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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7 views

Matrix approximation using only rows from the original matrix

Can someone please point me to the relevant literature for the following problem: Let $\mathbf{X}$ be a matrix comprised of $m$ different rows and $n$ columns, where $m\gg n$. I want to approximate $\...
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33 views

Approximating numbers using integer powers of primes

This is my first question here. Thanks for any advice. Here's the problem: For a positive integer $N$, define $P_N$ as the set $\{ p_1^{x_1}p_2^{x_2}...p_N^{x_N} \mid x_i \in \mathbb{Z}\}$, where $p_1,...
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Gaussian integral: $ \int_{-\infty}^{\infty} x^2\left\{ \Phi (a + b x) \right\}^2 \phi(x) \mathrm{d}x $

Is there an analytic solution for the following integral? $$ I = \int_{-\infty}^{\infty} x^2 \left\{ \Phi (a + b x) \right\}^2 \phi(x) \mathrm{d}x,$$ where $\Phi(\cdot)$ and $\phi(\cdot)$ are the cdf ...
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Approximation to the pth root of an integer N

I was working on the Introduction to Number theory textbook by CJ Bradley and encountered the following theorem . If $N>0$ and $\frac{u}{v}$ is a good rational approximation to $\mathrm{N}^{\frac{...
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62 views

Madelung Constant

I have been working in this series $$\sum _{m=0}^{\infty } \sum _{k=0}^{\infty } \sum _{j=0}^{\infty } \frac{(-1)^{j+k+m} \left((j+1)^2+(k+1)^2+(m+1)^2\right)}{\left((j+1)^2+(k+1)^2+(m+1)^2\right)^{3/...
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$\underset{n \to \infty}{\text{lim}} (1-p)^{n-d-1}$, given $p = o(n^{-(d+1)/d})$, and $d \geq 0$ a constant.

This is a step in a proof I’m reading. I did a bunch of calculations and it seems like the answer is $1$. But I’m not sure so I’d like to get it checked, or to see a cleaner argument. Also, my ...
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How can I approximate $t\to \sin(t)$ function for art purposes using computationally cheap alternatives?

Background : The other day I felt like updating my knowledge with GUI programming so I made a small application that rendered a multivalued periodic function $\mathbb R^3 \to \mathbb [0 ,255]^3$. The ...
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1answer
12 views

Order of Backward Euler Method

Say we have a function $y(t)$, that satisfies the ordinary differential equation $\frac{\mathrm{d}y}{\mathrm{d}t}= f(t,y) \quad\text{for } t\in (t_0, t_{\max}],$ where $t$ takes discrete values, $t_n$,...
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Approximation of time dependent matrix

The approximation theory of scalar and vector functions has a lot of famous results based on families of basis functions. I do not know similar results in the field of time-dependent matrices (or more ...
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Approximation of $\frac{1-x}{1+x}$ if $x \ll 1$?

In an exercise in general relativity, I am trying to show that in the limit as $\hat{r}\rightarrow\infty$, \begin{equation*} \frac{1-GM/2\hat{r}}{1+GM/2\hat{r}} \approx 1 - \frac {2GM}{\hat{r}},\ \ \...
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Approximate $\log_{10}$ values without a calculator

I've got this problem: $1,000,000^{{1,000,000}^{1,000,000}} < n^{n^{n^n}}$ What is the first positive integer value of n for which this inequality holds? I managed to reduce it to this: $6+\log_{...
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1answer
54 views

If $x \ll y$, then is $xy$ large enough to not be neglectable?

