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 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 ...
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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,...
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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 ...
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Nonlocal linear approximation to nonlinear ordinary different equation
Suppose I have a nonlinear ordinary differential equation, in several variables, with a stated initial condition. How would I go about finding a nonlocal linear approximation? What is known about such ...
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How to calculate the error of a taylor polynomial in multivariable
So I've tried googling but I am not finding any information about the error in mulitvariable, everything out there is about single variable. In my prof powerpoint it says for quadratic approximation ...
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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 ...
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Accuracy of linear approximation
I am working on Calculus I by Marsden and Weinstein and this is Exercise 43 from Section 1.6
Let $g(x)=-4x^2+8x+13$. Show that the linear approximation to $g(3+\Delta x)$ always gives an answer which ...
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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 ...
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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 = \...
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Solving a Stochastic Dynamic Programming with Vector State
Consider the following stochastic dynamic program (SDP):
$$
V_t(\textbf{s}_t)= \max_{\textbf{a}_t\in A_t(x_t)} \{(1-\lambda(a_t))V_{t+1}(\textbf{s}_t) + \lambda(a_t)(r_t(a_t)+V_{t+1}(\textbf{s}_t-\...
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Series approximation to $\frac{1}{1-(1+x)^{-n}}$
I'm attempting to find a series approximation to:
$$f(x)=\frac{1}{1-(1+x)^{-n}}$$
where $n\in\mathbb{N^+}$ and is a constant, and $x\in\mathbb{R}$ and $0<x<1$.
Using Wolfram Alpha, I noted the ...
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Is there a way to approximate a posterior distribution over markov matrix with restricted information capacity?
Let a matirx $M^\ast$ is a stochastic matrix with its steady state distribution $\mu(s)$ where the row of $M^\ast$ is a probability of transition to next state $s \in S$ among a finite set of states $...
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Approximating difference betwen $\log$ function
How should the difference between $\log$ at different points $a, b$ be approximated (say via the Taylor series)?
So, I'm trying to approximate the following term $\log(a) - \log(b)$ where $a, b \in [0,...
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Difference between Taylor polynomial of $f(r)=(1+\frac{a^2-2ay}{r^2})^{-3/2}$ at $r=m$ as $m\to \infty$ and the function $(1+s)^{-3/2}$ at $s=0$?
Consider the expression
$$E(x,y)=\frac{x\hat{i}+(y-a)\hat{j}}{(x^2+(y-a)^2)^{3/2}}\tag{1}$$
where $r^2=x^2+y^2$. I'd like to investigate what happens when $r>>a$.
Let's consider the factor ...
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How to correctly use Linear Approximation
I know about the formula L(x, y) = f(x, y) + fxdx + fydy, but I'm having issues applying them in this problem, any help would be appreciated.
In a room, the temperature is given by T = f(x, t) in ...
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small angle approximation of $\dot{\theta}^2\sin{\theta}$
In the equations of motion of vibrating systems, it often appears terms like
$$ \dot{\theta}^2\sin{\theta}$$
Usually, small angle approximations gives $\sin(\theta)\approx\theta$.
However, some ...
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Linear approximation of $\sqrt[7]{e}$
I need to find a linear approximation $\sqrt[7]{e}$. I know that for $x_0 = a + \Delta x$
$$
f(x_0) \approx f(a)+f'(a)\cdot \Delta x
$$
Thus my inital function is $f(x) = \sqrt[7]{x}$ and $e = 1 + (e -...
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Prove that a nonlinear function of x is a linear function of x for small x values
I have a function of $x$ defined as below
$$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\...
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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 ...
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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 ...
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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}...
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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 ...
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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:
$$\frac{1}{2} \text{tr} A \pm \frac{1}{2} [(\text{tr}A)^2 - 4\text{det}A ]^{\frac{1}{2}}\...
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Construct affine minorant for convex LSC proper
Context : I wish to show that if $X$ is a real normed space, $f : X \longrightarrow \overline{\mathbb{R}}$ is convex lower semi continuous and proper, then forall $(x_0,t_0) \in X \times \mathbb R$ ...
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How to linearize a max function about a small change??
My question is the following:
Given a vector field $\mathbf{v}$, we have the following functional: $f\{\mathbf{v}\}=1/|\mathbf{v}|_{\text{max}}$, where $|\mathbf{v}|_\text{max}$ is the maximum of the ...
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Choosing the parameter in a function to bring the function as close to 1 as possible when x is approximately equal to zero.
The problem states the following:
How should the parameter λ be chosen so that f(x) = e^(-λx)/(1+2sin(x)) remains as close to 1 as possible, when x ≈ 0?
I understand that the solution first simplifies ...
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Linear approximation of expected value
I am reading a paper where there is an approximation of an expected value. I am not sure what sort of approximation method they are using. Reminds me a bit of Taylors Theorem, but I am just not ...
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Bilinear approximation error bounds
Start with $F(Z,C) = Z^2+C$, perturb this using $$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$$ with $A=2Z$ and $B=1$ has negligible relative error ...
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What would be the partial derivative of $\frac{\partial x}{\partial t}$ with respect to itself $\frac{\partial \dot x}{\partial x}$ be?
For context, I'm trying to linearize the function
$$\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|...
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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 ...
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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
$$J=\|f(x)-\sum_{i=1}^K (m_ix + n_i)\|_{L^2([a,b]} + \...
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(Why) Do Functions have to be Twice Differentiable to use linear approximation?
According to Wiki on Linear Approximation
Given a twice continuously differentiable function $f$ of one real variable, Taylor's theorem for the case $n = 1$ states that
$$f(x) = f(a) + f'(a)(x - a) +...
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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 ...
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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 ...
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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 ...
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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(...
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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 ...
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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 ...
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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 \...
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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 ...
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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 ...
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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 ...
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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 $n$-...
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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 ...
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Finite element approximation of the cofactor matrix of the Jacobian
Let $\Omega = (0,1)^d$ and assume there is a tensor product mesh $\mathcal T_h$ of quadrilaterals, of maximum diameter $h$, covering $\overline \Omega$, i.e., $\overline{\Omega}= \bigcup_{K\in\mathcal ...
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These are two similar problems on estimating error; Why does one involve taking a linear approximation, and not the other?
I'm having a hard time understanding where to apply linear approximation. These two problems seem very similar to me, except one does not use linear approximation:
Problem 1: Suppose that you measure ...
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2
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In this error approximation, why can we "kill" the $\Delta x\cdot \Delta \theta$ term, but not the term with $\Delta \theta$ alone?
While trying to solve a problem I've come to this equation -- where $\Delta y$ is measurement error of $y$:
$$\Delta y = \Delta h+\Delta x \cdot \tan(\theta)+x \cdot \sec^2(\theta) \cdot \Delta\theta +...
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What is the purpose of taking the linear approximation of this tangent function for estimating the measurement error?
$h$ = distance from ground to eye of a person, who measured the angle to the top of the building with a protractor.
$θ$ = from the eye to the top of the building.
$x$ = distance of the person from the ...
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