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Questions tagged [linear-approximation]

For questions about linear approximations, $f(x) \approx f(a)+f'(a)(x-a)$ for $x$ around $a$.

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How do we determine what small angle and small $x$ are for a simple pendulum to justify linear approximation?

Consider a simple pendulum consisting of a point-like mass $m$ attached to a massless string of length $L$ from a fixed support and constrained to move in a vertical plane. Here is a picture of this ...
xoux's user avatar
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How to set up a convex concave procedure for the minimization of $abc$?

From this post, it seems that there are a lot of advantage of approximating nonconvex problem with the convex concave procedure. Out of curiosity, suppose that I have a simple problem that is $\begin{...
Tuong Nguyen Minh's user avatar
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Linearization that holds for any fixed parameter but not with limit

Consider this problem for $x\ll1$, and assume initially $a\neq 1$ $$ \frac{1}{1+(a-1)\frac{1}{x}} = \frac{x}{a-1} \frac{1}{1+\frac{x}{a-1}} = \frac{x}{a-1}\left (1 - \frac{x}{a-1} +\cdots\right)\...
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Linear approximation problem has me banging my head against the wall

A golfer hits a golf ball at an angle of $\theta=23^{\circ}$ with initial velocity $v=150$ ft/s. (a) Estimate $\Delta s$ if the ball is hit at the same velocity but the angle is increased by $3^{\circ}...
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Approximating the identity with two singular matrices

I am working on a ML project where I would need to find a $n \times m$ matrix $A$ and a $m \times n$ matrix $B$ such that $n<m$ (ideally $n=72$ and $m=429$) and $BA$ is as close to the identity as ...
user600785's user avatar
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2 answers
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Why is $\lim_{x\to a}\frac{E(x)}{x-a} = 0$, instead of $\lim_{x\to a} E(x) = 0,$ used to explain why linear approximation works?

In my calculus textbook, the author made the following remark in the chapter about linear approximation: Let $f$ be a differentiable function with $f'$ continuous. Define $E(x)$ to be the error in ...
ten_to_tenth's user avatar
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Solving ODE near a point by Taylor expansions

Consider a real function $f(r)$ with a real domain and ODE that has a form: $$ \Psi(f) \frac{\mathrm{d} f}{\mathrm{d} r}=\Phi(r,f) $$ where $\Psi$ and $\Phi$ are just some real functions that ...
atapaka's user avatar
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4 votes
1 answer
360 views

Particular Integral, with sequence

Subject: Seeking Help for a Computer Science Contest - Integral Estimation Hello everyone, I hope this message finds you well. I am currently preparing for an ongoing computer science contest, and I ...
Henry D's user avatar
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Is the product of two Jacobian matrices equal to the locally linear approximation of the original functions?

For two $\mathcal{C}^1$ functions $\vec g: \mathbb{R}^n \mapsto \mathbb{R}^m$ and $\vec f: \mathbb{R}^m \mapsto \mathbb{R}^k$ does the equality hold $$\mathbf{J}_{\vec f} \mathbf{J}_{\vec g} \approx \...
Galen's user avatar
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How can we prove that some function, .e.g., the hyperbolic tangent function, tends to be linear around zero?

I have found this interesting answer about increasing the linear range of the hyperbolic tangent function. Now, I am looking for a proof (or at least have a reference from literature if it shows up to ...
dawid's user avatar
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4 answers
120 views

Why does taking the tangent line improve the approximation in Newton's method?

I have gained a comprehension of the operational process through the discussion located at Why does Newton's method work?. Nevertheless, there is one aspect that remains unclear to me. To initiate,...
Nijat Hamidov's user avatar
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99 views

Jacobian of azimuth and elevation angles with respect to unit vector

We know that azimuth ($\theta$) and elevation ($\phi$) angles can represent a unit vector as $\mathbf{e}=\begin{bmatrix}\cos\theta\cos\phi \\ \sin\theta\cos\phi \\ \sin\phi\end{bmatrix}$. It is easy ...
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How to get exact value from multiple approximate values

So I was on amazon looking for measurement cups and stumbled upon this set that had a measuring table for converting between one measurement and another. Here is the link for reference Amazon ...
mathlover123's user avatar
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1 answer
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Linear approximation in multivariable

Find the tangenplane at (1,2,3) $f (x, y ) = \frac{x^2y}{y − 1} + 1$, then approximate the value of the function at f(1.2, 2.3) I partial differentiated and got $f_x$ = 4 and $f_y=-1$ then I used the ...
Need_MathHelp's user avatar
4 votes
3 answers
450 views

Contradiction in derivatives as linear approximations

From the definition of a derivative, we have that $$f'(a) = \lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$$ or $$\lim\limits_{x\to a}f'(x) = \lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$$ This leads me to ...
Ark1409's user avatar
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1 answer
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When using Taylor polynomials to approximate an expression, does approximating two different parts of the original expression lead to correct results?

