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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Approximating probabilistic event by Central Limit Theorem
We're throwing a die 3600 times. Let $X_i$ be the number rolled, and $S_n=X_1+...+X_n$. By the law of large numbers, we know $\mu_Χ=3.5$. We want to approximate the probability that $\frac{S_n}{n}$ ...
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Chebyshev approximation for bivariate function
I read the paper.
I am a litte bit confused regarding formulation of Chebyshev approximation for bivariate function(See photo). There is only one integral over variable x. Should it be in formula one ...
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Proof of transformation of Hypergeometric to Whittaker.
I am working on this paper, regarding the spectrum of a certain operator in the hyperbolic plane, and at a certain point are presented with an hypergeometric function
\begin{equation}
\text{}_2 F_1\...
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Approximation of the n'th prime
An old paper by Ernest Cesàro provides a suggested approximation of the n'th prime. The expression and the reference currently appears in the Wikipedia article on the Prime Number Theorem.
It is ...
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Compute $\int_0^{0,2}\frac{1-e^{-x}}{x}dx$ accurate to $\alpha=0,0001$
$\int_0^{0,2}\frac{1-e^{-x}}{x}dx$
I was trying to compute integral,cause it'll look like alternative series.
I'm stuck and have troubles to calculate this integral.
in first task,that looked like ...
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Best approximation of a line by a rational line [closed]
Let n be a positive integer and let $Q_n$ denote the set of rational numbers whose denominator is bounded above by n.
Given an arbitrary line L:
$Ax+By = C$, with $A,B,C \in \mathbb{R}$
I want to find ...
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Help to improve polynomial approximation with example of sine approximation
I found that composition of polynomials has interesting properties for approximation. (1e-20 error for 6 coefficients).
In pseudocode $\ p2=x+ax^3+bx^5$
...
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How this non-convex function innerly approximated as convex function???
The function is $ P= \left \| \hat{\mathbf{b}}^{H}\left ( \boldsymbol{\theta} \right )\mathbf{v} \right \|^{2}$ is non-convex.
And it has been approximated as follows:
$\left \| \hat{\mathbf{b}}^{H}...
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Approximating $\pi=4\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}$ [duplicate]
Consider the series
$$
\pi=4\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}
$$
How many terms of this series do I need to consider to have an approximation of $\pi$ accurate up to $10$ decimal places (for ...
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Show that the normalized sequence in the Power method for eigenvectors converges to the dominant eigenvector
Given a diagonalizable matrix $A \in \mathbb{C}^{n \times n}$ with ordered eigenvalues: $$|\lambda_1|> |\lambda_2| \geq \ldots \geq |\lambda_n|$$ and
corresponding eigenvectors $x_j$, we start with ...
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More efficient methods of approximating Bessel functions of the second kind
Looking at Bessel functions I see that Bessel functions of the first kind can be described by the equations
$$J_a(x)=\sum_{m=0}^\infty\frac{(-1)^m}{m!\Gamma(m+a+1)}\left(\frac{x}{2}\right)^{2m+a}$$
...
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Point in use of Taylor Series to approximate functions in an age with computers?
I hope this doesn't sound too vague or like I'm dismissing the use of Taylor Series entirely, I'm just curious about any proper real-world applications.
Many times Taylor Series are shown-off as a ...
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How to approximate $\ln \left( {{e^{{x_1}}} - {e^{{x_2}}}} \right)$?
In the Max-Log-Map algorithm for channel decoding, the approximation $\ln \left( {{e^{{x_1}}} + {e^{{x_2}}}} \right) \approx \max \left( {{x_1},{x_2}} \right)$ can be considered because $\ln \left( {{...
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Multiplicative Harmonic Series
Is there any approximation formula for this equation? I have been trying to find approximations for it so I can create an approximation formula myself for something else, but for some reason I can't ...
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"Formal" asymptotic approximation of an integral
I wanted to asymptotically evaluate the following integral:
$$
\mathcal{I}(\epsilon)\equiv\int_{-\infty}^{\infty}dx\int_{-|x|}^{|x|}dy f(y)\,\exp{\left(-\frac{|x|}{\epsilon}\right)}\,,
$$
for $\...
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How to approximate the median of the numbers in the first $n$ rows of Pascal's triangle?
How can we approximate the median of the numbers in the first $n$ rows of Pascal's triangle? (The top row is the $0$th row.)
Using Excel, I made a graph of the natural log of the median against $n$, ...
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Coupled non-linear PDE analytical solution to verify numerical solution.
as part of a computational module we were given the following coupled PDEs and solved them numerically using finite difference methods:
I got the following graphs representing the numerical solution ...
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How to evaluate $\sum_{i=1}^n i^{2 i}$?
