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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2
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2answers
65 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
votes
1answer
23 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 $x$ ...
0
votes
2answers
25 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
vote
2answers
60 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
votes
1answer
23 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: $\sum_{...
0
votes
2answers
40 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
vote
0answers
21 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
votes
1answer
45 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
vote
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 (...
0
votes
0answers
25 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 f(...
1
vote
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
votes
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 ...
0
votes
0answers
21 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
votes
2answers
34 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
60 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
votes
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 ...
0
votes
1answer
39 views

What is mean value of Jacobian in finite difference method?

I was reading a paper, where the author gives a method for solving a differential equations system using finite difference method. I am trying to simulate this result. The problem I am facing is that ...
2
votes
3answers
75 views

Curve-fitting using circles

I'm working for a firm, who can only use straight lines and (parts of) circles. Now I would like to do the following: imagine a square of size $5\times5$. I would like to expand it with $2$ in the $x$...
3
votes
0answers
65 views

Calculate integral curve through a discretely sampled vector field

given a finite sample of a smooth vector field, i.e. a set $$S=\{(p_i,v_i)\mid i\in \{1,\dots, n\}, p_i,v_i\in\mathbb{R}^2\}$$ where the $p_i$ are the points at which we have a vector $v_i$, how do I ...
0
votes
2answers
33 views

Does this approximation always overestimate?

So, I'm working on a method that calculates $\cos\theta : \theta \in [0, \frac{\pi}{2})$ using geometric methods, and I'm trying to work out the error of the approximation. I have a wonderful estimate ...
2
votes
1answer
41 views

Could anyone explain why this is a general case of Weierstrass Approximation?

Suppose $X_1, X_2 ...$ are independent Bernoulli random variables. with probability $p$ and $1-p$. Let $\bar{X}_n = \frac{1}{n} \sum\limits_{i=1}^nX_i$. If $U \in C^0([0,1],\mathbb{R})$, then $E(U(\...
1
vote
1answer
54 views

Error bound in the sum of chords approximation to arc length

We are currently covering arc length in the calculus class I'm teaching, and since most of the integrals involved are impossible to solve analytically, I'd like to have my students do some ...
1
vote
1answer
50 views

Approximation for f''(x) using forward, centered, or backward difference

I have a question with respect to deriving an approximation for $f^{''}(x)$ using the forward, backward, or centered difference. It goes as follows: "Using the method of your choice, construct an ...
2
votes
1answer
24 views

Approximating $\frac{a+\delta a}{b + \delta b}$

I have two quantities $a(t)$ and $b(t)$ that have a constant mean ($a$ and $b$) and some small fluctuating noise part with vanishing mean $\delta a(t)$ and $\delta b(t)$. I'll write them as $a(t) = a +...
1
vote
2answers
50 views

Obtaining exact decimals in bisection method

While studying the bisection for the approximation of roots of non-linear equations I was given the following bound for the error: $|x_n-s| \leq \frac{(b-a)}{2^{n+1}}$ where $x_n$ is the n-th ...
0
votes
3answers
46 views

Approximated second derivative

Approximated second derivative of $y(0)''$ function $y(x)$ at x = 0 difference quotient, using the values ​​of $y(x)$ at the sites of the three-point template $ {x}_{1} = \frac {-4h} { 5}, {x} _ {2} = ...
0
votes
0answers
21 views

Approximated second derivative

Approximated second derivative of $y(0)''$ function $y(x)$ at x = 0 difference quotient, using the values ​​of $y(x)$ at the sites of the three-point template $ {x}_{1} = \frac {-4h} { 5}, {x} _ {2} = ...
1
vote
0answers
29 views

Unbounded approximation algorithm for minimum vertex cover

Suppose we find the minimum vertex cover of a graph by repeatedly choosing the vertex with the highest degree and delete all edges incident on that vertex, until there are no edges left. How can one ...
3
votes
0answers
70 views

Infinite products of even analytic functions - highly accurate approximation

I discovered a way to evaluate infinite products of even analytic functions with high accuracy. $$ \prod_{k=1}^{\infty} f(k^2) \approx \prod_{k=1}^{\infty} \left(1-\frac{A_1}{k^2}+\frac{A_2}{k^4}-\...
1
vote
0answers
20 views

Step in using Stirling's formula to get an upper bound

I'm having trouble seeing why the following holds. Given the conditions that $N=\binom{n}{2}$ and $m$ is a function of $n$ such that $N-m \to \infty$ as $n \to \infty$, why is it that $$(1+o(1)) \...
0
votes
1answer
22 views

Reverse economization of Chebyshev series

Suppose I have some function which is represented as converging series of Chebyshev polynomials of first kind in $[-1;1]$: $$ f(x)=\sum\limits_{n=1}^\infty a_n T_{2n}(x) $$ I need to transform this ...
0
votes
1answer
33 views

Density of polynomials in $\cos t$ in $\mathcal{C}^0([0,\pi],\mathbb{R})$

I'm looking at the Fourier cosine transform, and as a preliminary I have to show that every $f$ in $\mathcal{C}^0([0,\pi],\mathbb R)$ is the uniform limit of a sequence of functions of the form $t\to ...
0
votes
1answer
74 views

Why is $\tan 3 + \pi$ a near-integer? [closed]

When playing with my calculator I found that $$\tan 3 + \pi \approx 3$$ Is there a mathematical reason for this?
4
votes
3answers
174 views

Something similar to the bizarre Koide formula?

