w(x,h)=0$$ $$w\left(\frac{\pm h}{\sqrt3},...
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Stationary Phase with controlled phase $|\int_0^Te^{\frac{i}{\epsilon}\phi_j(t)}dt| = O(\epsilon)$

My question concerns the stationary phase theorem when we only know control information on the phase. Suppose that $\phi_a(t)$ and $\phi_b(t)$ are phases such that $$\phi_a(t) = a_1 + a_2t$$ $$\...
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1answer
41 views

Can we perturb a low rank map to a full rank map in a smooth way?

Let $f:\mathbb{R}^n \to \mathbb{R}^n$ be smooth. Can we find, for every $\epsilon>0$, a $C^1$ map $\tilde f:\mathbb{R}^n \to \mathbb{R}^n$ of full rank such that $\|df-d\tilde f\|_{C^0}<\...
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1answer
32 views

Showing that $\lambda - (A + B)$ has dense range

Let $A$ be the generator of a $C_0$-semigroup $(T(t))_{t \geq 0}$ of contractions on a Banach space $X$ and $B \in \mathcal L(X)$ a bounded operator. To apply some approximation formula I want to show ...
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1answer
31 views

Perturbation expansion within trig function

I'm trying to find an approximate solution to a nonlinear differential equation. It involves something to the effect of $\frac{d\Psi}{ds} = \sin{\Psi} + \dots$ , where $\Psi$ is a small variable. If ...
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37 views

Perturbation solution Dirac equation

I'd like to know how to solve the Dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\...
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What should I do with: $\frac{d}{dx} \frac{d}{d \xi}$ in multiple scale perturbation for second-order eqs?

What should I do with: $\frac{d}{dx} \frac{d}{d \xi}$ in multiple scale perturbation for second-order eqs? I'm doing my first ever exercise in perturbation theory for "multiple scales". The eq. is: ...
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1answer
21 views

Perturbation of Diracs equation (first order)

I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\...
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1answer
114 views

Mandelbrot set perturbation theory: When do I use it?

I have read the post on Perturbation of Mandelbrot set fractal. I will also be referring to the PDF by K.I. Martin on this topic. My question is to do with the precision laid out in the Martin paper. ...
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1answer
38 views

How to find the critical point for this coulomb field

Two equal positive charges are at distance $d$, $-d$ from the origin on the $y$ axis. What is the distance on the $x$ axis beyond which a small perturbation in $y$ will move a particle away from the $...
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1answer
50 views

Effect of perturbation on eigenvalues

Given a diagonal matrix $A$ with diagonal elements $a_1,\dotsc,a_n$ what can we say about the eigenvalues of the perturbation $A+E(t)$ satisfying $E(t) = O(t^k)$ so that there exists a constant $K>...
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1answer
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In perturbation theory, does it make sense to consider added error to “shift” the function and alter no. of roots?

In perturbation theory, does it make sense to consider added error to "shift" the function and alter no. of roots? That is consider e.g. $f(x)=x^2+\epsilon$. For $\epsilon=0$ there's a single root $x=...
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1answer
54 views

Inverting a power series matrix

Let's assume I want to invert a matrix function $M=M(x)$, which is expressed as a power series of the small parameter $\epsilon$ $$M = M_0(x) + M_1(x) \epsilon + M_2(x) \epsilon^2 + \mathcal{O}(\...
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0answers
19 views

Perturbing a matrix with asymptotic constraints

I want to expand the following function in series of $\epsilon = a/l$: $$f=\frac{h \cos\alpha + l\sin\alpha}{l \cos \alpha + c \sin \alpha}$$ I know from the physics of the problem that also $$\frac{...
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2answers
149 views

A perturbed function has a stationary point, while the unperturbed has not

I am looking for extrema of the function $$ B(x,y) = \frac{1}{x} \Big[ F(y) - \epsilon [\log(x-y) +1] \Big]$$ limited to the the domain $\Omega = \{y \ge 0, x \geq y\} $ $F$ is a twice ...
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45 views

Perturbation theory of the eigenvalues about a symmetric matrix: Reality of Eigenvalues

Let $A$ be an $n\times n$ real symmetric matrix, and let $E$ be a real matrix. Is it true that if the perturbation matrix $E$ is small in some norm, then the eigenvalues of $\hat A := A+E$ are all ...
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1answer
156 views

System of equations and perturbation methods

I would like to characterise how the solution of a nonlinear system of equations change if a perturbation term is added. Namely, I have the system \begin{array}{lcl} 2y [F(x) - \varepsilon \frac{1}{y(...