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).
1,354
questions with no upvoted or accepted answers
22
votes
0
answers
2k
views
How to find the approximate basic period or GCD of a list of numbers?
I want to tell the number which act as the best approximate basic period (or wavelenght as pointed out by Eric) of a list of real numbers: e.g for {14, 21, 35} we should obtain 7 as the basic period, ...
17
votes
0
answers
244
views
Symbolic approximation through integration by parts
This is a slightly soft question. Suppose I have an integral $f(x) =\int_a^x g(t) dt $ which cannot be expressed in terms of elementary functions. One might still be able to integrate by parts to get ...
15
votes
0
answers
743
views
Using Padé approximants for the quaternion exponential
On a lark, I wanted to know if one can use Padé approximants to compute the exponential $\exp(z)$ of a quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$. Since Mathematica has a package meant for ...
14
votes
0
answers
319
views
Why Is $\ln 23+\cfrac{1}{\color{red}{163}+\cfrac{1}{1+\cfrac{1}{\color{red}{41}}}}\approx\pi$
I know from reading that the Heegner number 163 yields the prime generating or Euler Lucky Number 41. Now apparently $\ln23<\pi$ and this can be shown without calculators. I noticed that
$$
\pi-\...
14
votes
0
answers
758
views
An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function
I think, here, I found
$$
P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n}
\sum_{z\in\{...
12
votes
0
answers
383
views
Surprising approximation of exponential series?
Consider the following expression
$$
y_j= \sum_{k=0}^{L} \frac{e^{-\sum_{i=-k}^k(k-|i|)x_{j+i}}-e^{-\sum_{i=-k}^k(k+1-|i|)x_{j+i}}}{\sum_{i=-k}^k x_{j+i}}\tag{1}
$$
for $1\leq j \leq L$. Given smooth ...
11
votes
0
answers
159
views
The fair soup division and approximating numbers
Recently The Vee confessed:
Literally every time I'm serving some soup I'm thinking of this little mathematical problem I devised.
Solving the problem, I introduced the following notion.
A number $q\...
9
votes
0
answers
2k
views
Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)
While solving problems and exercises, so far I've only used Lagrange's form of the remainder.
Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
8
votes
0
answers
361
views
Approximation of integral of gaussian function over a parallelepiped
Given a multi-dimensional gaussian function, defined by
$$f(\boldsymbol{x})=\exp\left\{-\frac{1}{2} \boldsymbol{x}^T\boldsymbol{x} \right\}=\exp\left\{-\frac{1}{2} \sum_{i=1}^nx_i^2 \right\}$$
And an ...
8
votes
0
answers
400
views
Is there a integer that makes $e^{\pi\sqrt{n}}$ closer to an integer than 163?
$e^{π\sqrt{163}}$ is almost an integer about $262537412640768744$.
Let $\delta = -\log_{10}{\left|[x] -x\right|}$, where $[x]$ means round $x$.
$\delta(e^{π\sqrt{163}}) \approx 12.125$, I searched ...
8
votes
0
answers
313
views
What is the best rational approximation of $\frac{1}{x}$
Let $x \in \mathbb{R}$, $x \notin \mathbb{Q}$ and let the function $f:\mathbb{R}\setminus \mathbb{Q} \,\times \mathbb{N}\rightarrow \mathbb{Q}$ provide the best rational approximation for $x$ where ...
8
votes
0
answers
1k
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Are there any solvers to Chance Constrained Programming Problems?
I'm trying to solve a chance constrained programming (CCP) problem
$\min_x f_0(x, \xi), \text{ such that } \mathbb{P} ( f_i(x, \xi) \ge \alpha_i ) \le \epsilon_i, \text{ where } i = 1,2,\cdots, m$
...
8
votes
0
answers
379
views
Why does this fractal approximate so many others?
The Mandelbox is a three-dimensional fractal (definition below); the recursive sequence in its definition uses folding space in place of the squaring in the definition of the Mandelbrot set. It can ...
