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).

learn more… | top users | synonyms

0
votes
1answer
23 views

For small values of $x$, the approximation $\sin(x)\approx x$ is often used. Estimate the error using this formula with the aid of Taylor's Theorem.

For small values of $x$, the approximation $\sin(x)\approx x$ is often used. Estimate the error using this formula with the aid of Taylor's Theorem. For what range of values of $x$ will this ...
13
votes
2answers
847 views

Application de Stone-Weierstrass

Bonjour, J'ai rencontré le problème suivant dans le livre "Real and Functional Analysis" de Lang, au chapitre $3$. J'explique d'abord le contexte, puis j'en viendrai à la question précise. Il faut ...
0
votes
2answers
35 views

Binomial expansions question

In a physics book the autor make the following expansions, given the fact that $z>>d$ (much greater). However I didn't understand how he manage to get the final expression. ...
2
votes
1answer
29 views

Error estimation for the Wallis product

From the Wallis product we know $$\prod_{k=1}^{\infty} \left(\frac{2k}{2k-1} \cdot \frac{2k}{2k+1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot ...
2
votes
0answers
273 views
+50

Multivariate B-Spline Derivatives

To construct a multivariate B-spline, we simply take the Kronecker tensor product between the univariate basis functions constructed for each individual dimension. What I'd like to know is how do you ...
4
votes
0answers
86 views
+300

Radial Basis Functions Interpolation

$ \let\oldcdot\cdot \renewcommand{\cdot}{\!\oldcdot\!} \newcommand{\e}{\varepsilon} \renewcommand{\p}{\varphi} \renewcommand{\p}{\varphi} \renewcommand{\vp}{\vec{\boldsymbol\p}(x)} ...
4
votes
1answer
63 views

Yet another Gamma function approximation

I know I have asked a similar question a couple of days, ago, but I still have a problem. I need a upper bound for: $$ ...
1
vote
1answer
520 views

Smooth approximation of maximum using softmax?

Look at the Wiki page for Softmax function (section "Smooth approximation of maximum"): https://en.wikipedia.org/wiki/Softmax_function It is saying that the following is a smooth approximation to the ...
2
votes
0answers
49 views

Approximate an integral with Bessel functions

Given $r,a,\lambda\in\mathbb{R}$, $r<a$, how can I find an approximate solution for the following definite integral? $$ \int_0^\infty J_0 (\lambda r)J_1(\lambda a)\frac{1}{\sqrt{n+\lambda^2 ...
0
votes
1answer
10 views

Comparing a function and its estimate

What are some clever ways of comparing (visually) a function with its estimate? For regions where the function does not cross zero, plotting the ratio of the functions and plotting the relative error ...
2
votes
3answers
75 views

Symmetry Of Differentiation Matrix

I have a problem computing numerically the eigenvalues of Laplace-Beltrami operator. I use meshfree Radial Basis Functions (RBF) approach to construct differentiation matrix $D$. Testing my code on ...
1
vote
1answer
30 views

Lower and upper bound of the Stirling's approximation

Perhaps everybody has heard of the Stirling's approximation, namely: $$ \Gamma(z)\approx\sqrt{\frac{2\pi}{z}}\left(\frac{z}{e}\right)^z $$ Thus (the very basic example): $$ ...
0
votes
1answer
36 views

Coin Change Problem with Fixed Coins

Problem: Given $n$ coin denominations, with $c_1<c_2<c_3<\cdots<c_{n}$ being positive integer numbers, and a number $X$, we want to know whether the number $X$ can be changed by $N$ coins. ...
1
vote
0answers
51 views

Is there an analytic approximation to integral of this form?

Started working on trying to find an analytical approximation to this integral and not getting very far. Any assistance or direction is greatly appreciated! Thanks Vince $$\int_{0}^{t} ...
1
vote
0answers
21 views

Please check this perturbation solution of polynomial root and truncation order.

I have a quintic polynomial where the coefficients depends on a parameter $c$, i.e. $$ a_0(c)+a_1(c)x+a_2(c)x^2+a_3(c)x^3+a_4(c)x^4+x^5 $$ I know that the roots of the polynomial are real and ...
0
votes
2answers
62 views

Approximation of the Gamma function

I am having trouble obtaining a lower bound for the following formula: $$ \ln\frac{\Gamma\left(\frac{x}{3}\right)}{\Gamma\left(\frac{x}{4}+1\right)\Gamma\left(\frac{x}{12}+1\right)}. $$ I tried using ...
1
vote
2answers
31 views

Simple question, what is meant by 'as $x \to \infty$ the number of squares $\leq x$ is $\sqrt{x} + O(1)$?

For $x \to \infty$: the number of squares $n^2 \leq x$ is $\sqrt{x} + O(1)$. From here (page 6). More specifically, do they mean that... I'm confused now. I'm really not sure what they mean ...
0
votes
0answers
59 views

How to I approximate $I = \int_{-1}^{1} \sqrt{1-x^2}\cos(x)dx$ s.t. the error is bounded?

