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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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Log-Sum-Exp as an approximation of min function

I can prove that the function: $$f(\tau, x_1, x_2, ..., x_N) = -\tau \log \frac{1}{N} \sum_{i=1}^{N} \exp{\left(-\frac{x_i}{\tau}\right)} $$ converges to $\min(x_1, x_2, ..., x_N)$ for $x_i \geq 0$ as ...
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$\{a_n\}$ be a sequence such that $ a_{n+1}^2-2a_na_{n+1}-a_n=0$, then $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in…

Let $\{a_n\}$ be a sequence of positive real numbers such that $a_1 =1,\ \ a_{n+1}^2-2a_na_{n+1}-a_n=0, \ \ \forall n\geq 1$. Then the sum of the series $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in......
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2answers
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How can this binomial expansion result in two different approximations of root 2?

I have been working on a problem on approximating $\sqrt{2}$ using the first three terms of the following binomial expansion, and a substitution of $x = -\frac{1}{10}$ : $$(4 - 5x)^.5 = 2 - \frac{5x}...
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Looking for a specific differentiable function

I'm from computer science and not a mathematician, so i hope you can help me by finding a function with the following properties: In principle it is about the following function: \begin{equation} ...
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23 views

Comparing two curves

For my Bachelor thesis, I have to analyse the goodness of an Approximation. The exact formula is called $\mathcal{I}(\varepsilon)$ and the Approximation $\mathcal{I}_{approx}(\varepsilon)$. In order ...
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2answers
45 views

Finding leading order approximation to $\frac{d^2y}{dx^2}-\epsilon y=x$

The conditions provided were $y(0)=1$ and $y'(0)=1$. Since this equation is regular, I can neglect the term involving $\epsilon$, giving $\frac{d^2y}{dx^2}=x$. I tried to solve this in order to ...
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1answer
24 views

Inverse of the Gamma Function

I've come across an equation that needed use of the gamma function's functional inverse, and I know that you can't represent the functional inverse of the gamma function by any combination of ...
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45 views

What does it mean to say it “converges geometrically”

I've been reading up on some algorithms and I've come across the statement "it converges geometrically" but what does this mean? How does it differ from saying "it converges"? Is there some "rate" ...
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1answer
62 views

Find the value of $f''(0)$ to leading order

Question: Suppose that $f$ satisfies $$f''' + \frac 12 ff''=0$$ with $f(0) = f'(0) = 0$ and $f(x) \sim x$ as $x \rightarrow \infty$. Find the value of $f''(0)$ to leading order. Attempt: The ...
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Under what condition, a function can be upper bounded by its second-order Taylor expansion?

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$, and $g(x) = f(a) + \nabla f(a)^\top (x-a) + \frac{1}{2}(x-a)^\top \nabla^2f(a) (x-a)$ be its second-order expansion at $a$. We know that if $f$ is concave,...
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62 views

Gordon Growth Model (2-Step) Approximation

It is easy to prove that the sum of the infinite geometric series $$\frac{D(1+g)^n}{(1+r)^n}$$ is $$S = \frac{D(1+g)}{r-g}$$ where n = 1, 2, 3, ... and 0 < g < r < 1. Now, if the infinite ...
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1answer
46 views

Please take a look at my proof on Dirichlet's theorem

General/simultaneous Dirichlet's theorem : If $α_1,\dots,α_m\in \mathbb R$ and $n\ge 1$ is an integer, then there are integers $p_1,\dots,p_m$ and there exists an integer $q$ with $1\le q\le N^m$ such ...
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Elevation profiles approximation with polynomials/splines

enter image description hereI am to approximate natural ground elevation profiles with a mathematical function. Let say the elevation profiles can be approximated with ...
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1answer
113 views

Removable singularity in $\phi(\vec{x})=\int \left[\phi\frac{\partial (\ln r)}{\partial n} -\ln(r) \frac{\partial \phi}{\partial n}\right] ds$

If I have the following integral equation $$\phi(\vec{x})=\frac{1}{\pi}\int \left[\phi\frac{\partial (\ln r)}{\partial n} -\ln(r) \frac{\partial \phi}{\partial n}\right] ds$$ An approximate solution ...
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1answer
23 views

In search for a method to approximate a curve into horizontal segments

In order to model a 24-hour load profile I need to approximate it into smaller number of pieces. Is there any method to approximate such a curve? Would anybody help me to figure out this?
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Approximate a density function from sampled data

Let $(E,\mathcal E,\mu)$ be a measure space $E_0\in\mathcal E$ with $\mu(E_0)\in(0,\infty)$ and $\mathcal E_0\subseteq\left.\mathcal E\right|_{E_0}$ be finite and disjoint with $$E_0=\biguplus\...
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What does “quasi-optimal” approximation mean in Finite Element Method (FEA)?