8
votes
0
answers
153
views
Approximating intervals and squares by increasingly dense disjoint finite sets with special properties
Apologies for the length of the question.
Consider interval $I=[0,1]$. For any $n \in \mathbb{N}$ we can always find two finite sets $S_{1n} \subset I$ and $S_{2n} \subset I$ such that:
a) $S_{1n}\...
8
votes
0
answers
396
views
Show that the function is constant
Let $S^n$ be an $n$-dimentional unit sphere.
Consider $f: S^n \longrightarrow R_+$ even continuous function.
Denote
$$
F(f):=\int_0^{\infty}\int_{S^n}f(y)g\left(\frac{|xy|}{t}\right)dy\frac{dt}{t^{...
7
votes
0
answers
385
views
Need your thoughts on a faster approximation for the signum function I came up with.
I recently was kind of fascinated with the whole signum function out of the blue, and was generally really curious of the already existing approximations of it. I really don't want to lag this down ...
7
votes
0
answers
313
views
High dimensional integral of exponentials
I am attempting to marginalize a probability density function. But I got stuck on the following integral
$$
\int_{-\infty}^\infty\cdots\int_{-\infty}^\infty
\frac{\exp(\pmb x^T A\pmb z)}
{|\exp(A\pmb ...
7
votes
0
answers
132
views
Explain approximate lines in graph of this function
Sorry that this is a long question; the crux of it is that I want to know why lines appear in the graph of the function ($\varphi^\infty(x)$) I've defined.
Define $\varphi(x)$ as follows: If the ...
7
votes
0
answers
258
views
Understanding a medieval approximation
A medieval text (Maimonides's commentary to chapter 2 of Eruvin in my retranslation from the Hebrew) discusses a rectangle whose area is $5000$ square cubits. It reads in relevant part:
… that the ...
7
votes
0
answers
865
views
Response Surface Methodology using Moving Least Squares Method
I would like to obtain the response surface of a mathematical function for reliability-based design optimization (RBDO). To obtain a reliably response surface, I learned that moving least squares ...
7
votes
1
answer
568
views
Approximation of the exponential
Let $c>1,k\in\mathbb{N}$.
Let's consider two approximations of the exponential function :
The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$
and the second one is $...
7
votes
0
answers
339
views
Weighted sum of ratio of partial sum of binomial coefficients
I would like to approximate the following sum when $n \rightarrow \infty$ and $n \gg k$,
$$\sum_{x = k}^n \sum_{y > x}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m}}{\sum_{m = 0}^k {y - 1 \choose ...
6
votes
0
answers
172
views
Approximation of the sum of a series $S(t)=-\frac{2}{\pi t} \cos(\frac{\pi t}{2}) \sum_{m\ odd}^{\infty}\frac{m^2\alpha_m}{t^2-m^2}$ as $t\to +\infty$
The function S(t) has the following infinite series form:
\begin{align}
S(t) &=\frac 2\pi \int_0^{\pi/2} dx \sin(tx){\sum_{m\ odd}^{\infty} \alpha_m \cos[m(\frac \pi2-x)]}\\
&=-\frac{2}{\pi t} ...
6
votes
0
answers
189
views
How must have Liu Xin calculated $\pi$?
I know there won't be a definite answer to this question because of a lack of required historical evidence but there is so much we know of Liu Xin and of that time period. In that time, to get to an ...
6
votes
0
answers
198
views
Finding closed-form approximations of the solutions of $f(x,y)=0$
Consider
$$f(x,y)=\sum_{i=1}^n\dfrac{\sin(\omega_ix)}{\sin(\omega_iy)}r_i$$
where $n,\omega_i,r_i>0$ are known parameters.
Restrict to domains where $ \sin(\omega_iy)\neq 0$, and by symmetry $x,y&...
6
votes
0
answers
122
views
How to solve a distance problem inside of a picture?
sorry for my bad english. I have the following problem:
In the picture you can see 4 different positions. Every position is known to me (longitude, latitude with screen-x and screen-y).
