Questions tagged [approximation]

For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

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Polynomial approximation for small x

I found this approximation made in a paper. The equation was essentially, $H = \sum_{i=0}^{N-1}\frac{\lambda}{2}((x^2+a^2)^{1/2}-a)^2$ and that for small x this could be approximated as $H = \sum_{i=0}...
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How can you proove that every bounded function in $L^1[0;1]$ can be approximated by continuous function in $C[0;1]$?

Here my question, is this true that: Every bounded function in $L^1[0;1]$ can be approximated by continuous functions in $C[0;1]$ It seems to me true as we know that $C[0;1]$ is dense in $L^1[0;1]$, ...
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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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Approximating $\mathcal X \subset \mathbb R^n$ with the union of disjoint hyperrectangles

Let $\mathcal X$ be a bounded subset of $R^n$ that is generated by a set of linear inequalities. For example, let ${\bf x} = (x_1, x_2, x_3, x_4, x_5)^\intercal \in \mathcal X$ iff \begin{align*} 0 &...
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Approximating (projecting) a surface with bivariate polynomials

I'm working on something and got stuck. First some background: say we have a given continuous function $f(x,y)$ that we want to approximate with the best polynomial (discrete least squares) $z_{n}(x,y)...
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Why is $-cN/z^\alpha$ a good approximation for $\ln(1-F(z))^N$?

Bertin's Statistical Physics of Complex Systems, 3rd ed. p. 65 defines a "complementary cumulative distribution $\tilde{F}(z)$" equal to $\int_z^\infty p(x)\,dx$, where the density $p(z)$ is ...
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Approximation of PDF with summation to infinity, cos(x) and exp(x)

I would like to implement this probability density function in C++. However, on this current form, the algorithm takes a lot of time to return a result (especially because it include a summation). Do ...
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4 votes
3 answers
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For what values of $p>0$ is $\lim_{n\rightarrow\infty}\int_0^n\frac{(1-\frac{x}{n})^ne^x}{n^p}dx=0$?

For what values of $p>0$ is $\lim_{n\rightarrow\infty}\int_0^n\frac{(1-\frac{x}{n})^ne^x}{n^p}dx=0$? My thoughts: We know that $(1-\frac{x}{n})^n\leq e^x$, so the numerator is $\leq e^{2x}$. So, ...
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Let $f:\mathbb{R}\rightarrow \mathbb{R}$ with $f(x) = xm, m>1$ . Write Newton’s method for approximating root $x^∗ = 0$ of $f$ starting with $x_0= 0$. [closed]

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ with $f(x) = xm, m > 1$ Write down Newton’s method for approximating the root $x^∗ = 0$ of $f$ starting with initial guess $x_0= 0$. Express the $n^{th}$ ...
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What do you call it when, in the limit, two functions are proportional by a factor of 1?

Two functions $f(x)$ and $g(x)$ have the property that $$\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$$. What is this referred to as commonly? All I know of is Big-$\Theta$ notation, which says $$f(x)\in\...
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How can I optimize an experimentally derived objective function?

I have to optimize the result of a process that depends on a large number of variables, e. g. a laser engraving system where the engraving depth depends on the laser speed, distance, power and so on. ...
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Limit of integral product

I have an expression of the sort $$\displaystyle \lim_{N \to \infty}\prod_{i=1}^{N}\int_{0}^{1}\mathrm{d}\sigma(p_i)\sqrt{\frac{N\beta J}{2\pi}}\int_{-\infty}^{\infty}\mathrm{d}x \exp{\left [ -\frac{N\...
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In TSP where a modified triangle inequality holds, what is the approximation ratio of Christofides' algorithm?

In the metric TSP, we can use Christofides' algorithm to get a $\frac{3}{2}$-approximate solution. This is a consequence of the triangle inequality where $d_{ij} + d_{jl} \geq d_{il}$, that enables us ...
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Approximate first-order loss function with piecewise-linear functions in python

I'm trying to approximate the first-order loss function $$\mathbb{E}[max(d-y,0)]=\sigma \cdot (\phi(\frac{d-\mu}{\sigma})- \frac{d-\mu}{\sigma} \cdot (1- \Phi (\frac{d-\mu}{\sigma})),$$ where $d\in \...
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How to derive the following approximation for a set of coupled equations?

