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).
4,622
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Why does $y = \frac{2A\sin(x\pi \ell)}{\pi^2 \ell (1-\ell) x^2}$ simplify to $y=c/x$ as $\ell$ approaches $0$?
I am working on something using this equation, and I find something strange. I am manipulating $\ell$ here between $0$ and $1$. I note that as $\ell$ approaches zero ($\sim0.001$ or less) it becomes a ...
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6
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Find limit: $\lim\limits_{n \rightarrow \infty} \left( \cos\frac an \right)^n$
EDIT: since the proposed below "property" is incorrect in general, one can solve the limit by exponentiating the function.
When investigating this limit
$$\lim\limits_{n \rightarrow \infty} ...
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Question about the asymptotic expansion of the Exponential Integral $\int_z^\infty \frac{e^{-x}}{x}dx$: using Taylor expansion on the infinite domain
I am reading the argument on finding the first three terms of the asymptotic series of the Exponential integral $E_1(z)$ as $z\to \infty$, but I don't understand a step here.
The Exponential integral ...
- 5,511
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1
answer
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Approximation of $(1 - 1/n)^m$ when $m$ depends on $n$
Specifically, is it correct to approximate the expression $\left(1 - 2^{-t}\right)^{x \cdot 2^t}$ as $e^{-x}$? This answer from an older post suggests that for an expression of the form $(1 - 1/n)^m$, ...
- 147
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Approximate inverse Mellin transform
I need to evaluate the following complex integral (which is essentially an inverse Mellin transform):
$$\int_{-c-i\infty}^{-c+i\infty} \Gamma^2 (-s) \Gamma (s+1) \Gamma (a-s) \mbox{}_1F_1(s;1;-y) x^s{...
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finding $a,b$ such that the equation $\frac{a}{x} = \cosh{\frac{b}{x}}$ only has one solution
The equation can be rewritten (assuming I haven't made a mistake) as
$$\alpha y = \ln (y \pm \sqrt{y^2 - 1})$$
with $\alpha = \frac{b}{a}$ and $y=\frac{a}{x}$.
It seems that, for the +ve square root, ...
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MLMC Estimate, Implementation and expected value diverging
i'm having some doubts/problems implementing a Multi Level Monte Carlo method. The setting is the following, I have generated some samples (approximated solutions of a PDE) using various polynomial ...
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1
answer
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Best polynomial approximation does not exist
Let $f$ be a real-valued continuous function on $[0,1]$ which is not a polynomial. Prove that there does not exist a polynomial $p$ such that $||f-p||$=$\min_{q \in \mathbf{P}}||f-q||$, where $\mathbf{...
- 1,673
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approximate definite integrals using quadrature approximation
I understand that we are looking for the Q formula that has the highest possible degree of precision
Why do we stop at $x^2$?
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Simplifying the set of constraints of an optimization problem
I’m currently working on a constrained optimization problem (the problem comes from the European Power Market) where the constraints define a solutions space which forms a complex polytope with many ...
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1
answer
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Evaluating $\int_{0}^{\infty}{\frac{e^x}{x!} dx}$
I was trying to evaluate $\int_{0}^{\infty}{\frac{e^x}{x!} dx}$ or approximate it (WFA only gives an approximate result so maybe there is no closed form). I tried the Fourier transform :
$$\mathcal{F}\...
- 977
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Polynomial approximations to $e^{f(x)}$
Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is a bounded function on some compact subset of the real numbers, i.e. $|f(x)|\leq B$ for every $x$ in the domain $D = [-L,L]\cap \mathbb{R}$. For ...
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Is there a asymtopic bound to a confluent hypergeometric function of ${_1F_1}(n,1.5n,nz)$?
As the title mentioned, is there a good way to approximate
$${_1F_1}(n,1.5n,nz)$$
where $n \in \mathbb{N}$, and $z$ is a real positive number. If the direct bound is difficult, an asymptopic bound (as ...
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Dimension of maximum volumed unit ball
Let $V_n=V(B^n)$ be the volume of the $n$-dimensional unit ball $B^n$. By cross-sectioning $B^n$ along $x_n$-axis, $-1\leq x_n\leq 1$ and by means of similarity of hyper disks we have
$$V_n=2\int_0^1(\...
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Does the approximation related to the generalized quadratic Gauss sums hold?