Edit: Because the original question was pretty trivial, I want to ask the same question but with:$I = \int_{-1}^{1} \sqrt{1-x^2}\cos(x)dx$. How to I approximate $I = \int_{-1}^{1} ...
2
votes
0answers
54 views

Approximating sums

I got a general question, that is motivated by a recent problem. So let me first describe the problem and then add the general part: I got a rather simple (using only basic elements) equation, which ...
3
votes
1answer
33 views

$\sum_{n \leq x} \frac{1}{n} = \int_{1}^x \frac{dt}{t} + O(1)$ help deriving it

On page 5 of: Probabilistic Number Theory by Dr.J¨orn Steuding, there's $\sum_{n=2}^{[x]} \frac{1}{n} \lt \int_{1}^{[x]} \frac{dt}{t} \lt \sum_{n=1}^{[x] - 1}$ Therefore integration yields: ...
1
vote
2answers
69 views

Series Approximation How to evaluate $1/3+1/3(1/3)^3+1/5(1/3)^5+…$?

How to evaluate $$\frac13+\frac13(\frac13)^3+\frac15(\frac13)^5+...$$? I faced this particular sum in the website www.toppr.com .And it is given under the heading "Problems on Approximation"...but I ...
1
vote
0answers
18 views

How to compare experimental data with teorethical prediction

I would like to know, what is the method to approximate experimental data to teorethical one. I have heard about polynomial regression. After calculating particular matrices and solving set of ...
2
votes
1answer
59 views

Closed forms for two times series similar to geometric series, but with additional power

Does anyone know a close form solutions to any of the following time series? (approximate upper bounds might as well work). $$ \sum_{k=1}^T \frac{1}{2^{k^2}} $$ or $$ \sum_{k=1}^T k ...
13
votes
9answers
8k views

How to calculate $e^x$ with a standard calculator

Is there a simple method for calculating the $e^x$ ($x\in\mathbb{R}$) with a basic add/subtract/multiply/divide calculator that converges in reasonable time, preferably without having to memorize ...
0
votes
1answer
29 views

Approximating fractions

I have a fraction $\dfrac{a}{b}$ where $a$ and $b$ are both two large integers with $30$ digits each. I wish to approximate this fraction with a new fraction $\dfrac{c}{d}$ where $c$ and $d$ are both ...
2
votes
1answer
65 views

Upper bounding a tricky sum

For a problem in probability, I'm trying to find an upper bound for $$ \sum_{d=0}^k\binom{k}{d}\gamma^d(1-\gamma)^{k-d}\left(1-p^d(1-p)^{k-d}\right)^m$$ which will help me analyze what values of ...
0
votes
0answers
22 views

Minimum vertex cover of two edge disjoint perfect graphs

How well can the minimum vertex cover of the union of a perfect graph and bipartite graph (the two graphs are edge disjoint but not vertex disjoint) be approximated?
23
votes
0answers
256 views

Is there an integral for $\pi^4-\frac{2143}{22}$?

In Ramanujan's Notebooks, Vol 4, p.48 (and a related one in Quarterly Journal of Mathematics, XLV, 1914) there are various approximations, including the close (by just $10^{-7}$), $$\pi^4 \approx ...
2
votes
1answer
38 views

How to find solutions for this nonlinear equation?

I want to find an analytical solution $x$ as a function of parameters $(e,u,r,t)\in\mathbb{R}^4$ that satisfies the following condition: ...
0
votes
1answer
47 views

Boundary layer problem

This question is taken from Bender & Orszag "perturbation methods" $y' = (1 + X^{-2}/100)y^2 - 2y + 1$ ,$y(1)=1$ first we can see that if we set $\epsilon=100x^{2}$ we can translate the above to ...
0
votes
2answers
45 views

Find to how many digits is the value 355/113, an accurate approximation to $3.1415929204$

Find to how many digits is the value 355/113, an accurate approximation to $3.1415929204$ What i did was i computed using calculator value of 355/113 which came out to be $3.14159265$ Now i see up ...
1
vote
1answer
19 views

Approximate first order derivative without solving the best fitted polynomial

For example, I have $K$ points of the form $(x_k,y_k,f(x_k,y_k))$ for $k=1, ..., K$ near $0$. The distances between the points and $0$ are not the same. Is there an approximation for first order ...
0
votes
0answers
13 views

Universal polynomial approximation algorythm

I would like to ask, is there any universal algorythm to fill this matrix for any n value? $\textbf{A} = \matrix{n & \sum x_i & \sum x_i^2 & \cdots & \sum x_i^n \cr \sum ...
1
vote
1answer
54 views

Question about normal approximation on roulette

Problem: At roulette, you bet a dollar on red $30$ times in a row. Each time you win a dollar with prob. $18/38$ and lose a dollar with prob. $20/38$. Find approximately the probability that after ...
3
votes
4answers
98 views

Why does $\sqrt{3x} \left(\dfrac {x}{2} \right)^x$ approximate $x!$ pretty well?