Background I have the following set of coupled equations $\frac{d}{dt}\rho_{ab} = -i\omega A - \frac{1}{2}\Lambda B$ with $\omega, \Lambda$ parameters that are real and $A, B$ 4x4 matrices that are ...
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2 answers
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Finding exact/approximate value of a logarithmic function

I have an equation as below $$2\log_2(1+xC)=\log_2(1+yC)$$ what is the relation between $x$ and $y$. I mean I want to see the relation as $x=\alpha y$ Then what is the value of $\alpha$. Do we have ...
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2 votes
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Approximate value of hyperpolic tangent in certain case

Reading this interesting Book ( thé nature of magnetism) , I came across a particular approximation of hyperpolic tangent, while in first case T bigger than Tc , it is just Taylor series, in case T ...
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Bound KL divergence between two distributions by KL divergence of two Gaussian mixture models

I'm trying to bound the KL divergence between two continuous random variables with the KL divergence between two Gaussian mixture approximations motivated by the fact that the Gaussian mixture model ...
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approximating indicator function of threshold functions using 1 hidden layer of a neural network

I'm reading a paper about approximations using neural network and I'm trying to understand the next sentence (that isn't proven in the paper): I feel like I'm missing something simple here. The ...
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Finding Best approximation in space with standard inner product

I'm new to functional analysis. I do not know how to solve the below problem. I know that for g to be the best approximation of f, it has to satisfy the condition that (f−g)⊥G. But I could not apply ...
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Functional analysis - best approximation norm [closed]

I have no idea how to prove or disprove the parts of the below question. I intuitively feel they are correct but cannot form a rigorous mathematical proof for it. Given is a normed space E. Also, ...
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5 votes
2 answers
162 views

Issues with the Fourier Transform of $f(t)=(1-t^2)^4$ on $[-1,\,1]$, should be analytical but looks like having a singularity with noise-like rippling

Issues with the Fourier Transform of $f(t)=(1-t^2)^4$ on $[-1,\,1]$, should be analytical but looks like having a singularity with noise-like rippling Intro I was trying to made a compact-supported ...
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Finding a better approximation for $f(L) = \left\lfloor{\frac{1}{4}\sum_{n=1}^{L-1}\left\lfloor n+300\times2^{n/7}\right\rfloor}\right\rfloor$

I have the following equation: $$f(L) = \left\lfloor{\frac{1}{4}\sum_{n=1}^{L-1}\left\lfloor n+300\times2^{n/7}\right\rfloor}\right\rfloor$$ And im looking for either a closed-form solution or an ...
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1 answer
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Approximating $\sqrt[r]x$

So here's a question. What is the best way to approximate this: $$\sqrt[r]x$$ Here is a few method I found, but I am not sure which is faster: brute force. The most straight forward and easiest. Just ...
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Why would we expect integration by parts to generate an asymptotic approximation to the integral?

I'm doing a perturbation methods course where we have shown that often (but not always - e.g. if dominant contribution is not at an end point of the integral, or if boundary term diverges etc), doing ...
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Dimension of function derivatives space

It is an exercise on textbook named A course in approximation theory page 9. It says: Let $U$ be an $n$-dimensional subspace in $C^{(n)}[a,b]$. Let $D = d/dt$. Thus $D^{k}(U) = \{u^{(k)}: u\in U\}$. ...
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Generalization of Rayleigh quotient with $\mathcal O(\epsilon^3)$ error reduction

In my Numerical Linear Algebra class, I've learned about the Rayleigh Quotient $r(x)$, defined for vectors $x\in\mathbb R^m$ and matrices $A\in\mathbb R^{m\times m}$ as follows: $$r(x)=\frac{x^T A x}{...
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1 vote
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Simplifying a Kullback-Leibler divergence

I want to find an approximation of a mixture of probability distributions that minimises the Kullback-Leibler divergence (KLD). I need to verify my result, as it seems suspect. We have a joint ...
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approximation error of $p$-th moment

Suppose that $\delta>0$ is a small quantity and $p\in (1,2)$ is a constant. Suppose that $x_1,\dots,x_n,y_1,\dots,y_m\in (0,\delta)$ and $a_1,\dots,a_n,b_1,\dots,b_m\in (0,1)$ satisfy that $$ \...
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3 votes
4 answers
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Approximation of $\int_0^\pi \big[x(\pi-x)\csc (x)\big]^k\,dx \quad \forall k$

A recent post addressed the problem of the closed form of $$I(k)=\int_0^\pi \Bigg[ \frac {x(\pi-x)} {\sin(x)}\Bigg]^k \,dx$$ When $k$ is a positive integer, they seem to be known except that they ...
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Can a neural network with ReLU activation represents exactly all $B$-bounded and $L$-Lipschitz $K$-max-affine functions?

A max-affine function is defined as the maximum over a set of affine functions, which is always convex. More specifically, we define a $K$-max-affine function $f:\mathbb{R}^d\to\mathbb{R}$ that can be ...
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16 votes
1 answer
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approximation of integral of $|\cos x|^p$

Let $p\in [1,2)$. Let $$ \beta = \frac{1}{2\pi}\int_0^{2\pi} |\cos x|^p\, dx = \frac{\Gamma(\frac{p+1}{2})}{\sqrt{\pi}\Gamma(1+\frac{p}{2})}. $$ Consider the following approximation to the integral ...
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1 vote
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Probability density — odds of a tie in the California election

Assume, as seems likely, in the 2024 presidential election, Californians will cast 20 million votes. Make the simplifying assumption that that each Californian will vote Democrat with a probability ...
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The best approximation for $L^2$ is the partial sum of a fourier series?