I have a generalized quadratic Gauss sums, which is defiend as
$$S(a,b) = \sum_{n = 0}^Ne^{-j(an + bn^2)}$$
where $a\in(-\pi,\pi)$, $b\in(0,\pi)$ and $n$ is an integer. Now I am trying to approximate ...
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Estimate the sum of a series only with the recursion $a_{n+1}=\frac{a_n}{a_n^2+1}$
We have $\{a_n\}$,
$$
a_{1}=1,\quad a_{n+1}=\frac{a_n}{a_n^2+1}\quad \big(n\in\mathbf{N}^{*}\big)
$$
Note
$$
S_n=\sum\limits_{k=1}^{n}a_k
$$
How to estimate $S_n$? For example, I just want to know $\...
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Can smooth approximations always preserve injectivity?
Most generaly, does every continuous injective mapping $f:M\rightarrow N$ (these are smooth manifolds) have a smooth and regular injective approximation of arbitrary precision, given that $\dim(N)>\...
- 383
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Approximate solution to ODE potentially using perturbation theory
On the wikipedia article for stokes drift (https://en.wikipedia.org/wiki/Stokes_drift) they show that for:
$\dot{\zeta} = u\sin(k\zeta - wt)$ has approximate solution (by perturbation theory):
$\zeta(\...
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Derive a formula approximating the mean of logit-normal distribution
Let $X\sim N(\mu,\sigma^2)$ be a normal distribution and $Y$ a logit-normal distribution $logit(Y)=X$ of parameters $\mu$ and $\sigma$.
I need to calculate the mean of $Y$ but it does not have an ...
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Why is this for function specific range tending towards $e$
I was messing around with the binomial approximation method as per which $(1+x)^{n} ≈ (1+nx)$ for $x<<1$, so while entering values in the calculator I observed something strange that the value, $...
- 3
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47
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Equivalent of a special Generalized hypergeometric function
Is there any equivalent or tight upper bound with an "elementary function" of following generalized hypergeometric function:
${}_k F_{k-1}(2,\dots,2,1-m;1,\dots,1;-1)$
when especially
$m$ ...
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Could an equation like ${^{h}b} = b^h + $c together with polynomial interpolation be used to calculate tetration? Are there flaws in it?
As it is quite easy to calculate integer solutions for $\;^hb = b^h $ the question arose on how to find a solution for real heights.
One idea was of predicting the height by finding a formula ...
2
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2
answers
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How can I find the periods of difference of the sine waves that has irrational coefficients
I'm working on a project right now. And now I need to find periods of difference of the sine waves and i'm stuck.
In few resources I found that I can find the periods of summed or differenced sine ...
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An approximation to $\ln(1+e^x)$ and how to use it for splitting the logarithm of a sum [closed]
Trying to split the logarithm of the sum of two exponential functions (this question), I found the following approximation for the Softplus function $f(x)=\ln(1+e^x)$:
$$\ln(1+e^x) \approx \begin{...
- 1,390
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Given base $b$ what is the expected value of round off error in rounding to $d$ digits?
Given base $b$, what is the expected value of round off error (relative, not absolute) when we round $x$ to $d$ digits, where $x$ is a random variable? Assume $x$ is drawn from a uniform distribution ...
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How did Feynman produce solutions in under a minute?
As detailed here and elsewhere, Feynman and others at Los Alamos could calculate many problems to 10% accuracy in minutes:
When I was at Los Alamos I found out that Hans Bethe was absolutely topnotch ...
- 4,360
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It is possible to represent the sum of two geometric Brownian paths as another geometric Brownian motion?
It is possible to represent the sum of two geometric Brownian paths as another geometric Brownian motion?
Intro_________________
I am trying to understand how it would behave a weighted composition of ...
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Closed form error of inner-product-based low-rank approximation of binary matrix
Given a binary square matrix $A \in \{0, 1\}^{n \times n}$ and $1 \leq k \leq n$, we aim to compute the error of outer-product-based low-rank approximation.
Formally, we aim to compute
$$
\min_{X \in \...
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best polynomial approximations to $f$ vanish at $0$ imply $f$ is an odd function
It is known that if $f\in C[-1, 1]$ is odd/even, then the best polynomial approximation (in $L^{\infty}$ norm) of degree $n$, denoted by $p_n$, must also be odd/even. This has been asked on MSE before....
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Contradiction occurred trying to calculate the value of $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)$ [closed]
Today I was trying to calculate the value of $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)$ (i.e., $\sqrt{2}$ plus the real part of the tenth tetration of the base $-\sqrt{2}$) up ...