Now i want ...
6
votes
0
answers
750
views
Low-rank approximation to the Graph Laplacian matrix of a regular grid.
As mentioned in the title, does anybody know any methods of efficient low-rank approximation $LL^T$ to the Graph Laplacian matrix $A$ corresponding to a square lattice? (except PCA)
6
votes
0
answers
299
views
Watson's Lemma Extension
We all know that Watson's Lemma is used to approximate the integral
$$
F\left( s \right)=\int_0^\infty {{e^{ - st}}f\left( t \right)dt}
$$
for large $s$.
However, for arbitrary $s$, are there any ...
5
votes
0
answers
206
views
A guess on two increasing rational approximations to $\frac{\pi}{4}$
When investigated the Wilf function
$$W(z)=\frac{\arctan\sqrt{2\operatorname{e}^{-z}-1}\,}{\sqrt{2\operatorname{e}^{-z}-1}\,},$$
see the preprint [1] below, I proposed a guess which reads that the ...
5
votes
0
answers
117
views
Approximation of harmonic numbers and their analytical inverse.
In the same spirit as DeTemple–Wang for a series expansion of harmonic numbers, I tried to approach the problem as
$$H_n\sim\frac 12 \log(n^2+n+a)+\gamma-\frac 1{b(n^2+n+a)+\Delta}\tag 1$$ hoping to ...
5
votes
0
answers
240
views
General form of the solutions to a PDE
I have encountered the following linear PDE in the context of reaction-diffusion processes
\begin{equation}
\partial_t (\nabla^2f) - \nabla\cdot \left( \nabla (\nabla^2 f) - m^2 \, \nabla f \right) =...
5
votes
0
answers
131
views
Approximate solution of $\Gamma(x+a)=k\, \Gamma(x+1)$ for $0 \leq x \leq a$
As the title says, I am looking for good approximate solution of the equation
$$\Gamma(x+a)=k\, \Gamma(x+1) \qquad \text{for} \qquad 0 \leq x \leq a$$ for a given real and positive value of $a$ and $...
5
votes
0
answers
99
views
Show that the sum $\sum_{k=0}^nC_n^k x^k (1-x)^{n-x}(-1)^k f(\frac{k}{n})$ tends to $0$ when $n\to \infty$
For $f\in C[0,1]$ show that for all $x\in[0,1]$ $\sum_{k=0}^nC_n^k x^k (1-x)^{n-x}(-1)^k f(\frac{k}{n})\to 0$ when $n\to \infty$
I don´t have a idea of how start with the proof, for one hand since our ...
5
votes
0
answers
244
views
Abel's summation formula and approximating an integral of Jacobi theta functions
As my previous post remains unanswered, I thought I would post a more complete form of the problem in case it would be more practical to work on/ solve. I am trying to compute the following integral
\...
5
votes
0
answers
100
views
"Solving" for $n$ the equation ${2n\brack n}=k$ (Stirling numbers of the first kind)
Interested by this question, I wondered how easily we could "solve" for $n$ the equation
$${2n\brack n}=k$$ where the left hand side is the unsigned Stirling number of the first kind.
I ...
5
votes
1
answer
109
views
How does rounding affect Fibonacci-ish sequences?
I'm curious how one might account for rounding in simple recurrence relations.
$\textbf{Explanation}$
For a specific problem, suppose we have a sequence of positive integers $a_1, a_2, a_3,...$ ...
5
votes
0
answers
208
views
Correctness of an Approximation involving the Log-Normal Distribution using Taylor (Maclaurin) Series
Question
I am looking to find a simplification of the expression below. There seems to be a mistake in my attempt (steps shown below) using the Taylor (Maclaurin) series.
Could someone please verify ...
