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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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20 views

Chebyshev approximation for large interval

In the context of neural networks and cryptography, I would like to approximate some activation functions. However, I need to approximate them into polynomial forms for my purpose. It seems that ...
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1answer
25 views

Determine the (2,2) Pade approximation to sqrt cube to x+8 and estimate its error

here is the full question its about pade approximation and I serch the internet and couldn't find the answer anyone can help it will be great
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Unique Rational Approximation With “Small” Denominator

Suppose we have some irrational $x > 0$ and some $\epsilon > 0$. I want to show that there is at most one rational approximation $\frac{a}{b}$ such that both $| x - \frac{a}{b}| < \epsilon$ ...
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3answers
25 views

Reciprocal using Newton Raphson

Say you want to calculate 1/R using Newton-Rapshon method. Then we let, $$f(x) = 1/x - R$$ This means the root of the this function is at $f(1/R)$. So to find $1/R$, you can find the root of this ...
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2answers
72 views

How to show that $\left| \Gamma \left(x + iy \right) \right|^{2} \approx (\pi y^{(2x - 1)}) /(\cosh(\pi y))$?

I found the approximation: $$\left| \Gamma \left(x + iy \right) \right|^{2} \approx \frac{\pi y^{(2x - 1)}}{\cosh(\pi y)} $$ for $y \gt 2$, within an answer for another question, but I could not ...
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18 views

How do you find two curves that enclose a set of data points?

I'd assume you'd use least squares here to first get the best fit line for the data, but I need to specifically minimize the vertical distance between the two curves. Originally, I was thinking I'd ...
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1answer
83 views

Nice result that I can't prove: $\int_{-2}^{2} \tan^{-1} \bigg( \exp(-x²\text{erf}(x)) \bigg) \;dx=\pi$

I'm always trying to find the integral representation of $\pi$ using some interesting special function, at this time I have got the below representation $$I=\int_{-2}^{2} \tan^{-1} \bigg( \exp(-x^2\...
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Approximate a ($y²-x²=1$) hyperbola with line segments and elliptic ($a(x-x_0)²+b(y-y_0)²=1$) arcs

On some IT graphic systems, you have tools which draw line segments and circle or elipse arcs, but which do not draw parabolas or hyperbolas, and in many cases, those system keep track of graphical ...
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1answer
43 views

Computing approximation of cos function

i have an assignement in which the whole point was to approximate $\cos$ function using 2 methods : Using series expansion using a more algebric method with a linear system The teacher also defined ...
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1answer
24 views

Functions that Tend To Non-Smooth Functions as Some Parameter Tends to Infinity

I recently saw a post in which the query was about a function that tends to the Dirac delta function as a parameter in it tends to infinity. The function chosen was $${(1+\cos x)^n\over C}$$ as $n\to\...
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Approximation of one function with smooth functions

There is a continuous but non-smooth function at $x=0$: $$ f(x)=\left[\frac{2}{1+e^{-2x}}-1 \right]_{+},$$ where $[u]_+\equiv \begin{cases} u,\quad u \geqslant 0\\ 0, \quad u<0 \end{cases} $ So, $...
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3answers
179 views

An Inconsistency in Numerical Approximation

Consider the expression $$ 10^5 - \frac{10^{10}}{1+10^5}. $$ Using the elementary properties of fractions we can evaluate the expression as $$ 10^5 - \frac{10^{10}}{1+10^5} = \frac{10^5 + 10^{10} ...
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0answers
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increment function for the implicit midpoint rule

Assume that $h \rightarrow \phi(s; x; h; f ) $ is continuously diffrentiable in a neighborhood of 0. Then, $\phi$ is consistent if and only if there is a continuous increment function $h \rightarrow \...
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1answer
46 views

Can we refine this asymptotic for Laguerre polynomials?

I just found an interesting and useful limit for Laguerre polynomials: $$\lim_{n \to \infty} L_n \left( \frac{2r}{n+1/2} \right)=J_0(2 \sqrt{2r})$$ I'm using specifically this form of the argument ...
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1answer
45 views

Approximate a solution for a single variable exponential equation

Can anyone please help me fined (if it is possible) a closed-form solution or an approximation for the solution for the following equation (x is the only variable): $$\frac{((a-1)b^{x+2}-(b-1)a^{x+2}+...
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1answer
38 views

Root of an quadratic equation

I have the following quadratic equation : $m^2 + m(p-1/l) - (\Omega_x^2 + \Omega_y^2)=0$ I would like to get the solution in terms of $\Omega_x, \Omega_y$ with some approximations i.e. neglecting $(...
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0answers
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Proof of the variance for a normal approximation

For large sample sizes the Wilcoxon signed rank test can be approximated by a Normal distribution: $X\sim N(\frac{n(n+1)}{4},\frac{n(n+1)(2n+1)}{24})$. And for the Wilcoxon rank-sum test: $X\sim N(\...
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1answer
46 views

Continued fractions approximation using golden ratio

Hello today my friend helped me with my problem, but he did not give me any additional informations why it works like that. Let's suppose that I need to get ln(n) using continued fractions. He told ...
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3answers
39 views