Let,
$$\mathcal{S}(n) = \sum_{i=1}^n i^{2 i}$$
for $n \in \mathbb{N}$
I will be completely honest. When I was returning from my physics tuition center, and suddenly this popped up in my head from ...
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Convergence rate for Chebyshev polynomials to approximate $\text{erf}(x)$ on a subset of $\mathbb{R}$
Let $[-\alpha, \alpha] \subset \mathbb{R}$, and let
\begin{equation}
\text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}dt.
\end{equation}
given projections of $\text{erf}(x)$ onto the first $k$ ...
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Expansion of $1/\log\left(\dfrac{f(a)^2 - f(x)^2}{f(a)^2 + f(x)^2}\right)$ around $a$
I want to know what's the order of the second leading term in the expansion of
$1/\log\left(\dfrac{f(a)^2 - f(x)^2}{f(a)^2 + f(x)^2}\right)$ around $a$. For now, I have:
$$1/\log\left(\dfrac{f(a)^2 - ...
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Tight bound for a sum of binomial coefficients
I am looking for a tight upper bound for the following sum:
$$\sum_{m=1}^k \binom{n}{m} \binom{k-1}{m-1} $$
with $k \leq n$ and $m \leq n$.
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Second-order Taylor expansion for Operators
Let $u(t)$ and $v(t)$ be functions in $C^{\infty}$. Then let $A(u)$ be an operator.
A valid reference mentioned that the second-order Taylor expansion of the operator $A$ is:
$$A(u+v) = A(u) + dA(u)[v]...
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Bounding $\int^{\infty}_{0} \int^{\infty}_{0} \frac{dx}{1+x^2} \frac{dy}{1+y^2} \frac{\sinh(\pi x) \sinh(\pi y)}{\sinh(\pi (x+y))}$ close to $1$
I've come across an integral that when numerically evaluated is quite close to $1$. The integral is in fact loosely related to a recent question that similarly was suspiciously close to a simple ...
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Explanation for standard deviation rule-of-thumb, $s\approx \text{range}/4$
In several introductory statistics books, for a list of data $X = \{x_1,\ldots,x_n\}$, I have frequently seen the following rule-of-thumb:
$$
s \approx \frac{\text{range}(X)}{4} = \frac{\max(X)-\min(X)...
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Approximation of angle relationships in arbitrary triangle
I am trying to approximate the expression
$\sin \varphi_s - \sin \varphi_e = 2 \cos\left(\frac{\varphi_s + \varphi_e}{2}\right) \sin\left(\frac{\varphi_s-\varphi_e}{2}\right)$
to include the angle $\...
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Asymptotic study of eigenvalues
I have the differential equation
$$(a + x - b(1+x^2))\frac{dg}{dx}(x)+(1+x^2)\frac{d^2g}{dx^2}(x) + \lambda g(x)=0 \, $$ where $a, b >0$ and $x$ is a real variable. I want to find out the bounded ...
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approximating permanent using Huber-Law approch
In a paper by Huber and Law on approximating the permanent of a dense matrix, in equation (3.13) they show that $$\frac{M(f(A,i,j))}{M(A)} = \frac{e}{h(r(i)-A(i,j))}\prod_{i'=1}^{n}h(r(i')-A(i',j))/h(...
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Is there a self-correcting iterative method for approximating pi without using transcendental functions?
The Newton-Raphson method is an iterative method for finding a root of a function, and it is self-correcting in the sense that any error in the initial input is reduced with each iteration so that it ...
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derive a depth of plate contacting a particle by using a curvature
I am learning physics theory called Hertz contact theory which considers a small particle contacting a plate. But I have been stuck in the geometry problem. I would like to get $z$ in the figure. When ...
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Approximating functions when one value is much bigger than another.
When trying to approximate the behavior of a function in some particular limits, we can use things such as binomial approximations or Taylor expansions. For example, if I have the following formula $$\...
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Error in difference of matrices
Suppose that I have two matrices $A$ and $B$ and I know that $A^n\to B$ as $n\to \infty$. Clearly, when we pick some norm one will find that $\|A^n-B\|\to 0$ as $n\to \infty$.
Now let’s say that I ...
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Approximating the number of numbers with a given divisor count
I want to estimate the number of number with $k$ divisors up to $n$, $\mathcal{D}_k(n)$. I am mostly interested in estimating $\mathcal{D}_k(n)$.
We will also find the "$k$-prime" counting ...
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Why can't this approximation be made
Let's say we have a function $f_{\varepsilon}(x)$ and we know that
$$\lim_{\varepsilon \to 0} f_\varepsilon(x)=g(x).$$
This is equivalent to saying that $f_\varepsilon(x)=g(x)+o_{\varepsilon\to 0}(1)$....