This question regards using a Taylor polynomial to approximate the expression $$V_P = \frac{kq}{\sqrt{r^2-2ra\cos{\theta}+a^2}}\tag{1}$$ We can rewrite the right-hand side as $$\frac{kq}{\sqrt{r^2(1-\...
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Activation function and its slope in Neural network

I have the following question about the Neural network. In the following paper https://arxiv.org/pdf/2210.05189.pdf The first layer output for each neuron is given by the following equation $$ o_i = \...
GGT's user avatar
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Solving a Stochastic Dynamic Programming with Vector State

Consider the following stochastic dynamic program (SDP): $$ V_t(\textbf{s}_t)= \max_{\textbf{a}_t\in A_t(x_t)} \{(1-\lambda(a_t))V_{t+1}(\textbf{s}_t) + \lambda(a_t)(r_t(a_t)+V_{t+1}(\textbf{s}_t-\...
EagleEdge0423's user avatar
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2 answers
97 views

Series approximation to $\frac{1}{1-(1+x)^{-n}}$

I'm attempting to find a series approximation to: $$f(x)=\frac{1}{1-(1+x)^{-n}}$$ where $n\in\mathbb{N^+}$ and is a constant, and $x\in\mathbb{R}$ and $0<x<1$. Using Wolfram Alpha, I noted the ...
Akyidrian's user avatar
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Approximating difference betwen $\log$ function

How should the difference between $\log$ at different points $a, b$ be approximated (say via the Taylor series)? So, I'm trying to approximate the following term $\log(a) - \log(b)$ where $a, b \in [0,...
Peeveey's user avatar
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Difference between Taylor polynomial of $f(r)=(1+\frac{a^2-2ay}{r^2})^{-3/2}$ at $r=m$ as $m\to \infty$ and the function $(1+s)^{-3/2}$ at $s=0$?

Consider the expression $$E(x,y)=\frac{x\hat{i}+(y-a)\hat{j}}{(x^2+(y-a)^2)^{3/2}}\tag{1}$$ where $r^2=x^2+y^2$. I'd like to investigate what happens when $r>>a$. Let's consider the factor ...
xoux's user avatar
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1 vote
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How to correctly use Linear Approximation

I know about the formula L(x, y) = f(x, y) + fxdx + fydy, but I'm having issues applying them in this problem, any help would be appreciated. In a room, the temperature is given by T = f(x, t) in ...
Pablo Lopes Teixeira's user avatar
2 votes
3 answers
158 views

Linear approximation of $\sqrt[7]{e}$

I need to find a linear approximation $\sqrt[7]{e}$. I know that for $x_0 = a + \Delta x$ $$ f(x_0) \approx f(a)+f'(a)\cdot \Delta x $$ Thus my inital function is $f(x) = \sqrt[7]{x}$ and $e = 1 + (e -...
Adam Bogdański's user avatar
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Prove that a nonlinear function of x is a linear function of x for small x values

I have a function of $x$ defined as below $$f(x)=\sqrt{\frac{1}{N^2}\sum_{l=0,l\neq k}^{N-1}\frac{\sin^2(\pi(l-k+x))}{\sin^2(\frac{\pi}{N}(l-k+x))}+\left|\frac{1}{N}\sum_{l=0}^{N-1}\exp(j2\pi lx/N)-1\...
zahra's user avatar
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0 answers
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Complex Differentiability at a Point Equivalent to Having an Affine Part

I’m trying to prove that a function $f: \mathbb C \rightarrow\mathbb C$ continuous at some point $a$ has affine part $Az+B\overline z+C$ if and only if it is differentiable at $a$. Note: The affine ...
emiliom's user avatar
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1 vote
1 answer
113 views

What does a 'good' approximation mean?