I was just messing around and trying out things in the desmos calculator and found that $\sqrt{3x} \left(\dfrac {x}{2} \right)^x$ is pretty close to $x!$ most of the time, here is a graph. Why does ...
1
vote
1answer
58 views

Is there a “greater than about” symbol?

To indicate approximate equality, one can use ≃, ≅, ~, ♎, or ≒. I need to indicate an approximate inequality. Specifically, I know A is greater than a quantity of approximately B. Is there a way to ...
0
votes
1answer
30 views

How to approach to fitting curve?

I'd like to approximate fitting curve some kind of curves like below. (1, 3.5), (2, 4.3), (3, 7.2), (4, 8) which is having 4 points. and I heard that this solver is PINV() of matlab function. But ...
0
votes
1answer
38 views

What are some applications of “separable” spaces?

A separable space is a space that contains a countable dense subset. For example, the space of continuous functions $C[a,b]$ is separable. Are there some practical applications arising out of this ...
0
votes
0answers
14 views

What is the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A company has a one-year group life policy that divides its employees into two classes as follows: Class, Probability of Death, Benefit, Number in Class, A, 0.01, ...
0
votes
1answer
63 views

Finding the closest function describing a “magnetic line” (given magnetic readings)

I'm collecting data from a smartphone magnetometer while I move a magnet along a straight line (a slider). I am collecting the values of the magnetic field strength along the three axes. I would like ...
1
vote
1answer
57 views

Approximation for $\sin(\beta\sin(x))$

Can someone explain why, assuming $\beta\ll 1$, we have $$\cos(\beta \sin(2\pi f_mt))\approx 1$$ and $$\sin(\beta \sin(2\pi f_mt))\approx \beta \sin(2\pi f_mt) $$ The equations are part of a FM ...
0
votes
2answers
59 views

How to find a equation from approximate curve? [closed]

I want to know a way to find equation from a curve. for example, if I have 4 point (1, 3.5), (2, 4.3), (3, 7.2), (4, 8) then how to find a good approximation equation ? What if I got above curve ...
2
votes
1answer
140 views

Mysterious subleading corrections to sums with internal dependence on limit

Is there a standard method for finding expansions in $N$ of sums like $$S(N)=\sum_{n=0}^N \sqrt{N^2-n^2}$$ beyond the first term? It is easy to compute here that $$S(N)=N^2 \int_0 ^1 \sqrt{1-x^2} ...
5
votes
3answers
808 views

How to find an approximation to $1 - \left( \frac{13999}{14000}\right )^{14000}$?

I want to find an approximation to the expression $$ 1 - \left( \frac{13999}{14000}\right )^{14000} $$ I tried by taking logarithm $$ \ln P = \ln\left(1 - ...
-1
votes
1answer
19 views

Numerical integration of functions over computable Cauchy sequences

I'm interested in exact real arithmetic (and by extension constructive analysis). A nice representation of real numbers is via Cauchy Sequences. The basic idea being that you have a function which, ...
13
votes
1answer
458 views

Approximation of Semicontinuous Functions

Assume that $k \in \mathbb{N}$ and $f : \mathbb{R}^d \rightarrow [0,\infty)$ is lower semicontinuous, i.e. $f(x) \leq \liminf_{y \rightarrow x} f(y)$ for all $x \in \mathbb{R}^d$. Does there exist a ...
0
votes
5answers
186 views

best approximation of $\sqrt{2}$

The approximation \begin{align} \sqrt{2} &\approx \frac{1}{8} \operatorname{csch}\left(\frac{3\pi}{2}\right) \operatorname{sech}^3(\pi) \, \left[2+3 \, ...
1
vote
1answer
26 views

How is the approximation justified and how to improve it?

In an attempt to find the solution to the equation $Mx=e^x$ with $M$ being a large real number and the solution $w \gt 1$ I was asked to justify why $\ln M$ is a reasonable approximation to ...
2
votes
5answers
118 views

Approximating $\sqrt{1+\frac{1}{n}}$ by $1+\frac{1}{2n}$

I was wondering how to approximate $\sqrt{1+\frac{1}{n}}$ by $1+\frac{1}{2n}$ without using Laurent Series. The reason why I ask was because using this approximation, we can show that the sequence ...
0
votes
0answers
12 views

Approximate CDF of arbitrarily aggregated random variable

I would like to know if my solution for the following is mathematically correct in general: I have a random variable $Z$ that is an arbitrary function of two other rvs $X$ and $Y$, so: $Z = f_{arb}(X, ...