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Questions tagged [linearization]

On the many different ways to turn non-linear systems of equations into linear ones.

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linearization problem of $y^2(x) + y(x) - 2.8 = 0$

Problem: Suppose that a function $y(x)$ satisfies the following equation for small values of $|x|$: $$y^2(x) + y(x) - 2.8 = 0$$ Also, $y(0)= 1$ and $y'(0) = 2$. A) Find linearization of $y$ at $x = ...
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24 views

Linearization of nonlinear dynamic system

I am reading the following paper: ergodic theory of chaos and strange attractors, by J.-P. Eckmann (can be easily downloaded) My question is around equation (1.7)...
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1answer
18 views

Linearising an Equation to the Form y=mx+c

I have been working on some data analysis stuff and I have to linearise this equation so I can plot it as a straight line with form y=mx+c $$T=2\pi\sqrt{\frac{(k^2 + h^2)}{gh}}$$ Where, T will be ...
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1answer
26 views

Linearizing the following equation

I'm working through a section in a book I'm reading about delay differential equations (Semi-discretization for time delay systems, Springer), and the authors are discussing the following equation $$\...
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21 views

Linearizing squareroot equation with x in both numerator and denominator

I am having trouble linearizing the following equation in order to find the relationship of y and x. $$y=\sqrt{\frac{x*6}{x+0.5}}$$ I was thinking about converting the equation above into the standard ...
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12 views

Linearization of two binary variables with multiple indexes

At the moment I'm working on a scheduling problem and I'm trying to linearize the following constraint: $\sum_{k \in K} (s_{j,t} * y_{i,j,k,t} + \sum_{t \in range(t, t+p_j)} \sum_{jj \in J\j} y_{i,jj,...
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39 views

Application of Lyapounov's Theorem

I have the following exercise to fullfill: Given the system of differential equations $x'=f(x)=-\nabla{g}$ , where $x(t)\in\Bbb{R^3}$ and $g$ is $C^1$ and $f(0)=0$ and $0$ is a total maximum for $g$,...
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26 views

Linear approximation of $3$ variable function and its maximum error?

I've got a problem which asks me to find linear approximation of multi-variable function and its maximum error. Here's the problem : By about how much will $$g(x,y,z)=x+x\cos(z)-y\sin(z)+y$$ ...
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1answer
27 views

Linearization of a function: can someone explain me this last step?

I'm studying a linearization of a differential equation. $x(t)$ and $r(t)$ are really small signals and G, K, B and M are constants. I understand everything until I reach $$ \frac{d^2x(t)}{dt^2}=G+\...
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24 views

linearization of constraints

I am dealing with an optimization model where my binary variables xi have to follow this type of constraint (in the attached link): ...
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1answer
14 views

Linearization of Autonomous Equations

Consider $y'=y(y-50)(y-100)$ (a) Let $y_e$ be the stable equilibrium solution and let $u=y-y_e$, rewrite the equation as a differential equation for $u$. $y_e=50, u=y-50, u'=(u-50)(u)(u+50)$ (b) ...
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11 views

linearizing dynamics about non fixed point for LQR implementation.

I am trying to implement LQR control for the cart pole system. I am curious if I can maintain a constant non-zero pole angle. So, I need to linearize my dynamics about my goal state. I know we can use ...
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22 views

Linearization between two ranges

We are currently trying to solve a problem where the input is range of variables x and y. y $\in$ {min, max}. x $\in$ {0, 255} min & max are integers of range $\{-2^{32}, 2^{32} \}$ Need to ...
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32 views

Determining the original, unapproximated form of a rotation vector

a bit of an odd one here: I'm reading a paper that supposedly rotates a basis in a symmetric way, using two scalar rotation quantities, $\theta_1$ and $\theta_2$. The idea, I believe, is for each ...
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36 views

Carleman matrix of multivariable functions - Carleman tensor?

Recently I learned about a matrix called Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying. Carleman linearization is a technique used to embed a finite ...
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1answer
85 views

How to solve binary nonlinear programming problems?

I have written binary nonlinear programming problem: Now I want to solve this problem. My decision variables are $x_{i,j}, y_{i,j}$ and $z_{i,j}$. The other terms are constants. N=30 and K=4. I ...
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1answer
59 views

How do I linearise the rational function to analyze the critical points?

For the system $$\frac{dx}{dt}=\frac{3xy}{1+x^2+y^2}-\frac{1+x^2}{1+y^2}\\\frac{dy}{dt}=x^2-y^2,$$ the point $\begin{pmatrix}1\\1\end{pmatrix}$ is A. an unstable node B. a stable node C....
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44 views

Linearizing two variable function

I have another linearization question similar to the one in here. This time, I have got two variables in my equation and I am in search of an "$A+B\rho$" or possibly "$A+B\rho+C\theta$" approximation. ...
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38 views

how I can linearize or simplify these complicated and non-linear constraints?

