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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38 views

$1/x$ approximation for small $x$

I am having a problem to make and approximation of $1/x$ for small $x$. If I try to use Maclaurent series I get $1/0$ as the function should be evaluated at $f(0)$. Is there a way to make this ...
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0answers
4 views

Determine $(g \circ f)'(1, 1)$ and an approximate value of $(g \circ f)(1,01; 1,01)$. Approximation by differentials and chain rule.

First time posting. Excuse me for the formatting or grammar. Question Let $f: R^2 \to R^3$ be a differential function, such that $f(1,1) = (3, 1, 2)$ and $f'(1,1): R^2 \to R^3$ is given by the ...
4
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1answer
62 views

The 9 most significant digits in Fibonacci series (Project Euler 104)

With regards to project Euler, problem 104: https://projecteuler.net/problem=104 The essence of the question here is how to keep track of the 9 most significant digits of a Fibonacci series (Keeping ...
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1answer
29 views

approximation of binomial coefficient by exponentiation

Show that inequality holds for every n, k. $$\ { n \choose k} \leq \frac{n^k}{k!}. ( \, 1-\frac {k}{2n} ) \,^{k-1} $$ I used Stirling's formula but I stucked, which is $$\ { n \choose k} = ...
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2answers
32 views

How to arrive at this approximation? [closed]

I encountered an equation: $$\frac{1}{(ja + \delta{z_{n}} - \delta{z_{n-j}})^2} + \frac{1}{(-ja + \delta{z_{n}} - \delta{z_{n+j}})^2}$$ can someone tell me how it approximates to: ...
1
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1answer
24 views

Stuck in using Stirling's approximation to show and justify an approximation of the number of permutations with and without ordering

This is a problem from my applied mathematics class where we are currently working on using Stirling's approximation which is: $ n! \sim (\frac{n}{e})^n \sqrt{2 \pi n} $ and the context of this ...
2
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2answers
41 views

Rational approximation of square roots

I'm trying to find the best way to solve for rational approximations of the square root of a number, given some pretty serious constraints on the operations I can use to calculate it. My criteria for ...
2
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0answers
50 views

Does the continued fraction for $e^{3/n}$ have a pattern?

Wikipedia has patterns for the simple continued fractions $e^{1/n},e^{2/n}$, which made me wonder whether there is one known for $e^{3/n}?$ (by pattern, I mean that the partial quotients $a_n$ can ...
1
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1answer
22 views

propagation of error from product of Taylor Series

Say I have two functions $f(x)$ and $g(x)$, both of which I will be approximating with Taylor series $T_f(x)$ and $T_g(x)$ respectively. Lets say $f(x)$ is order $O(x^{n_1})$ and $T_f(x)$ has error of ...
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0answers
19 views

Approximating number of partitions of $n$, denoted $p(n)$, by $p(n)\ge e^{c\sqrt n}$

I was to show that $p(n)$, the number of partition of a positive integer $n$, satisfy: $p(n)\ge \max_{1\le k\le n}{{n-1\choose k-1}\over k!}$, which was obvious because every unordered collection of ...
1
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1answer
36 views

Linear Approximation for functions with first derivative as $0$

Linear approximation around a point through Taylor series requires the first order derivative to be non-zero unless you want to get only the value at that point. However this is only true when you are ...
2
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2answers
42 views

Approximating $f(1)$ given $f(0)=2$ and $\frac{1}{2}x^2≤f'(x)≤x^2$?

This is an exercise using the mean value theorem: Approximate $f(1)$ given $f(0)=2$ and $\frac{1}{2}x^2≤f'(x)≤x^2$ for $x≥0$. I've found (using MVT): $$\frac{1}{2}x^2+2≤f(1)≤x^2+2$$ and I can ...
3
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2answers
25 views

Minimal Subset that sums up to

Let $X \subsetneq \mathbb{N}$ be a finite set, and $c \in \mathbb{N}$ we are looking for a subset $$ Y \subseteq X $$ such that $\sum_{y \in Y} y \geq c$. Assuming a subset that satisfies the ...
2
votes
3answers
107 views

Approximating $\exp(\sin(x))$ with polynomials

I want to find explicit formula for the sequence $f_n$ of polynomials which uniformly convergent to $\exp(\sin(x))$ on $[0,2014]$. Taylor's expansion is terrible for this function so i think that ...
12
votes
1answer
219 views

Where am I violating the rules?

Being fascinated by the approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ proposed, more than 1400 years ago by Mahabhaskariya of Bhaskara I (a ...
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0answers
26 views

Help with basic arithmetic involving Big Oh

I'm trying to determine the resulting "Big Oh" when arithmetic operators are applied between two different functions, but I'm a bit unsure after looking at even the basic operators shown on wikipedia ...
0
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0answers
27 views

Upper-bounding $\mathbb{E}\left[ \frac{\tilde{p}}{p} a \right]$ with $D(p, \tilde{p})+ \mathbb{E}[a]$?