Say that we have some number $x$ that is much less than $y$. My question is, if we have some expression $x + y$, then $x$ could be assumed to be small enough such that the expression is approximately $...
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1answer
22 views

$X\sim\text{Bin}(n,p)$. Lower bound on $\Pr(X=np)$ that is a function of $m=np$, and not dependent on $n$

Consider the binomial distribution with parameters $n$ and $p$, $\text{Bin}(n,p)$. It has mean $m=np$, and the probability that a random variable $X\sim\text{Bin}(n,p)$, equals its expected value, is $...
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1answer
50 views

Approximating a summation by partial summation+integral

Say, for example, you want to approximate the sum: $$\sum^{\infty}_{n=1}\frac{1}{n^2}=\frac{\pi^2}{6}$$ The most intuitive first step would be to truncate the summation at $N$ $$\sum^{N}_{n=1}\frac{1}{...
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1answer
26 views

Approximation of third derivative

Let $f(x)$ be the investigating function. Let's say that we have the values of $f(x)$ at equidistant locations $x_0, x_1, x_2$ and $x_3$ (consecutive points are separated by $h$ apart) as $f_0, f_1, ...
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How to use the Backwards Euler Method

Given the ODE $\frac{dy}{dt} = f(t,y)$ and the function $f(y) = -y^3$, with the initial condition $y(0)=1$, I want to use the backward Euler Method with $h = \frac{1}{2}$, combined with the Newton-...
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Confusion about factorial $\dfrac{(N!)^2}{(2N)!}$

I am confused with the expression $\dfrac{(N!)^2}{(2N)!}$. Suppose we write $(2N)!=p_1^{\alpha_1}p_2^{\alpha_2}\dots p_\ell^{\alpha_\ell}$ and by Hardy-Ramanujan theorem that states that almost all ...
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1answer
19 views

Bounding the coefficients of the approximating polynomial of a $1$-Lipschitz function

Let $f$ be a $1$-Lipschitz function on the interval $[0,1]$. Then, by Jackson's theorem we get that there exists a polynomial $P_n$ of degree $n$ such that $$ |f(x)-P_n(x)|=O\!\left(\frac{1}{n}\right)....
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Deriving an approximation for the Riemann zeta function

By analyzing the graph of the function $\zeta \left(\frac{1}{2}+it\right)$ I've noticed a circular shape for $-3.3<t<3.3$ as shown below. Based on that, an approximation for such function on ...
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1answer
82 views

Approximate $\sin 29^\circ$ using differentials

Using differential find approximate value of $\sqrt[3]{1.02}$ I did this. We know $f(x_0+\Delta x)-f(x_0)=\frac{df}{dx}(x_0)\cdot\Delta x$ $$f(x)=\sqrt[3]{1+x}$$ $$f(0.02)-f(0)=\frac{df}{dx}(0)\cdot0....
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1answer
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two-term asymptotic expansion for each root of the polynomial equation

two-term asymptotic expansion for each root: $\epsilon z^8-z^3-1=0$ as $\epsilon \rightarrow 0$ how to find other solutions than the one near -1? first, I check when $\epsilon=0$, I get the root of $z=...
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Estimating the number of infected people

Given a set $U$ of people, there is a subset $S\subset U$ of infected people. You only know the size of $U$ but want to estimate $\dfrac{|S|}{|U|}$. However, you don't have enough time to test every ...
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2answers
61 views

Sine cosine function approximation [closed]

I have originally posted this question in electrical stackexchange. I am just interested to know the derivation of the $\sin \phi$ approximation.
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Guessing the correct subset of a universe

Let $U$ be a universe, s.t. $|U|=\text{exp}(n)$ (exponential in $n$). Let there $G_i \subseteq U$ for $1\le i\le n^2$. Also let $f_i: U \to \{0,1\}$, s.t $$ f_i(e) = \begin{cases} 0 & e \not \in ...
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pattern in a certain approximation of square roots

I was playing around with approximating an irrational number $x>1$ as the product of an integer and terms of the form $\frac {n+1}{n}, n\in \mathbb{N}$. In particular I investigated the sequence ...
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An approximation for log(x!) [duplicate]

During development of a data compression routine, I needed to implement $log(x!)$ as it came up in entropy estimation. As $x!$ is impractical to represent as an intermediate result due to its ...
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1answer
38 views

Why does this Gamma function approximation fail?