In FEA, we know that optimal approximation usually refers to that the approximate solution converges to the exact solution with optimal rates. However, what is "quasi-optimal" approximation mean ...
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1answer
28 views

What is a good estimate for this log sum?

Given $n\gg0$ what is a good estimate for $$\sum_{i=1}^K\log\Big(\frac{n}{3^i}\Big)?$$ I am particularly interested in case of $K=O(1)$ and $K=O((\log n)^c)$ at a fixed $c>0$.
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Approximating a discrete function with continuous function. [closed]

$$\sum^n_{i=1}\frac{x^i}{(n-i)!}$$ I tried approximating it but i am not able to find any series similar to this . I need to find a function which can be used in place of this series. Thanks!
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Bernstein approximation error for functions with Lipschitz continuous derivatives

Consider a Lipschitz continuous function $f$ on $[0,1]$ with constant $L_f$, and its Bernstein polynomial $B_{n,f}$ of degree $n$. It can be shown that $$\sup_{x\in[0,1]}|B_{n,f}(x) - f(x)| \leq \...
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Dirichlet's approximation on two numbers

Show that for any real α,β and any positive integer N, there are integers p and p′, and an integer q satisfying 1 ≤ q ≤$ N^2$, such that $$|a-p/q|<1/q^N$$ and $$|β -p'/q|<1/q^N$$. I understand ...
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1answer
21 views

Write logical operator all(x<a) in terms of Heavyside functions

I am currently solving a complex optimisation problem, with constraints that take the form: $1 - all(g(x)<a) <= 0$, meaning I require all values $g(x)$ (for some function $g$) to be below some ...
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1answer
32 views

Find the composite solution to this problem

I want to find a composite solution to the boundary value problem: $$ \epsilon y'' - y' + y^2 = 1, \text{ for }0<x<1 ,\text{ where }y(0) = 1/3,\,y(1) = 1 $$ where $\epsilon\ll 1$. My approach: ...
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1answer
50 views

Average approximates definite integral.

In my studies, I read a result similar to the following. Let $g:[0,1]\rightarrow \mathbb{R}$ be twice continously differentiable on $[0,1]$ (this is probably stronger than necessary). Then \...
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0answers
40 views

Elementary approximations to $\zeta(s)?$

What are the best approximations in terms of elementary functions of one real variable for: $$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s},$$ for $Re(s)>1?$ There is not an elementary function that ...
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3answers
51 views

Showing $\frac{1-\cos \left((n-1)\pi/n\right)}{1-\cos \left(\pi/n\right)} \approx 4n^2/\pi^2$

Problem Show $\frac{1-\cos \left((n-1)\pi/n\right)}{1-\cos \left(\pi/n\right)} \approx 4n^2/\pi^2$ Try I have noticed that the numerator can be approximated $$ 1-\cos \left((n-1)\pi/n\right) \...
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1answer
42 views

Find a value of $\ln 1.2$ with accuracy of $10^{-4}$

I know the formula that helps find an approximate value. In this instance it would be like $\ln 1.2 = \ln (1 + 0.2) \approx 0 + 1 \cdot 0.2 = 0.2$. But I need to find the value more precisely. I ...
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1answer
25 views

Infinitesimal approximations in triangle yielding differing results

I am trying to prove the equations of motion of a pendulum from its energy equation, and I am obtaining different results depending on which infinitesimal approximation I choose. The idea here is to ...
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1answer
49 views

How does WolframAlpha or other software get such precise values for zeta(3), etc.?

I've been looking with a friend at the values of zeta at the odd integers. WolframAlpha can give us over 100 digits in a second or two, but it seems that if you take the sum out to n, say, then you ...
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26 views

Uniform approximation of $L^2$ basis by smooth functions with bounded derivatives of all orders

Let $\mathcal{F}=\{f_i\}_{i\in\mathbb{N}}$ be an orthonormal Hilbert basis of $L^2[0,1]$. I am wondering whether it is possible to approximate the $f_i$ uniformly across $i$ in the $L^2$-norm by ...
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2answers
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Inconsistencies when approximating to the nearest whole number

Does 22.449 approximate to 22 or 23? If we see it one way $22.449≈22$ But on the other hand $22.449≈22.45≈22.5≈23$ Which one is correct?
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1answer
16 views

Multiplying a normal distribution by a log-normal distribution

I need direction to approximate the resultant probability distribution of the product of two independent distributions: $N(\mu, \sigma^2)$ and $lognormal(\mu_{N}, \sigma_{N}^2)$, where $\mu_{N}$ is ...
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Approximate $\sum\limits_{i =1}^{2t-1}\left(\prod\limits_{j=0}^{i-1}\frac{\left(q^{3t}-q^{j}\right)^{2}}{q^{i}-q^{j}}\right)$