Define two slightly different probability distributions: $$ p_k = \frac{ \exp \left[ - \eta L_k \right] }{ \sum\limits_{i=1}^K \exp \left[ - \eta L_{i} \right] }, \quad \tilde{p}_k = \frac{ \exp ...
0
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0answers
18 views

What “approximation factor” means for approximation algorithms?

The paper I read starts with the following text: One of the most appealing open problems in the graph streaming area (see Problem 60 in [1]) is to close the gap between the approximation factor ...
3
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1answer
51 views

Deriving Stirling's approximation formula via the definition of the Gamma function

In my asymptotic analysis and combinatorics class I was asked this question: We first remember the definition f the Gamma function $ \Gamma(n+1) = n! = \int_{0}^{\infty} t^{n} e^{-t} dt $ and ...
8
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2answers
842 views

Sum of inverse of Fibonacci numbers

If $F(n)$ is the nth Fibonacci number, How can I prove that: $$\sum_{i=1}^{\infty} \frac{1}{F(i)}\approx 3.36\, .$$
16
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2answers
202 views

Elegantly Proving that $~\sqrt[5]{12}~-~\sqrt[12]5~>~\frac12$

$\qquad$ How could we prove, without the aid of a calculator, that $~\sqrt[5]{12}~-~\sqrt[12]5~>~\dfrac12$ ? I have stumbled upon this mildly interesting numerical coincidence by accident, ...
0
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1answer
48 views

Limit of regular polygons approaching pi - earliest proofs

Archimedes used areas of regular polygons to approximate pi. He calculated both inner and outer polygons and realized that more sides yielded closer results to each other. There's surviving proof that ...
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0answers
23 views

Constraint on curvature of polynomial approximation for camera calibration

OpenCv uses the following polynomial for estimating lens distortion. More information can be found here. $\frac{1+k_1r^2+k_2r^4+k_3r^6}{1+k_4r^2+k_5r^4+k_6r^6}$ There is a lot more to the ...
2
votes
3answers
49 views

How to show that $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$

How can we show that: $2 ^\binom{N}{2} \sim \exp(\frac{N^2}{2}\ln(2))$ for large N.
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0answers
33 views

$\int_{-\pi/2}^{\pi/2} |\vec{v}-\vec{V}|(\vec{v}-\vec{V}) d t$, with $\vec{v}=(\vec{i} \cos t +\vec{j} \sin t) (\sqrt{1-p^2 \cos^2 t}+p \sin t)$

I need to solve this integral: $$\int_{-\pi/2}^{\pi/2} |\vec{v}-\vec{V}|(\vec{v}-\vec{V}) d t$$ $$\vec{v}=(\vec{i} \cos t +\vec{j} \sin t) \sqrt{1-p^2 \cos 2 t+2 p \sin t \sqrt{1-p^2 \cos^2 t}}=$$ ...
1
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2answers
64 views

How to evaluate $\int^\pi_0 cos((n+1)x)(cosx)^jdx$ for $j=0,1,…,n$

I have spent quite a time trying to calculate the integral below: $\int^\pi_0 cos((n+1)x)(cosx)^jdx$ for $j=0,1,...,n$ I applied many trigonometric idenetities, but after some iterations they got ...
0
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1answer
45 views

Radial Basis Function RBF Gaussian based Interpolation

Based on short description below (an image), how do I find the highlighted f function value? I understand that it is a value associated with the vertex, sorry I am not a good math student to ...
0
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1answer
15 views

Solving a low rank symmetric system

I'm working on a problem where I need to solve a large set of systems of equations, where each has a structure that looks like: $\left( M^\top_{n\times p}\Sigma_{p\times p} M_{p\times n} + ...
1
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0answers
26 views

Quality of $E(f(X))\approx f(EX)+\frac 1 2 f''(EX)\sigma_X^2$ approximation

For convex $f$, we have Jensen's lower bound $Ef(X)\ge f(EX)$. What conditions do we need to put on $f,X$ so that the second order expansion in the title would be an upper/lower bound/good ...
1
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1answer
23 views

Approximation of continuous functions

Let $f$ $\in$ C([0,1]), $f(0)=0$ and $\epsilon > 0$. Prove there exists a polynomial $p$ such that $p(0)=f(0)=0$, $p´(0)=0$ and $||p-f|| < \epsilon$ . The norm is sup-norm I Know that by ...
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5answers
228 views

Is $22/7$ an often used approximation for $\pi$?

It is $\pi$-day and the internet is full of stories about $\pi$. One story mentions that "an approximation -- $22/7$ -- is used in many calculations." I have never actually used $22/7$ as an ...
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0answers
13 views

Approximate 2D convolutions as a sum of separable convolutions

Just like this 3D question, but for 2D. I have a set of 2D convolution kernels, not separable. Is there a good methods to approximate them as a sum of a relatively small number of separable arrays? ...
1
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0answers
84 views

A double inequality for $\frac{\pi}{2}$

Approximating $\frac{\pi}{2}$ from above Since $$\left(\frac{\pi}{2}\right)^9\approx 58.220897$$ the root $$58^\frac{1}{9}\approx 1.5701$$ is not far from $$\frac{\pi}{2}\approx 1.570796$$ This ...
3
votes
2answers
39 views