In 1981, Koide found the empirical relation, $$\frac{m_e+m_\mu+m_\tau}{\big(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau}\big)^2} = 0.666659\dots\approx \frac{2}{3}\tag1$$ where $m$ are the masses of the ...
0
votes
0answers
27 views

Find the linearization of the function at 0

The problem asks: Find the linearization of $f(x)= \sqrt{a+bx} $ at $0$ To get all parts of $L(x) \approx f(c) - f'(c)(x-c)$ I've done: $$f(0) = \sqrt{a}$$ $$f'(0) = {b\over 2\sqrt{a}} $$ Now: $$...
0
votes
1answer
42 views

Find alternative shortest paths given extra properties

This is a follow-up question for a question I asked at here. The problem is mapped to a graph with say non-negative weights on edges (no preference if it can be directed or not). However, along with a ...
0
votes
0answers
28 views

Interpolating sequence problem.

Below is a question which I cannot quite figure out. Any tips would help appreciated!I've been working on this for about a month. "Let $(z_j)$ be a sequence of distinct points in a domain $D$ that ...
0
votes
1answer
19 views

Taylor expansion of this electric field

I'm trying to determine what happens when R>>z for the below equation $\frac{z\sigma}{2\varepsilon }\left ( \frac{1}{z}-\frac{1}{\sqrt{R^{2}+z^{2}}} \right )$ Like most books, which is a great ...
0
votes
1answer
12 views

Minimal error given when making an approximation of $f(x)$ by sines and cosines

I am studying by myself Fourier analysis and have encountered the following problem: We are trying to approximate a function by a finite sum of sines and cosines with general constant coeficients: $$...
7
votes
3answers
71 views

Approximating $x=\sqrt{2}+1$

Suppose $y>1$ is some approximation to $x=\sqrt{2}+1$. Give a brief reason (not a proof) why one should expect $(1/y)+2$ to be a closer approximation to $x$ than $y$ is. After testing this out ...
1
vote
1answer
26 views

Show a bound on the error

Suppose $g \in C[a,b]$, and $M(b-a) < 1$, and let $y$, $y_N$ $\in C[a,b]$ be the unique solutions of the equations, $y = g + Ky$ and $y_N = g + K_N y_N$. $y_N$ has been previously defined as an ...
0
votes
3answers
48 views

A simple approximation algorithm?

I'm not sure if this method works perfect, but I have found it to work in approximating things easily, that is, you need no more than simple algebra to understand this method. Suppose you are trying ...
1
vote
0answers
42 views

How can I prove two empirically derived graphs are topologically equivalent?

I have two graphs that I've derived from an empirical data set and I suspect that they're topologically equivalent. It seems much easier to show that these graphs are not equivalent than to show that ...
0
votes
0answers
39 views

approximation of a trigonometric sum

I have a trigonometric sum as below $$\frac{1}{N^2}\sum_{r=0}^{N-1}\frac{\sin^2(\pi e)}{\sin^2(\frac{\pi(r-n+e))}{N})}\frac{\cos^2(\frac{\pi(Ne-e-r+n)}{N})}{\cos^2(\frac{\pi(Ne-e)}{N})}$$ and I want ...
9
votes
3answers
591 views

A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
0
votes
2answers
47 views

About Taylor series

Suppose $f(0) = 0, f'(0) = 2, f''(0) = −1$ and $|f''' (x)| ≤ 0.024$ for $0 ≤ x ≤ 2$. Estimate $f(1)$ to $4$ significant figures by using a Taylor polynomial. Compute a good bound for the absolute ...
13
votes
2answers
427 views

Is $\pi^k$ any closer to its nearest integer than expected?

Particular questions such as Why is $\pi$ so close to $3$? or Why is $\pi^2$ so close to $10$? may be regarded as the first two cases of the question sequence Why is $\pi^k$ so close to its nearest ...
1
vote
1answer
71 views

Another way to calculate $\cos(x)$ and $\cosh (x)$

I usually don't see any expressions for approximate calculation of trigonometric functions aside from their Taylor series. But Taylor series work better for small angles, so in general they are not ...
6
votes
1answer
191 views

Is there a series to show $22\pi^4>2143\,$?

This extends this post. I. For $\pi^3$: $$\pi^6-31^2 =\sum_{k=0}^\infty\left(-\frac{63}{(2k+2)^6}+\frac{31^2}{(2k+3)^6}\right) =\sum_{k=0}^\infty P_1(k)\tag1$$ As pointed out by J. Lafont, when ...