I have some constraints which are in the form $$ \dfrac{x_{1}-x_{2}+x_{3}+\cdots+x_{n}}{(x_{i}-x_{j})^{2}+\cdots+(x_{l}-x_{k})^{2}}+\cdots+\dfrac{x_{1}+x_{2}-x_{3}+\cdots+x_{n}}{(x_{j}-x_{i})^{2}+\...
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3answers
90 views

Linear Approximation of x/ (1-x)

I am trying to linearize the following function, but, having difficulties. Let, $x = \frac{l}{m},$ where $l,m \in R^+$ and $l<m$ Assume $l$ is a variable, while, $m$ is a constant (parameter), ...
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1answer
48 views

Linearized Pitot system

I am trying to create a linearized model of a compressible pitot tube system with altitude $h$ as the input and velocity as the output. When I take the derivative and try and linearize around a point ...
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1answer
114 views

Linearization of differential system of equation

I would like to ask if I understand correctly the process of linearization for analyzing critical points. I was given differential equation: $\dot x = xy+1$ $\dot y = x+xy$ And my task was to ...
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56 views

Linearization of nonlinear Schrodinger equation

Picture below is from the 56th page of Lyapunov stability of ground states of nonlinear dispersive evolution equations. The nonlinear Schrodinger equation (NLS) is $$ i\phi_t(x,t)+\Delta \phi(x,t)+ ...
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83 views

About linearization around an equilibrium point

Consider the following non-linear differential equation $$\label{star} \dot{x}(t) = f(x(t))+g(x(t),u(t)), \quad t\ge 0, \ x(0)=x_0\in\mathbb{R},\tag{$\star$} $$ where $f(\cdot)$ and $g(\cdot)$ are ...
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1answer
129 views

LQR Robotic Arm

I have a robotic arm model in Simulink and I'd like to control the position of the end-effector such that it follows a given trajectory. This is done by inputting joint angles and comparing the output ...
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0answers
34 views

(Checking) derivatives of determinant, cofactor, trace

I'm considering an $n\times n$ matrix $T_\epsilon$ such that there exists a unique inverse for $\epsilon \in (-\delta,\delta)$ for some $\delta > 0$. I'm trying to check whether I've correctly ...
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1answer
21 views

Properties of matrix $T_\epsilon$ such that there exists a unique inverse for $\epsilon \in (-\delta,\delta)$

I'm considering an $n\times n$ matrix $T_\epsilon$ such that there exists a unique inverse for $\epsilon \in (-\delta,\delta)$ for some $\delta > 0$. I'm trying to determine several quantities of ...
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4answers
215 views

Newtons method for finding reciprocal

Define a function 1 which is $f_1(x)=a-1/x$ and function 2 which is $f_2(x)=1-ax $ If I set both to zero I am looking for when $x=1/a$ as the root using Newtons method. When I do this I get two ...
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How to linearize a system of ordinary differential equations, ODEs, around a periodic solution (using numerical methods)?

I am trying to linearize the thermal analysis of a spacecraft model with N nodes over M working points in one orbit. The non-linear equation is given by the following system of first order ODEs, In ...
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How to linearize this function around a working point?

How can I linearize the following function, where my variables are $\phi_d$, $\dot{\phi_d}$ and $\dot{L}_1$: $\dot{L}_1=\frac{1}{2\sqrt{(l_d\cos{\phi_d}-x_Q)^2+(l_d\sin{\phi_d}-x_Q)^2-R_2^2}}\small(2(...
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25 views

If $ \|G(x+ty)\|<\|G(x)\| $, is then $ \|G(x) + tG'(x)[y]\| <\|G(x)\| $?

It seems intuitive that given a $ C^1 $ function $ f : \mathbb R \to \mathbb R $, where for some $ t_0 > 0 $ we have $ |f(x+t)| < |f(x)| $ for all $ t \in (0,t_0) $, and $ f'(x) \neq 0 $, ...
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Similarity in the temperature coefficient of resistivity and coefficient of linear (length) expansion

This was posted on physics stackexchange, https://physics.stackexchange.com/questions/390785/why-change-in-resistivity-is-proportional-to-the-original-resistivity? , but got downvotes and close votes. ...
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1answer
126 views

Stability of Euler's Method for non-linear ODE

Consider the ODE $$y'(t) = \lambda y(t), \quad y(t_0) =y_0.$$ Euler's method $y_{i+1}=y_i+h\lambda y_i $ is stable (meaning that the solution decays or stays constant as $ i \to \infty$) provided that ...
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1answer
1k views