We need to find the maximization of the following summation/product: $\frac{1}{q^{9t^{2}}}\cdot\sum\limits_{i =1}^{2t-1}\left(\prod\limits_{j=0}^{i-1}\frac{\left(q^{3t}-q^{j}\right)^{2}}{q^{i}-q^{j}}\...
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1answer
33 views

Least Squares Cubic Approximation

Hello I am trying to create a least square approximation of the 4 points (-3, 1/10),(-1, 1/2),(1, 1/2),(3, 1/10), The problem is I don't understand how the method works, every pdf I view online ...
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1answer
22 views

How to rewrite kronecker products as linear combination of matrices

Is it possible to write a matrix $U$ such that $U \otimes U \approx A\Lambda A^T$, and $U \approx AB\Lambda_1B^TA^T$? Here $U \in R^{n \times n}$, $A \in R^{n \times k}$, $B \in R^{k \times k_1}$, and ...
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0answers
34 views

Approximate dependence of polynomial roots on coefficients

I have a polynomial equation $f(x) = x^8 - x^6 - 2 k a x^3 - a^2 = 0$. I know that this cannot be solved analytically using radicals. However, I would like to approximate the dependence of the ...
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0answers
20 views

Taylor polynomials in Fourier Domain?

Which types of functions admit Taylor polynomials in the Fourier domain? Ok, this may sound cryptic. Therefore I will try to explain it more in depth. We want to approximate function $$x\to f(x) \...
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0answers
24 views

Can $x^n$ be uniformly approximated by the combination of $x^{k^2}$?

For each $n\in\mathbb{N}$ , can $x^n$ be uniformly approximated by the linear combination of $\left(x^{k^2}\right)_{k\in\mathbb{N}}$ ? In order to facilitate a solution, we might as well try to ...
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1answer
31 views

Approximate solution for high order polynomial (order 12) [closed]

I'm trying to get an approximation value of y by x from the following equation $\ x = $$\sum_{i=1}^{12} y^i$ The current suggestion is to take y=1+z and z tending to zero or y tending to 1 any ...
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37 views

Making sense of a probabilistic existence argument (set cover-like problem)

The following are excerpts from a paper by Karger and Motwani. Given this, they prove the following result: My question is: how do they derive the conclusion in the fourth sentence of their proof ...
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1answer
21 views

Strategies for approximating $e$ with tangent planes

The exercise says we have to find an expression of a tangent plane to a function that gives us a good approximation of $\alpha =(0.99e^{0.02})^8$. I defined $f(x,y) = ((1-x)e^{2x})^y$, $0 \leq x < ...
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1answer
40 views

$1435\binom{20000}{10000}*0.515^{10000}*0.485^{10000} \leq 0.01$

This inequality is part of a larger probability question and must be shown without explicitly calculating it. The only hint I have is to use Stirling's Approximation so $1 \leq \frac{n!}{\sqrt{2\pi n}...
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Kalman filter parameters

I use a 1D Kalman filter for my task. Picture with filter. I get the source signal (orange) and filter it (black). Is it possible to set the filter parameters so that it display the this result - ...
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0answers
22 views

Triangular waveform modelling of scatterplot

I have a scatter plot of Voltage vs. Time that looks like this: scatterplot of voltage vs time. The points were collected with analog to digital conversion with a data logger sampling at a constant ...
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2answers
36 views

Small angle approximations - different answers

I would like to approximate $$\frac{\cos^2{x}}{\sin(x) \tan(x)}$$ using the small angle approximations. Throughout I will use $\sin(x) \approx x$, $\tan(x) \approx x$, $\cos(x) \approx 1 - \frac{x^2}...
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0answers
28 views

Unbounded approximation ratio

Suppose that there is a specific instance of a graph for which the approximation ratio of an algorithm polynomially increases with the number of nodes of the graph, say the approximation ratio is $n^2$...
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0answers
28 views

Question related to sigmoid function

I have been going through the Deep Learning course from http://neuralnetworksanddeeplearning.com/chap1.html I have come across this approximation in the function $w_jx+b$ $\Delta output \approx \...
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1answer
68 views

Approximating a Banach space “vector- valued function” by “simple functions”

I'm trying to prove the following claim: Let $T$ be a compact (metric) space, and let $\mathcal{X}$ be a Banach space over $\mathbb{K}$. Let $f : T \longrightarrow \mathcal{X}$ be a continuous ...
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0answers
23 views

Approximating square root for long expressions

I'm currently working on a problem which asks me to calculate the potential energy of a three spring system arranged in an equilateral triangle constrained to move in the x-y plane. As a consequence ...
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2answers
33 views

Small angle approximation for $\frac{1+\sin\theta}{5+3\tan\theta-4\cos\theta}$

Please could somebody explain how the expression involving $\theta$ that $$\frac{1+\sin\theta}{5+3\tan\theta-4\cos\theta}$$ approximates to for small values of $\theta$ is $1-2\theta+4\theta^2$?