I am a graduate student and currently studying functions of several variables.I am mainly following Paliogiannis and Moskowitz.When they are introducing differentiability for function of several ...
Kishalay Sarkar's user avatar
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1 answer
59 views

Approximation of Woodbury matrix identity

I am trying to find the inverse of the following matrix \begin{equation} \mathbf{X} = \sum_{r=1}^R \xi_{r} \mathbf{a}_{N, r} \mathbf{a}_{N, r}^H + \chi \mathbf{I}_N \end{equation} where $ \mathbf{a}...
Mahdi Eskandari's user avatar
1 vote
1 answer
143 views

Best piecewise with $n$ points for a portion of a log function.

Let $f(x) = |\log_2(x)|$ for $x$ belonging to the domain $(0,1]$. I would like to know if there is an algorithm to fit $f(x)$ using a piecewise linear function g(x) in an optimal way?. That is, on ...
juaninf's user avatar
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0 answers
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First order Taylor expansion of matrix function

In the textbook "Macroeconomic Theory" of Wickens, p. 372, I found the following approximation: $$\frac{1}{2} \text{tr} A \pm \frac{1}{2} [(\text{tr}A)^2 - 4\text{det}A ]^{\frac{1}{2}}\...
Leonardo's user avatar
1 vote
1 answer
77 views

Construct affine minorant for convex LSC proper

Context : I wish to show that if $X$ is a real normed space, $f : X \longrightarrow \overline{\mathbb{R}}$ is convex lower semi continuous and proper, then forall $(x_0,t_0) \in X \times \mathbb R$ ...
blamethelag's user avatar
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0 votes
1 answer
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How to linearize a max function about a small change??

My question is the following: Given a vector field $\mathbf{v}$, we have the following functional: $f\{\mathbf{v}\}=1/|\mathbf{v}|_{\text{max}}$, where $|\mathbf{v}|_\text{max}$ is the maximum of the ...
Joseph Robert Jepson's user avatar
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1 answer
22 views

Choosing the parameter in a function to bring the function as close to 1 as possible when x is approximately equal to zero.

The problem states the following: How should the parameter λ be chosen so that f(x) = e^(-λx)/(1+2sin(x)) remains as close to 1 as possible, when x ≈ 0? I understand that the solution first simplifies ...
Sam C.'s user avatar
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1 vote
0 answers
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Linear approximation of expected value

I am reading a paper where there is an approximation of an expected value. I am not sure what sort of approximation method they are using. Reminds me a bit of Taylors Theorem, but I am just not ...
kpr62's user avatar
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0 votes
1 answer
73 views

Bilinear approximation error bounds

Start with $F(Z,C) = Z^2+C$, perturb this using $$f(Z,z,c)=F(Z+z,C+c)-F(Z,C)=2Zz+z^2+c$$ Claim 1 The bilinear approximation to $f$ $$g(z,c)=Az+Bc$$ with $A=2Z$ and $B=1$ has negligible relative error ...
Claude's user avatar
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What would be the partial derivative of $\frac{\partial x}{\partial t}$ with respect to itself $\frac{\partial \dot x}{\partial x}$ be?

For context, I'm trying to linearize the function $$\dot x^2 \sin x$$ about $a = 0$ using Taylor Series. I can't come up with how to solve the second term, $$\left.\frac{\partial f}{\partial x}\right|...
Benjamin's user avatar
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34 views

Approximation of a function using n intervals

Consider the function 𝑓 𝑥 = 𝑠𝑖𝑛 𝑛𝜋𝑥 and divide the domain 0 ≤ 𝑥 ≤ 1 into m intervals. For the “exact” approximation of the function, we will use 𝑚 = 100 intervals. Plot the “exact” function ...
jondon96's user avatar
1 vote
0 answers
22 views

How to find the best piecewiselinear approximation with fewer intervals

Given a continuous function $f(x)$ on the interval $x\in[a,b]$. How could we find the best piecewise linear approximation of $f(x)$ such that $$J=\|f(x)-\sum_{i=1}^K (m_ix + n_i)\|_{L^2([a,b]} + \...
SAM's user avatar
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0 votes
1 answer
318 views

(Why) Do Functions have to be Twice Differentiable to use linear approximation?