Looking over my notes, I've found the following lemma Define $f_n$ as the $n^{th}$ partial sum of the Fourier series of $f$ and let $g_n$ be an arbitrary trigonometric approximation to $f$. Consider $...
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Gamma functions approaches solely n to a negative power

I have read $[x^n] (1-x)^\alpha\sim \frac{n^{-\alpha-1}}{\Gamma(-\alpha)}$ as $n\rightarrow\infty$ where $[x^n]$ means coefficient of $x^n$ in what follows. But I have never seen a proof of this ...
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1 vote
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How the green function for the relativistic heat equation converges to the green function of the heat equation?

The relativistic heat equation or telegraphers equation is: $$ (\alpha\partial_t^2 + \beta\partial_t - \omega\,\nabla^2_{\text{3D}})G_R = \delta $$ if $\alpha \rightarrow 0$ the solution must ...
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1 vote
1 answer
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(Polynomial approximation) Why does the nth derivative of both the function and the polynomial has to be equal at center?

I've watched many videos about Taylor/Maclaurin polynomial but no tutor ever explained why it has to be f(n)(c) = p(n)(c) at center x = c. I've seen the behavior of the graph of p(x) and f(x) when ...
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1 vote
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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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The special formula for order of convergence

In numerical analysis, for the problems (mainly PDE) for which analytical solution is not available, the error computation can be done often with 'double mesh principle'. For instant, Let $\mathcal U_{...
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Bounding sum of independent samples from random distribution

Context Given a discrete variable $X$, and $Y = \sum_{k=0}^N X$, the sum of $N$ independent samples of $X$, I want to an upper bound for $y_b$ so that $p(Y < y_b) \ge 1-l$. So I start with a ...
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1 vote
1 answer
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Select a subset to Minimize a continuous unimodal function

I want to find an approximation algorithm for the following problem. $\qquad$ Find a $S\subseteq N$ such that $\rho(S) = \frac{\sum_{i\in S}\ V_i}{(1+\sum_{i\in S}\ V_i)(4+\sum_{i\in S}\ V_i)}$ is ...
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Probabilistic bounds on approximation of nonlinear functions via Volterra functionals (and related methods)

I'm working on a nonlinear systems identification problem, and as far as I can tell variations on Volterra functionals are the best approach known for this problem - barring deep learning. The latter ...
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How could I construct a sequence of functions in $C^1$ such that satisfies the following requirements?

$\Omega=(-1,1)$ and $\ell=1$. Think of $f(x)=|x|$ and $g(x)=\operatorname{sgn} x$ as functions in $L^{2}(\Omega)$. Find a sequence $\left\{f_{n}\right\}_{n \in \mathbb{N}} \subset C^{1}(\Omega)$ such ...
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Approximation of coupled differential equations with recursive integration

I am currently reading the book "the quantum theory of light" link: http://rplab.ru/~as/2000%20-%20R.Loudon%20-%20The%20Quantum%20Theory%20of%20Light%20-%203rd%20ed%20Oxford%20Science%...
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2 votes
2 answers
73 views

Use differentiable function to approximate continuous function

Let $f$ be a continuous monotone increasing function on $[a,b]$. I want to show for any $\epsilon>0$ there exist a monotone increasing function $g\in C^1[a,b]$ such that $$ \max_{x\in[a,b]}|g-f|<...
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Approximation of $\|x\|_2$ via mollification

Let $f\colon \mathbb{R}^3\to \mathbb{R}$, $f(x)=\|x\|_2$. I want to approximate $f$ in $L^p$ by $C^1$ functions which are vanishing in a neighborhood of zero. Let $\varphi_{\varepsilon_n}=\frac{1}{\...
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  • 570
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1 answer
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Approximating the roots of an exponential polynomial

I am currently working with an exponential polynomial of the form $$f(x)=a_1 \cdot\exp(\,\lambda_1\cdot x\,) + a_2\cdot \exp(\,\lambda_2 \cdot x\,) + a_3.$$ In my application it is always true that $...
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Limit Behaviour of Binomial Distirbution Likelihood Function with Sterling's Approximation

The equation for the maximum likelihood estimator for the binomial distribution is $$\mathcal{L}(p|n,y)=\binom{n}{y}p^y(1-p)^{n-y}$$ For a statistical problems I am working on, I am interested in the ...
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Approximation for maximum space 2D irregular bin packing

So usually bin packing algorithms compute the tightest packed solution. I want to calculate the opposite, in my case the solution with the most space between the packed objects is needed. I tried ...
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How do I prove that this complex function can be approximated uniformly by rational functions?

I have the following problem: We have $K\subset \Bbb{C}$ a compact subset and $f:K\rightarrow \Bbb{C}$ a continuous function. We take $a\in \Bbb{C}\setminus K$. We assume that there exists a sequence ...
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