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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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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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$0 \cdot x=1$ has “no finite solution”? What does this mean? [closed]

$0 \cdot x=1$ has "no finite solution"? What does this mean? I've only used to refer to $0=1$ and similar as untrue. Context: In some text about perturbation theory (https://en.wikipedia.org/wiki/...
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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
42 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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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
126 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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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
139 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(...
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2answers
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Is the set of matrices with distinct singular values connected and dense?

$\newcommand{\GLp}{\operatorname{GL}_n^+}$ $\newcommand{\SO}{\operatorname{SO}_n}$ Consider the set $$ X=\{ A \in \GLp \,\,|\,\, \text{ all the singular values of } A \text{ are distinct } \},$$ where ...
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Consider the singularly perturbed cubic equation $\epsilon x^3+x^2-1=0$

Question: "Consider the singularly perturbed cubic equation $\epsilon x^3+x^2-1=0$ where $\epsilon >0$ is a small parameter. Obtain three-term astymptotic expansions for the two regularly ...
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Find approximation for two independent solutions to ODE using WKB method: $ϵ(1+x)y'' + y' +(1+x)y=0$

The boundary condition: $\epsilon (1+x)y'' + y' +(1+x)y=0 $, $y(0)=0, y(1)=0$ So far I got: $O(1): (1+x)(s'_0)^2 + s'_0 = 0$ a) $s'_0 = 0, s_0=0$ b) $(1+x)s'_0=-1, s_0=-ln(1+x)$ $O(\epsilon): ...
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first order Taylor expansion term of a function multiplied by a dot product of gradients.

I need to do the following Taylor expansion. Let's suppose we have a function $f(n,|\nabla n|^2)(\nabla (|\nabla n|^2),\nabla n)$, where by $(\nabla (|\nabla n|^2),\nabla n)$ I mean the dot product ...
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44 views

Regular Perturbation to obtain solution to: $y''+y= \epsilon y^2$

The following perturbation problem: $y''+ y = \epsilon y^2, y(0,\epsilon)=1, y'(0,\epsilon)=0$ So far I have deduced that: $y_0''+y_0=0, y_0(0)=1, y_0'(0)=0$, and so $y_0(x)= cos(x)$ For the second ...
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2answers
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Using Regular Perturbation, obtain approximate solution:

Consider the following ODE: $y'+ y = \epsilon y^2, y(0,\epsilon)=1$ Assuming the perturbation amsats gives: $y'_0 + y_0 =0, y_0(0)=1,$ $y'_1 + y_1 = y^2_0$, $y_1(0)=0$ How is: $y_0(x)=e^{(-x)}$ ...
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1answer
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Solve by means of regular perturbation to obtain an approximate solution up to and including $\mathcal{O}(\epsilon^2)$

I have to solve the following ODEs: $y''+ y = e^{\epsilon \cos x}$, $y(0,\epsilon)=y(1,\epsilon)=0$; $y''+ y = \epsilon y^2$, $y(0,\epsilon)=1$, $y'(0,\epsilon)=0$. I am having trouble ...
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Lower bound for the speed convergence of the spectrum

Let $A$ be a (complex unital) Banach algebra and $a \in A$ with spectrum $\sigma(a) = \{ 0 \}$ ($a$ is quasinilpotent). For $b \in A$ and $\varepsilon > 0$ consider the linear perturbation $a + \...
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Obtain first two terms in the asymptotic approximation $\epsilon x^3+x^2+2x-3=0$

Consider the singularly perturbed cubic equation $$\epsilon x^3+x^2+2x-3=0$$ where $\epsilon >0$ is a small parameter. This equation has two regularly perturbed roots and one singularly perturbed ...
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1answer
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WKB Approximation to Schrödinger Equation

Consider the Schrödinger equation: $$y''(x)+EQ(x)y(x)=0,\tag{1}$$ where $E>0, Q(x)>0, y(0)=y(\pi)=0. $ Use WKB approximation to obtain $$y(x) \sim CQ^\frac{-1}{4}(x)\sin\Big(\sqrt(E) \int_{0}^{...
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Conditions for Boundedness of Spectral Measures of Perturbations of Self-Adjoint Operators?