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Can I approximate $\rho_n$ somehow so it satisfies this Bessel function conidition? [closed]
While solving a PDE using the eigenfunction expansion method, I came across the following condition for $\rho _n$:
$$ J_0(\rho_n r_{max}) = J_2(\rho_n r_{max}) \tag 1$$
where $r_{max}$ is known. Is ...
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German online explanation of Schwarz lantern
I'd like to share this thing I just came upon with a friend who doesn't read English.
Schwarz lantern is a polyhedral approximation to a cylinder, used as a pathological example of the difficulty of ...
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Interpolation and general Gaussian quadrature
I just finished a course on numerical mathematics, and have become quite interested in interpolation and how it ties into numerical integration. What suprised me while studiyng quadrature was the fact ...
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What is the error of reconstruction of a smooth function observed only on a fixed grid by projection on a wavelet basis?
Context
I'm a PhD student in Statistics and I have evaluations of a $L_2([0,1])$ function $f$, that is $m$ times derivable, on a regular grid of $[0,1]$
$$f\left(\frac{k}{p-1}\right), 0\leq k \leq p-1....
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Approximating $\pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}$ with stable decimal places
Consider the Leibniz formula for $\pi$
$$
\pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}.
$$
What is the minimum number of terms needed to calculate $\pi$ accurate to $k$ decimal places, in the sense ...
- 3,427
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Additive approximation to the subset-sum problem
In the subset sum problem, the input is a list of positive numbers summing up to $S$, and a target value $T<S$. The goal is to find a sub-list with the largest possible sum that is at most $T$.
...
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Theory of sequence appoximation
There's plenty of literature about function approximation, both uniform and pointwise. Moreover, there are typically results on the speed of convergence of a given basis to the approximated function ...
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Approximations to elliptic integrals
For my physics research, I need to do some series expansions of complete elliptic integrals of the first kind. When I tell Mathematica to approximate it, I get...
$$\int_0^{\pi/2}d\theta \frac{1}{\...
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1
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Approximating the area of a washer [closed]
In my textbook the area if a washer is appropriated by a washer is $2\pi r_i\Delta r$ while what I know the area if a washer to be is $\pi(r_i + r_{i-1})\Delta r.$
Here's the segment of my textbook ...
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2
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Computing Gaussian-Weighted Integrals for Small Lengthscales
I am attempting to compute the following integral:
$$ I(l) = \int_{[-1,1]^2}dxdy \sqrt{1-x^2} \sqrt{1-y^2}e^{-\frac{(x-y)^2}{l^2}} $$
I am using Mathematica for symbolic calculations and translating ...
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Little-o notation for limits at zero
Very quick question which will help me clarify my understanding. Let's say I have an expression that uses little-o notation like the following
$$o_{h\to 0}(h)(1-h)+o_{h\to 0}(h)$$
Would I be correct ...
- 1,108
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Finding the leading order asymptotic approximation
Find the leading order asymptotics of $$I_n(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} e^{x\cos(\theta)}\cos(n\theta)d\theta$$ as $x \to +\infty$.
Here's my work:
I am using the Laplace method: $\cos(\theta)...
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0
answers
54
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Is my Taylor series expansion correct?
I am trying to approximate a function $f(r) = \sqrt{r^2 +a^2 +b^2 -2ra\cos \theta \sin \phi -2rb \cos \phi}$.
I want to use $r\times\sqrt{1+x} \approx r(1 +x/2 -x^2/8)$ to approxiimate this.
Here is ...
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On a First Order Nonlinear Differential Equation
Consider the ordinary differential equation $\dfrac{\text{d}y}{\text{d}x}=\dfrac{y-3}{x^2+y^2},$ with $y(0)=1$. My question is about determining the graph of $y$. Here is most of the information that ...
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votes
2
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Minimum number of terms to approximate $\pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}$
Consider the Leibniz formula for $\pi$
$$
\pi=4\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}.
$$
What is the minimum number of terms needed to calculate $\pi$ accurate to $k$ decimal places?
My attempt: ...
- 3,427
4
votes
1
answer
360
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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 ...
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1
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0
answers
45
views
Approximating probabilistic event by Central Limit Theorem [closed]
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 $\frac{S_n}{n}$ ...
- 167
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0
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43
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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 ...
- 1
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vote
0
answers
49
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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\...
6
votes
2
answers
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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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