I've spent a lot of time poring over this equation from Wikipedia: I was curious about its accuracy for low $N$, so I plotted it on Desmos, against my reference curve which I described using the ...
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28 views

Approximative projection

Let $\mathcal{A}$ be the subspace matrix of $A$ s.p.d., such that $\mathcal{A}=Q^{\top}AQ$. ($Q$ is a rectangular matrix.) We define $ p_1 := Q \mathcal{W}^{-1} Q^{\top} A$ $\mathcal{W}$ is a non-...
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1answer
33 views

Maximum Likelihood estimator question, completing the rest [closed]

I have $X_1,X_2,\dots,X_n$ as independent continuous r.v.s w/ p.d.f. $f(x) = \theta x^{\theta-1} $ if $0 < x < 1$, $0 $ otherwise The question states to find the maximum likelihood estimator (...
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1answer
64 views

How to bound the error of an approximation for ln|k+x|?

$$\forall x\in\left(-1,1\right):\space\ln\left|1+x\right|=\int\frac{1}{1+x}\space dx=\int\sum_{n=0}^{\infty}\left(-x\right)^n\space dx=x-\frac{x^2}{2}+\frac{x^3}{3}\dots=\sum_{n=1}^{\infty}\left(-1\...
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2answers
101 views

Can $ \int \frac{1}{x^5} dx $ be used to approximate $ \int \frac{1}{x^5+0.001} dx $?

The following similar indefinite integrals both have very different answers: $$ \int \frac{1}{x^5} dx $$ $$ \int \frac{1}{x^5+0.001} dx $$ Is there way to approximate an answer to the second integral, ...
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1answer
65 views

Backwards Euler Method/ Implicit Euler Scheme - Difference Equation + Stability

The backwards Euler method (implicit Euler scheme) is a numerical method for the finding the solution of ordinary differential equations, which is defined as follows, $$ y(t_{n+1}) \approx y(t_n) + hf(...
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Proof by self reduction that there is no approximate algorithm for SetCover with constant upper bound

I have to proof by self reduction that assuming $P \neq NP$ there is no polynomial approximation algorithm for the SetCover problem such that $$|OPT(I) - A(I)| \leq k$$ with $k \in \mathbb{N}$. This ...
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1answer
32 views

Calculating Definite Integral Using Taylor's Series

I'm an amateur so please go easy on me if this is a bit low-level. I'm working on a paper and derived the following which looks like it's just a restatement of Taylor's approximation. It's not exactly ...
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For $x,y>0$ then under what assumption for the approximation $\frac{{1 + x}}{{1 + y}} \approx \frac{x}{y}$ to be valid?

For $x,y>0$ then under what assumption for the approximation $\frac{{1 + x}}{{1 + y}} \approx \frac{x}{y}$ to be valid ? I see some researcher use this approximation in their paper, without ...
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27 views

Approximating an integral over a bell-shaped measure with a Gaussian

Denote the Gaussian tail function $H(x)=\int_x^\infty \frac{dt}{\sqrt{2\pi}}e^{-x^2/2}$. I would like to approximate an expression of the form: $$E_P(a,b) = \int dxP(x) H(a x + b)$$ for a specific PDF ...
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2answers
38 views

Best linear approximation for x(x-y)

I have a non-linear term in the form of $$x(x-y)$$ I need to replace it with a linear term in the form of $$c_1x-c_2y$$ I have seen the following linear approximation in literature but I need to know ...
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Summing the total error of a Taylor series over an interval

Just for fun, I played around with a Maclaurin series for $\sin(x)$. I did this on Desmos, which I know will have numerical precision errors. I discovered that: $$T(x) = x-\frac{x^{3}}{3!}+\frac{x^{5}}...
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1answer
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Phase differences approximation