Suppose $A$ is an unbounded self-adjoint operator in a Hilbert space $H$ with discrete spectrum $$\lambda_0 < \lambda_1 < \cdots$$ bounded below with lowest eigenvalue $\lambda_0$, lowest ...
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Change in eigenvalues due to perturbation to a correlation matrix

Let $A$ be a $m \times n$ matrix defined as $ A = \Big[\frac{a_1}{\|a_1\|} \cdots \frac{a_n}{\|a_n\|}\Big]$ and $a_k \in \mathbb{R}^{m\times 1}$ where $k \in [1,\dots,n]$. Now, we define a ...
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1answer
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Solve for Differential Equations by using regular perturbation method

carry out regular perturbation calculation for $\epsilon$ satisfying $$x''(t)+x(t)=\epsilon x^2(t)$$ correct to the second order in the small parameter $\epsilon$. then use the result to perform a ...
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1answer
69 views

Obtain the leading order uniform approximation of the solution

Obtain the leading order uniform approximation of the solution to $ \epsilon y′′-x^2y′-y=0$. The boundary conditions are $y(0)=y(1)=1$. Since $a(x)<0$ the boundary layer is at $x=1$. The outer ...
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1answer
39 views

Eigen-vector and eigen-function perturbation

In the following question we assume the eigenvalues are sorted in descending orders and the eigen-vectors and eigen-functions are sorted accordingly. Given a symmetric $n\times n$ matrix $P$ and a ...
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1answer
84 views

Perturbation of complex square root function

Let $z$ and $w$ be complex numbers. Than it is claimed that $$ |\sqrt{z+w} - \sqrt{z}| \leq \frac{|w|}{\sqrt{|z| + |w|}} $$ or $$ |\sqrt{z+w} + \sqrt{z}| \leq \frac{|w|}{\sqrt{|z| + |w|}} $$ I am ...
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1answer
23 views

Boundary layer type with initial value problem

Consider the initial value problem $\sqrt{\epsilon} \, u'' + u' - u = e^{2t}$ , with $u(0)=1$, and $u'(0)=1/\sqrt{\epsilon}$. I am trying to use a matched asymptotic expansion to find the leading ...
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1answer
125 views

Perturbation theory to speed up Julia fractal drawing

I have a really underpowered platform here and want to draw a Julia (and possibly Mandelbrot and Burning ship) fractals using a 8.24 fixed point class. I use iteration count for coloring and need to ...
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29 views

Newton Polygon Method (Newton-Kruskal)

$ϵ^{1-α}z^3-z^2+ϵ^{α}z-ϵ^{2α+\frac12}$. The solution to this in my notes now says: "The corners of the curve $\min\{1-α,0,α,2α+\frac12\}$ are at $α=-\frac12, 0, 1$." How does one arrive at this ...
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1answer
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Leading Order Behaviour using Watson and Laplace

Find the leading order behaviour of $$I(x)=\int_0^1 \sin te^{-x\sinh^4 t}dt, \enspace x\rightarrow\infty$$ Thus evaluate $I(10)$ and compare it with its numerical value $I(10)\approx0.13564$ So far, ...
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1answer
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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 $n$ ...
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Is it valid to partially neglect some terms in a higher order contribution?

Suppose I have a function $f(x,\alpha)$ which has a convergent expansion in the small parameter $\alpha$: $ f(x,\alpha)=\sum_{n=1\dots \infty,i=1\dots k} g(x,n,i)\alpha^n $ This expansion has $k$ ...
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1answer
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Does a norm ball around a real matrix contain all possible pairs of complex spectrum?

We consider the normed vector space $(M_n(\mathbb R), \|\cdot\|_F)$, i.e., real matrices with Frobenius norm. Let $A \in M_n(\mathbb R)$ be a diagonalizable matrix and have all eigenvalues to be real. ...
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Picking the correct Ansatz for valid solutions in Asymptotic Methods

I am trying to find the solution to the following equation, $\epsilon x^3 -x^2 +x-\epsilon^{\frac{1}{2}}=0$, for the first two non-zero solutions as $\epsilon \to 0^+$. I have used the principal of ...
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1answer
32 views

Conditions necessary for a boundary layer to exist

Determine values of $a$ for which the problem: $\epsilon y^{''} + y^{'}+ae^y=0,$ $ y(0)=y(1)=0$ has a solution with a boundary layer structure. I am familiar with the procedure for tackling this ...
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Analytic perturbation theory and its applications - Avrachenkov

I am reading Avrachenkov's book - 'Analytic perturbation theory and its applications'. One of the problems contained in books is as follow: If we substitute the series expansions $$A(z)=A_0+zA_1+z^...
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How can apply a perturbation method in a system of equations?