5
votes
0
answers
97
views
Methods to approach the non-linear bvp $y''(x) - \lambda e^{y(x)} - \alpha=0$
I am searching for (semi)-analytical methods to approach the following non-linear boundary value problem
\begin{align*}
y''(x) - \lambda e^{y(x)} - \alpha&=0, \qquad \lambda,\alpha>0 \\
y(0)=y(...
5
votes
0
answers
470
views
Error in computing $\frac{e^x-1}{x}$ for $x$ near $0$.
My book says that if we want to compute $\frac{e^x-1}{x}$ for $x$ near $0$ the following algorithm is a bad idea:
z1 = (exp(x) - 1)/x
while the following one is ...
5
votes
0
answers
68
views
Approximation the sum $\sum\limits_{n=0}^\infty (c+n)^{k-1} e^{-\frac{(c+n)^k}{2*a}}$
I would like to find lower and upper bounds on the following sum
\begin{align}
\sum_{n=0}^\infty (c+n)^{k-1} e^{-\frac{(c+n)^k}{2*a}}
\end{align}
where $c,a>0$ and $k>1$.
Note that if we ...
5
votes
0
answers
469
views
What's the most accurate way to estimate a percentile from multiple partial percentiles?
There exists 3 sets of numbers. I have the 99th percentile (p99) of each set and the cardinality of the set, but not the values in the set themselves.
p99: 540, cardinality: 215
p99: 288, cardinality:...
5
votes
0
answers
123
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\}$$
where the $p_i$ are the points at which we have a vector $v_i$, how do I ...
5
votes
1
answer
111
views
How can I use Stirlings inequality to prove this inequality?
Let $p,k$ be natural numbers with $p\ge k$, show that
$$
\frac{(p-k)!}{(p+k)!} \le \frac{1}{p^{2k}} \left(\frac{e}{2} \right)^{2k^2/p}.
$$
The text where I come across this says to use Stirling's ...
5
votes
1
answer
157
views
Asymptotic behavior of many derivatives
To compute the residue of a pole of very high order $M$ at $z=0$, one needs to compute
$\frac{d^M}{dz^M} g(z)$
Suppose that $g(z)$ is a reasonable but not trivial function, that itself may depend on ...
5
votes
1
answer
141
views
Given a vector $x\in \mathbb R^n$, how can we find $z\in \mathbb Z^n$ which is closest to a scalar multiple of $x$?
I am looking for how to find integer approximations to scalar multiples of real valued vectors. This is close to the problem of finding a best rational approximation to a real number, but kind of ...
5
votes
0
answers
607
views
Taking a stationary phase approximation of a multidimensional integral
I'm looking for a way to take a stationary phase approximation of an integral of the following form:
$$ \int_{-\infty}^\infty d\vec{q} \exp\left(2 \pi i N \left(S(q_{n+1}, \vec{q}, q_1) - \vec{K}^T\...
5
votes
0
answers
203
views
Can we use a sum of residues to develop an asymptotic expansion for this unknown function?
In the course of solving a particular physical problem, I have derived a relationship between two unknown functions:
$$ f(s) = \frac{s \sinh{\frac{\pi s}{2}}}{2 \pi i \beta} \int_{-c- i \infty}^{-c+i\...
5
votes
0
answers
158
views
Using formal power series to solve nasty equations
Consider a function $f:[0,\infty)\times \mathbb R\to\mathbb R$, and suppose that given some $a>0$, I would like to solve for $x\in\mathbb R$ satisfying
\begin{align}
f(\delta, x) = a.
\end{align}
...
5
votes
0
answers
1k
views
Approximations of the incomplete elliptic integral of the second kind
For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a ...
5
votes
1
answer
588
views
How to calculate (or approximate) "trimmed" (a+b)^n?
$a^n + C_n^{1}a^{n-1}b + ... C_n^{n-1}a^{1}b^{n-1}+b^n = (a+b)^n$
But how to calculate (maybe approximately)
$a^n + C_n^{1}a^{n-1}b + ... C_n^{i}a^{n-i}b^{i} = ?$
For info, the underlying problem ...