Linearizing Logarithmic Function

I have a given set of data points (y,x) with uncertainties. When I plot those points on a graph, the trendline appears to follow the equation y = c + a*ln(x). I want to be able to find the ...
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1answer
53 views

Finding contstant in remainder term for linearization of $exp(x)$

I've managed to find the remainder term for the linearization of $\exp(x)$ about $x=0$ in Lagrange form: $$ R_1(x)=\exp(θ_Lx)\frac 12x^2 \text{ ,where } θ_L∈[0,1]. $$ My question is how would I ...
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1answer
147 views

Product and Quotient Rule proof using linearisation

So I've recently been introduced to the concept of linearization and now I'm beginning to apply this concept to prove certain differenation rules. I've managed to prove the chain rule so far, but I ...
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1answer
109 views

What is the significance of the linearization of a non-linear PDE?

This may be too general a question so please let me know if I need to make it more specific. I am a first year graduate student in PDEs, and as such have not had much exposure to non-linear PDEs. I ...
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2answers
71 views

Taylor approximation with 2 variables, not sure how to solve this

This is the problem I'm trying to solve: Let $f(x,y)=ay+sin(bx)+c$. Evaluate the Taylor polynomial at $P(0,0)$ and find the values for $a$, $b$ and $c$ if $P(x,y)=-1+2x-y$ I do know the ...
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1answer
946 views

How to linearize a constraint including product of two binary variables in summation with different indexes?

I am trying to linearize the following two expressions: $\sum_{k=1}^K \sum_{t=1}^T\sum_{h=1}^W x_{ijkt} a_{hjt} =\sum_{k=1}^K \sum_{t=1}^T x_{ijkt} k , i\in N, j \in M$ $\sum_{k=1}^K \sum_{t=p_{ijk}...
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1answer
86 views

Show that $L(x,y)=ax^2+by^2$ is a Lyapunov function for the equilibrium at the origin

I am looking at past paper questions and I'm a little stuck on this one. I have the following system of ODEs: $\dot{x}=(\epsilon x+2y)(x+1)$ $\dot{y}=(-x+\epsilon y)(x+1)$ where $\...
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37 views

A function that turns a power to a product/sum?

I just recently started learning about the logarithm functions, and its concept is quite amazing (f(x.y)=f(x)+f(y)). Now I'm asking for a similar function but instead of x.y ; we use ...
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1answer
56 views

What were the steps in the linearization of the following equation?

I'm verifying a linear form of an equation, i.e. I can't find how the equation 6 leads to the equation 7
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1answer
278 views

Stationary points and linearisation of non-linear system

So, the problem is: Find and discuss the behavior of the stationary points of the system : $$ x'=-y+x\cdot (x^2+y^2)\cdot \sin\sqrt{x^2+y^2} =f(x,y)$$ $$ y'=x+y\cdot (x^2+y^2)\cdot \sin\...
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2answers
191 views

Linearization of Differential Equation

Find a linearization of the differential equation for $x$ near $0$. $$x''(t) + x(t) e^{0.05x} = 0$$ Not sure what to do here. My book isn't any help either.. Any help would be appreciated :)
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1answer
135 views

Linearization of a Differential Equation

Can someone please help me to linearize this system, which is given by the differential equation shown in the picture below. All variables are expressed as deviations from initial values (0, for ...
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1answer
12 views

Equation of tangent plane at the solution of an ODE

I have the following differential equation: $$P_0-P_n = \int_0^L f(x,P(x),Q)\,dx$$ In practice, $P_0$ is the pressure at the start of a pipe $(x=0)$, $P_n$ is the pressure at the end $(x=L)$, and $Q$...
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325 views

Exact discretization of non-linear dynamical system which was linearized at a non-equilibrium point. [closed]

0. Question tl;dr: just see section 2. c) I'd like to know how a non-linear system of first order ODEs (non-linear dynamical system) which was linearized at a point which is not the equilibrium of ...
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1answer
105 views

Linearization/Directional Derivative relationship?

I'm a little confused by the following question: You know that a function f(x, y, z) satisfies f(0, 0, 0) = 33 and $D_{<1,1,1>/ √ 3}$ f(0, 0, 0) = 4/√3 and $D_{<1,1,0>/ √2}$ f(0, ...
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2answers
238 views

Limit of 4x + 1 as x approaches 0

Okay, so I was approached with this question in my math class and I can answer the first two parts correctly but the third is throwing me off... ${f(x) = 4x + 1}$ a) Table of Values ${x = -0.0001 ...
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1answer
75 views

Understanding a proof about Riemannian metrics in three dimensions always being diagonalizable

I've recently been working through Deturck's and Yang's Existence of elastic deformations with prescribed principal strains. First and formost, I'm interested in it's proof that Riemannian metrics can ...