If a have a system of equation $$ \frac{d u}{dt} = 1- u e^{\epsilon(q-1)}$$ $$ \frac{d q}{dt} = u e^{\epsilon(q-1)}-q$$ $u(0)=q(0)= 0$ How Can I apply the perturbation methot $\epsilon$ small ...
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2answers
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Differentiability of largest eigenvalue for a $C^1$ function

I encountered following interesting statement: If $f: [a, b] \to M_n(\mathbb C)$ is a $C^1$ ( $C^1$ in the interior and left/right differentiable over the end points) function over an interval $[a, b]...
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Block matrices inequality

I want to find the relation between 1 with 2 and 3. I know the relation between 2 and 3 but I want to know how 2 is smaller than 1, also whats the relation between 3 with 1.
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Estimate of the perturbation of the spectral radius of a positive matrix

Let $A$ be a positive matrix. $B$ is a small perturbation of $A$, and $B$ is still a positive matrix. By Perron-Frobenius Theorem, it is known that $r(A)$ and $r(B)$ are algebracially simple ...
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Asymptotic expansion for roots of singular cubic equation

Find the first three non-zero terms in the asymptotic expansion for each of the three roots of the cubic equation $$\epsilon x^3+x^2-1=0, |\epsilon|<<1.$$ I found the expansions $1-\frac{\...
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1answer
45 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 ...
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1answer
54 views

Is it possible to determine the largest number $\tau$ such that the spectral radius $\rho(A\pm \tau ee^T) < 1$

Let $A \in M_n(\mathbb R)$ with no particular structure assumed and the spectral radius $\rho(A) < 1$. Let us denote the all $1$ vector by $e = (1, \dots, 1)^T$. I would like to determine a number $...
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Cholesky decomposition of $A+\epsilon I$, where $A$ is symmetric and posdef

Let $A$ be a real square symmetric positive definite matrix with the Cholesky decomposition $A=U^TU$, where $U$ is upper triangular with positive diagonal. What's the Cholesky decomposition of $A + \...
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2answers
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Matrix inverse of $A + \epsilon I$, where $A$ is invertible

Let $A$ be a square invertible matrix, and $\epsilon$ a small positive quantity. To first-order in $\epsilon$, what is the inverse of $A + \epsilon I$, where $I$ is the identity matrix?
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Second order homogenous differential equation with variable coefficients

I have a complicated non-linear first order homogeneous differential equation for coherent states $\psi(t)$. Via perturbation theory I obtained a linear non-homogeneous first order recursive ...
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A matrix M (smallest singular value >1) perturbatiion by diagonal matrix D(-1<=di<=1),then angle of (M+D1)y , (M+D2)y is sharp?

A matrix M (smallest singular value >1) perturbed by diagonal matrix $D1(-1<= d_i <=1)$ or $D2(-1<=di<=1)$,then for any y, the angle $<(M+D1)y,(M+D2)y> < \pi/2$. this is true? ...
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1answer
31 views

Finding a closed formula for $\sum_{i=0}^n 2^i \cdot (n-i)$ through the perturbation method

I need to find a closed form for $$\sum_{i=0}^n 2^i \cdot (n-i)$$ Through the perturbatino method. How could I start? May I reduce the summation in multiple simpler summations?
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26 views

How to expand an exact solution which involves error terms?

The problem begins as a non-dimensionalized two-body problem, represented by $\frac{d^2y}{d\tau^2} = \frac{-1}{(1+\epsilon y)^2}$. The exact solution to this problem is represented as $\tau_{apex}(\...
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40 views

Quantum mechanics: First order perturbation to eigenstates

On page 254 of "Introduction to Quantum Mechanics Second Edition", by David J. Griffiths he writes the first order correction $\delta u_n(x)$ to the eigenstates $u_n(x)$ as $$\delta u_n(x)=\sum_{m\ne ...
2
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1answer
80 views

Perturbation of ODE system

I am trying to apply perturbation theory to this system to approximate $y_1(t)$, which is oscillatory when $y_1(0)$ is complex: $y'_1(t)=\epsilon y_2(t)+2y_1^2(t)$ $y'_2(t)=y_1(t)$ But the solution ...
2
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1answer
48 views

Eigenvector Approximation

Consider symmetric matrices $A$ and $E$ (both with dimensions $n \times n$), and let $\hat{A} = A + E$. If $\mathbf{u_1}$ is the eigenvector of $A$ corresponding to the largest eigenvalue of $A$ (...