Approximation of square root of sum of two squared terms

I have the following equation $\sqrt{(x_a-x_n)^2+(y_a-y_n)^2}$. I want to get rid of square-root and find an approximation which contains only $x_a,x_n,y_a,y_n$ (there should not be any other non-...
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1answer
70 views

Aryabhata's sine approximation : Conversion for use with interval of $[-\pi,\pi]$

There's this sine approximation (mentioned in title) which works over the interval $[0, \pi]$: $$ \sin x \approx \frac{16x(\pi-x)} {5\pi^2-4x(\pi-x)} $$ With little changes it can be put work ...
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0answers
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Clarification for a proof about Lipschitz approximation in $W^{1, p}(\mathbb{R}^n)$

I was reading a proof about the Lipschitz approximation of functions $u\in W^{1, p}(\mathbb{R}^n)$. There the author defines a set $$E_{\lambda}=\{x\in\mathbb{R}^n:M|\nabla u|(x)\leq \lambda\}, \quad ...
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0answers
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Approximation to $1.05^{1.02}$ with Taylor's Theorem $T_2f(a;(x,y))$ in the point $a = (1,1)$

Let $f : (0, \infty)^2 \to \mathbb{R}$ with $f(x,y) := x^y$. How can one find out an approximation to $1.05^{1.02}$ with Taylor's Theorem, i.e. $T_2f(a;(x,y))$ in the point $a = (1,1)$. And how can ...
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0answers
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Piecewise linear robot motion (with obstacles)

The problem is best described with the image below. An robot (the diamond shape) is only allowed to move in a piecewise linear fashion, with each halt being a point of a curve (the red curve). The ...
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1answer
40 views

How to show $\frac{\Gamma((n-1)/2)}{\Gamma(n/2)} \approx \frac{\sqrt{2}}{\sqrt{n-2}}$

Show $\frac{\Gamma((n-1)/2)}{\Gamma(n/2)} \approx \frac{\sqrt{2}}{\sqrt{n-2}}$ Try Using the facts: $(1 + \alpha/m)^m = e^\alpha ( 1+ r_m)$, where $\lim_{m \to \infty} \sqrt{m}r_m = 0$ $\Gamma(n+1) ...
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3answers
35 views

How to approximate the answer of a polynomial of degree 4?

I want to solve this polynomial analytically. I know the useful answer is between 0 and 1. Is there any way I can write the answer based on a, b, and c? $$ 6\cdot a \cdot x^4 + 2 \cdot b \cdot x^3-b \...
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0answers
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Bounds for the error of this approximation to the Bessel function

I found a nice explicit approximation to the Bessel function today, using the integral: $$J_0(x)=\frac{2}{\pi} \int_0^1 \frac{\cos x u}{\sqrt{1-u^2}}du$$ With Chebyshev-Gauss quadrature we can see ...
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1answer
20 views

farthest point in metric space [closed]

For all $x,y\in \Bbb{R},$ defined $d(x,y)=1.$ So $(\Bbb{R},d)$ is a metric space. Let $W=\{\frac{1}{n}\}\subseteq \Bbb{R}.$ If for $x\in \Bbb{R},~~ F(x)=\{w_0\in W, ~d(x,w_0)=\sup_{w\in W}d(x,w)\}.$ ...
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0answers
31 views

Approximation of Cantor function by piecewise constant function in $L^1$

Let $c(x)$ be Cantor function. How can we prove that constant $\frac{1}{2}$ gives the best approximation in $L^1$ metric? Let $ h(x)= \begin{cases} \frac14 \quad\text{for}\quad x\in[0,\frac13]\\ \...
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72 views

$0<|\sqrt a-\sqrt[3]b|<\epsilon$ for $a,b\in\Bbb Z_+$

I'm trying to solve the following problem: Given $\epsilon>0$, are there positive integers $a,b$ such that $0<|\sqrt a-\sqrt[3]b|<\epsilon$ ? My solution: given $n\in\Bbb N$, $$|\sqrt{n^2}...
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0answers
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Discrete norm approximation of the $L^p$ norm for spline functions

In Theorem 5.2 in Lynche (1988) "A data reduction strategy for splines with applications to the approximation of functions and data", a bound for the difference between the $(l_2,t)$ and $L^2$ norms ...
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3answers
198 views

Evaluating the integral $\int_0^1 \frac{\cos bx}{\sqrt{x^2+s^2} }dx$

I'd really love to evaluate this integral exactly in terms of known functions, because for large $b$ it becomes a pain numerically. $$I(b,s)=\int_0^1 \frac{\cos bx}{\sqrt{x^2+s^2} }dx$$ Didn't get ...
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0answers
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What does it mean for a function to be semi-monotonic?

I mostly understand monotonic functions as described by wikipedia. However, I do not understand what it means for a function to be semi-monotonic as described in the java math class. This page helped ...
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Using Picard-Lindelof to find a solution to $y'(t,y(t))=t+\sin(y(t))$ where $y(2)=1.$

Consider the initial value problem $y'(t,y(t))=t+sin(y(t))$ with initial condition $y(2)=1$. Find the largest interval $\mathcal{I}\subset \mathbb{R}$ containing $t_0=2$ such that the problem has a ...
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0answers
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Family of graphs that have approximation ratio = 2

My question today is about the approximation algorithms. Well, for Approx-Vertex-Cover problems , we know we can get ratio of 2 just by picking an edge and taking 2 endpoints of the same and ...
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1answer
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How to understand the order of convergence $\|x_{k+1} - x\| \le C \|x_k - x\|^p$ (Convergence of a power function form)?