I'm sitting and trying to solve the equation of the phase difference given by: $\Delta \phi = k (\sqrt{a^2+d^2} -d ) \approx \frac{ka^2}{2d}$ Where $a$ is the size of an aperture and $b$ is the ...
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Approximating $\sqrt{2 \pi} \approx 5/2$

I noticed when working with the Gaussian distribution that $\sqrt{2 \pi} = 2.5066 \approx 5/2$. Is there any use in other approximations or geometric intuition to this? I suppose if you plot a ...
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2answers
35 views

Find an approximation of the integral $\int_{0}^{1}\frac{\sin\left(x\right)}{x}dx$ With an error less than $10^{-5}$

Find an approximation of the following integral $$\int_{0}^{1}\frac{\sin\left(x\right)}{x}dx$$ With an error less than $10^{-5}$. This is the first time I've been asked such a question and I don't ...
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1answer
29 views

Good approximation of the Lambert function for $(-\tfrac{1}{e},0)$

I am looking for an $\epsilon > 0$: $$\epsilon < \frac{1}{a}\left(1+R\,W_0(\beta)\right)$$ where $$a = 1 + \frac{kv}{g}\,\sin{\alpha}, \hspace{20pt} R = \frac{v}{k}\,\cos{\alpha}, \hspace{20pt} \...
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1answer
93 views

Global approximation by functions smooth up to the boundary

Consider the proof of Evans: My questions: Why the ball $B(x^{\epsilon})$ lies in $U\cap B(x^0,r)$, for all $x\in V?$ I think that this is true for an neighbourhood of $V$. How can i prove in a ...
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Approximating a transformed volume

A solid ellipsoid $E$ which contains $(1,2)$ is transformed to a region $T(A)$ using $T(x,y) = (x^2 +y^2, xy)$. Approximate the volume of $B$ if the volume of $A$ is $0.1$. I have absolutely no idea ...
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28 views

Simplify the expression $\log_2({x_1}^{y_1} + {x_2}^{y_2})$ for large values of $x,y$

Following this thread: Simplify the expression $\log(2^x+2^y)$ for large values of $x,y$ Is there a way to simplify (or get a good approximation) for the $\log_2$ of a sum of powers with different ...
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35 views

Approximate matrix inverse by Fourier approach

Given a hermititan matrix $A$ with the possibility to generate $e^{-iAt}$ for $t\geq 0$ how would I proceed to approxiamte: $A^{-1}\approx\sum_j\alpha_je^{-iAt_j}$ $\quad$ with $\alpha_j\in\mathbb{C}$...
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2answers
58 views

Approximating Lipschitz Functions by Sigmoidal Functions

We define a sigmoidal function $\sigma: \mathbb{R} \to [0,1]$ as a non-decreasing, continuous function with $\lim_{x\to\infty}\sigma(x) = 1$ and $\lim_{x\to-\infty}\sigma(x) = 0$. I want to show that: ...
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1answer
31 views

Why is $\exp(n/2 * (\log 2)^2) \approx 1.27^n$

I saw this approximation made in https://stats.stackexchange.com/questions/473496/infinite-coin-toss-probability by the accepted answer. What inspired this approximation?
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43 views

Show that $\forall x \in [a,b]$ $\frac{f(x)-f(a)}{x-a}\le \frac{f(b)-f(a)}{b-a}-\frac{k}{2}(b-x)$

Let $f\in C^2([a,b])$ such that $f''\ge k \ \forall k \in \mathbb{R}$. Show that $\forall x \in [a,b]$: $\frac{f(x)-f(a)}{x-a}\le \frac{f(b)-f(a)}{b-a}-\frac{k}{2}(b-x)$. My attempt was to use Taylor ...
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1answer
39 views

Approximation for a system of linear functions

There is a list of food and it's composition. For example, salmon: protein - 64 g, fats - 14 g, carbohydrate - 0 g per 100 g of food; nuts: protein - 20 g, fats - 53 g, carbohydrate - 21 g per 100 g ...

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