Validity of an Approximation

I am considering the approximation of the following integral: $$\int_{-W/2}^{W/2}\exp[-p( x + iq)^2 ] dx \approx \int_{-\infty}^{\infty}\exp[-p( x + iq)^2 ] dx = \sqrt{\frac{\pi}{p}}$$ For large ...
2
votes
2answers
64 views

Montecarlo estimate of a integrand from 0 to $\infty$

I have a question about monte carlo estimation of integrals. Suppose I am told to estimate using monte carlo, the integral: $$f(y) = \int_{0}^{y}\frac{4}{1+x^{2}}dx$$ I want to estimate $f(\infty)$. I ...
2
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1answer
21 views

Confusion about principles of decimal approximation

I ran into this paragraph in the introductory chapter on Taylor series of Morris Kline's "Calculus: An Intuitive And Physical Approach": "Thus, if the value of $\sin(x)$ for a particular value of $x$ ...
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2answers
24 views

approximating functions via a piecewise combination of linear and constant functions

I am curious if anyone has encountered any literature on approximating functions via a piecewise combination of linear and constant functions. I have seen a couple of papers which use piecewise ...
1
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2answers
58 views

Approximating statistics for huge dataset

I'm investigating users accounts statistics for Vkontakte social network. There are $N\approx2 \cdot 10^8$ accounts that have different metrics along them – boolean, discrete and continuous. I found ...
0
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1answer
21 views

Alternate (approximate) form for Hypergeometric function 1F1(0.5, 1.5, -x)

I have the following Hypergeometric function of the first kind: $_{1}F_1(\frac{1}{2}, \frac{3}{2}, -x)$ where $x$ is not negative. This function can also be written as the following series: ...
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2answers
33 views

Newton Raphson - Reciprocal Square Root Convergence

I'm attempting to use Newton Raphson method to calculate the square root of fixed point numbers. The mathematics I understand - and, using this question I easily managed the normal; $x_{n+1} = ...
1
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0answers
20 views

Approximation of a hypergeometirc-like distribution

Fix $0<\varepsilon<1$. For $m\in\mathbb{N}$, let $$c_m=\max\left\{\frac{{m-1\choose s-1}{m\choose k-s}}{{2m\choose k}}:\;k=1,2,\dots,2m(1-\varepsilon)\;\mbox{and}\;s=1,2,\dots,k\right\}$$ Prove ...
2
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1answer
42 views

Explain why $\exp(-7 \log_{10} n)$ approximates $1/n^3$ so well

I was graphing a few functions, and discovered that the graphs of $\exp(-7 \log_{10} n)$ approximates $1/n^3$ are almost the same. Can anyone explain why this is so? Is there a general result for this ...
1
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0answers
28 views

Intergral approximation?

For the integral: $$I=\int^{t/2}_{-t/2} e^{i(\omega_1-\omega_2)t'}dt'$$ What would be the lower limit on $t$ for the approximation: $$I\approx 2\pi \delta(\omega_1-\omega_2)$$ to hold? I would guess ...
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0answers
24 views

Approximate product by product

Let $\mathbb A _n = \{a_1, \ldots, a_n\} \subset \mathbb R_+$. For given $n, K \in \mathbb N$ can we bound from above the following: $$\left|\prod _{k=1}^K x_k - \prod _{\ell=1}^Lb_\ell \right| \leq ...
1
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2answers
28 views

Given a sequence for the cubic root of a number $Y=\sqrt[3]{X}$

Given a sequence for the cubic root of a number $Y=\sqrt[3]{X}$, if $a>0$, show that Y always lies between a and $X/a^2$ (if $a<Y$, then $X/a^2 > Y$, etc) I'm thinking use Newton's Method ...
0
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0answers
63 views

Why $(\alpha\frac{e}{t})^t e^{-\alpha}$ is an approximation for $P(X > t\alpha)$ for Poisson distribution $\frac{\alpha^ke^{k}}{k!}$?

I am reading Section 3.4 of Algorithms, 4th Edition. Page 466 is a proof of the following proposition: In a separate-chaining hash table with $M$ lists and $N$ keys, the probability (under ...
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0answers
17 views

Gradient Approximation Methods

I am trying to find a way to approximate the gradient of a multivariate function. This relates to gradient-based optimization problems. My assumptions are as follows: Implicit function (FEA) Very ...
2
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2answers
30 views

Technical challenge: Limit of von-Mises distribution approximates normal. How to take the limit?

Background: In psychophysics or the study of ant navigation it's important to represent random variables on a circle. The most popular distribution for doing so is the von-Mises distribution (the ...
2
votes
4answers
50 views

Finding the closest distance between a point a curve for multiple Points (n>1000)

I am trying to compute the closest distance between a point a curve (polynom of 2rd degree) : $$f(x)=a*x^2+b*x+c$$ $a,b,c$ are established. So if we denote that D(x) is an distance from $(x,f(x))$ ...
0
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0answers
19 views

Show that the Bernstein operator is not a projection

I'm currently trying to show that the Bernstein operator is not a projection, but I can't find a good counter-example to show that it's not a projection. I was thinking about starting with some ...