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How to minimize the maximum absolute difference between 2 functions?: example $\min_a\{\|\text{erf}(x)-\tanh(\frac2{\sqrt{\pi}}(x+a x^3))\|_\infty\}$
How to minimize the maximum absolute difference between 2 functions?: example $\min_a\{\|\text{erf}(x)-\tanh(\frac2{\sqrt{\pi}}(x+a x^3))\|_\infty\}$
Intro_______________
In this other question I ...
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Is there a way to simplify this expression $ (i\cdot T+ e^{-i\cdot T} -1)$
I've been using Maple to solve some problems at hand. In particular, when I solve this integral
$$ \int_{0}^{T} {(e^{i\cdot (t - T)} - 1)\cdot (N-a) \over b} dt = S $$
I get the following solution:
$$ ...
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How to approximate a curve passing through the surface of a mesh to a curve passing through the edges of the mesh (not passing any face of the mesh)?
I have a 3D triangle mesh. A curve is found by some algorithm. The curve does not pass through the edges of the mesh. The curve intersects the edges of the mesh, the locations of which are given (P0, ...
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Estimate $\sqrt{(10.1)^2 − (2.9)^3 − (3.05)^2}$ via differentials
This is a question from my Math for Economics assignment.
If it was a single variable question, like $\sqrt{(2.9)^3}$, I would use tangent line equation $y-y_1=m(x-x_1)$ where $x$ is $2.9$, $x1$ is $3$...
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Can asymptotic approximations of terms be used to give asymptotic approximations of a sum? [closed]
Let $a_n \sim b_n$,
More formally I mean, $\displaystyle{} \lim_{n \rightarrow \infty}\frac{a_n}{b_n} = 1$
Would that then imply that for some $n_0$, $\displaystyle{} \lim_{T \rightarrow \infty} \...
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Integrable approximation of error function over Gaussian measure
I am interested in a problem that involves computing the expectation of of the CDF $\Phi$ (or equivalently erfc shifted and scaled) for the standard normal distribution, for $x$ normal distributed ...
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3
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Approximating for the Error function $\text{erf}(x)$ through an Hyperbolic tangent function $\text{tanh}\left(\dfrac{4x}{4-x^2}\right)$
Approximating for the Error function $\text{erf}(x)$ through an Hyperbolic tangent function $\text{tanh}\left(\dfrac{4x}{4-x^2}\right)$
I was plotting some functions and I found that the function
$$f(...
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Help understand this solution/Approximation
Good day everyone. In this appendix, particularly this part, the author approximated $1-aR$ with $e^h$ where $h=2.5$. Can anyone explain to me how did the author get this outcome? Variable $a$ is a ...
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Approximation of a Generalized Hypergeometric Function
Is there any approximation with an "elementary function" for the following generalized hypergeometric function, especially for very large values of $n$ and $0 < p < 1$? I mean, without ...
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How is the Chebyshev polynomial approximation defined when using the spectral method to numerically solve PDEs?
I am studying spectral methods for numerical solutions for PDEs. Currently, I am on a chapter that explains how to use Chebyshev polynomials to solve non-periodic boundary value problems.
I understand ...
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Polynomials approximate $L^2$ function
Suppose $(\Omega, \mu)$ is measurable space with $\mu(\Omega)<\infty$, here $\Omega$ is a compact subset of $\mathbb{C}$. $f,g\in L^2(\Omega)$ with $\mu(\{x:f(x)=0\})=0$. I want to find polynomial $...
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Why does this approximation for binary entropy of $\cos^ {2}(t)$ hold?
I stumbled on an approximation I found surprising while working on a bipartite entanglement entropy problem (which isn't particularly relevant). Alas, I got the following messy result:
$f(x) = -\cos^2(...
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32
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2
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2
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55
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Linear approximation using $2D$ Linear Interpolation
Probably it is a simple question. but even after several hours of research, I could not find anything relevant.
I have a function $f(x,y) = xy$, with $x$ and $y$ that belong to bounded, continuous ...
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Efficient and Accurate Formulas for Approximating sin x , cos x , tan x and ln x.
Summary:
I am currently studying various mathematical functions and their real-world applications. I'm particularly interested in trigonometric functions $( \sin, \cos, \tan )$ and the natural ...
3
votes
2
answers
360
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Formula for the perimeter of an ellipse
So despite being rather amateur when it comes to this level of math, I tried my hand at this and am quite satisfied with the answer. I've checked the numbers and it's very close to Ramanujan's ...
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1
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Deriving small angle approximation $ \cos\delta_a\cos\delta_b\ \approx \cos^2\delta_a $
I'm trying to understand the derivation for the small angular distance approximation formula for the Angular Distance, as explained in Wikipedia
Almost to the end of the derivation, I'm presented with ...
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