According to Wiki on Linear Approximation Given a twice continuously differentiable function $f$ of one real variable, Taylor's theorem for the case $n = 1$ states that $$f(x) = f(a) + f'(a)(x - a) +...
Ben G's user avatar
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5 votes
1 answer
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$\zeta(1 + 2/x)$ has a strange, nearly linear behaviour

I was messing around with some infinite sums in $\ell^p$ spaces and I encountered a strange result: $\zeta\left(1 + \frac{2}{x}\right)$ looks like it is linear in $x$ for $x > 1$! A simple linear ...
NoOneIsHere's user avatar
2 votes
1 answer
98 views

Why is the linear approximation of my function, after I remove negligible terms, more accurate than the linear approximation not removing the terms?

I have the function $ F(x) $ where $ x >> a $ and I have derived two linear approximations of $ F(x) $: $ L_{1}(x) $, where I take that $ \frac{\left(a^{2}-2xa\right)}{x^{2}} $ is way way less ...
Jon's user avatar
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2 votes
0 answers
69 views

Taylor approximation of $\phi_1(\lambda) = \frac{1}{\sqrt{\psi(\lambda)}} - \frac{\sigma}{\lambda}$

I am reading this paper. In some point of the analysis the non linear equation $$\phi_1(\lambda) = \frac{1}{\sqrt{\psi(\lambda)}} - \frac{\sigma}{\lambda} \tag{A}$$ is studied, i.e. $(6.7)$ in the ...
darkmoor's user avatar
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0 answers
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Number of fixed points in a fixed point iteration function

I'm trying to understand if a fixed point iteration function can have more than 1 fixed points. In theory I tried to find an f(x) such that I could generate an iteration function g(x) in a way that g(...
Sasha Uzelevsky's user avatar
1 vote
3 answers
83 views

How exactly is the value of $\frac{1}{y\sqrt{y^2+\frac{l^2}{4}}}$ as $y \gg l$ calculated?

I am having trouble understanding if a particular calculation is or is not a limit calculation. I suspect it is not, but a particular set of notes from an MIT OCW physics course (problem starts on ...
xoux's user avatar
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0 votes
1 answer
79 views

linear approximation of a function at a number

question: Find linear approximation of the function f(x) = √x at x=9. Use it to approximate √9.1 linearisation formula L(x) = f(a) + f'(a)(x-a) In this question, a=9 right? and when we substitute ...
students's user avatar
1 vote
0 answers
154 views

What is the computational complexity (in flops) of nonnegative least square optimization?

Suppose I have a vector $x\in \mathbb{R}^D$ and a matrix $U\in\mathbb{R}^{m\times D}$. I would like to solve the following nonnegative least squares optimization problem: $a = \text{argmin}_{y\in \...
spolk17's user avatar
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1 vote
0 answers
89 views

Using the tangent line of a value to approximate a value

Good Day, For some reason, my brain is failing to craft a reasonable solution to this equation. Question: Use the tangent line of $f(x) = \sqrt x$ at $a=16$ to approximate $\sqrt 17$ I believe we are ...
BitlyTwiser's user avatar
0 votes
1 answer
138 views

Approximating $\ln(1+2x) = 2x$ using linear approximation

The question is as follows: Use the linear approximation formula $$\boxed{f(x+\Delta x) \approx f(x) +f'(x)\Delta x}$$ to show that $\ln(2x+1) \approx 2x$ for small values of $x$. I am having ...
Justuraveragemathsstudent's user avatar
1 vote
0 answers
92 views

Efficiently approximate Ax=b if you can choose values for A

Here is the situation. You have a black box x, whose values are not known to you, but you can choose values for A and generate the corresponding b value using Ax=b. There is some noise in the system ...
Heiko's user avatar
  • 11
2 votes
0 answers
62 views

What is the reason for this step in proving the Borsuk-Ulam Theorem (by triangulation)

I'm reading Using the Borsuk-Ulam Theorem, which presents several proofs of the theorem. Since this is a book for combinatorialists, the first involves a triangulation $\mathsf{T}$ of the $n$-...
While I Am's user avatar
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0 votes
1 answer
78 views

Why does the linear approximation of $f \circ g$ near $a$ imply $f$ gets linearized by $g$ for a small enough neighborhood of $g$ near $a$?

I've noticed that, if we input the linear approximation of $g$ near $a$ into $f$, we get a decent approximation (althought non-linear) of the linear approximation of $f \circ g$ near $a$. In order to ...
shintuku's user avatar

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