By definition, a sequence $x_k \in \mathbb{R}, k \in \mathbb{N}$ converges with order $p \in [1,\infty)$ to $x := \lim_{k\to\infty} x_k$ if \begin{align} \exists C \in [0,\infty): \forall k \in \...
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3answers
49 views

Approximating $\cos(47^{\circ})$

Given that $\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$, what would $\cos(47^{\circ})$ be. Using differential approximation, I get $\cos(47^{\circ})$ is about $\cos\left( \frac{45\pi}{180}\right)-2\sin\...
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1answer
33 views

Choose X s.t. $\Pr[X \geq k] = 1/k$

I am trying to implement an algorithm to approximate the weight of a Minimum Spanning Tree, as described here (p. 8-10). In the pseudo-code ApproxConnectedComps(G, s), p. 9, there is a passage where ...
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1answer
22 views

Approximation of a series expansion

In the approximation of a series expansion for the parameter $\delta$, I take only the first two terms of $$\delta=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}z^n}{n^{\frac{3}{2}}}$$ and have $$\delta\...
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2answers
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Approximating a random variable versus approximating probability statements about a random variable?

After formally stating the central limit theorem my statistics textbook says this: Interpretation: Probability statements about the sample mean $\overline{X}_n$ can be approximated using a Normal ...
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1answer
63 views

How to approximate exponential function or under approximate this function

I am trying to find the minimum of this function analytically: $$ G(x)=\frac{c(a.x^3+b)}{x}+2(a.x^3+b)\sum_{m=1}^{\infty}\frac{e^{-\beta^2m^2(T-\frac{c}{x})}-e^{-\beta^2m^2T}}{\beta^2m^2} $$ where $0&...
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2answers
50 views

Manual approximation of $\operatorname{sech}(0.7)$

In the archive of a midterm exam collection there are some question like the one above. How can we approximate expressions like $$\operatorname{sech}(0.7)$$ without a calculator? Thanks in ...
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3answers
46 views

Harmonic series sum approximation

Question: $ 1 + 1/2 + 1/3+\dots+ 1/n > 4.$ Find the range of smallest value of n. Answer: $n$ lies in $(20,60)$. Source: KVPY 2017. To the best of my knowledge I find this series to ...
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1answer
34 views

Can I find an infinitely differentiable function of of bounded moments closest to triangle wave?

Based on this question regarding existance of closest function in Schwarz class, where answer was negative. What if we add a new constraint. Not only infinitely differentiable compact support but with ...
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2answers
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Which function in the Schwarz class of functions is “closest” to triangle wave in $L^2$ sense?

Would it be possible to calculate which function in the Schwarz class of infinitely differentiable functions with compact support is closest to triangle wave? Let us measure closeness as $$<f-g,f-...
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1answer
13 views

Show that if $f_i = O(g), i = 1,…,n$ as $x \to 0$ and..

Show that if $f_i = O(g), i = 1,...,n \space as \space x \to 0 \space and \space |g_i| \leq |g| i = 1,...,n,$ then $$\sum^n_{i=1}{a_if_i}=O(g), as \space x \to 0,$$ where $a_i, i=1,...,n,$ are ...
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2answers
77 views

Picking the correct Ansatz for valid solutions in Asymptotic Methods

I am trying to find the solution to the following equation, $\epsilon x^3 -x^2 +x-\epsilon^{\frac{1}{2}}=0$, for the first two non-zero solutions as $\epsilon \to 0^+$. I have used the principal of ...
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0answers
25 views

Probability of at least two of n independent events occurring subject to some conditions

Given a set of independent Bernoulli random variables $\{x_1, \dots x_n\}$, let $p = \sum_{0<i\leq n}\Pr[x_i = 1]$ and $X=\sum_{0<i\leq n} x_i$. We know that for any $i$, we have $\Pr[x_i = 1]\...
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4answers
71 views

Convergence to $\sqrt{2}$

It is a very good way to approximate $\sqrt{2}$ using the following; Let $D_{k}$ and $N_{k}$ be the denominator and the numerator of the $k$th term, respectively. Let $D_1=2$ and $N_1=3$, and for $k\...
3
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1answer
29 views

Approximating the first moment of h(x) where $x$~Lognomal($\mu, \sigma$)

What is the best way to approximate $E(h(X))$, where $X$ ~ Lognomal($\mu, \sigma$). So far, I can think of Monte Carlo Methods and Gaussian Hermite quadrature as below: \begin{align} E(h(X)) &= ...
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0answers
21 views

Approximation of trig functions

I need help to understand how following terms can be approximated: $\cos(a \sin\omega_mt)≈1$ $\sin(a \sin\omega_mt)≈a \sin\omega_mt$ if $0 < a < 0.4$