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

4
votes
1answer
327 views

Find the order of the error for the approximation $f' '(x)$

Given $$f''(x) = \frac{ f(x+h) - 2f(x) + f(x-h)}{h^2}.$$ I realize that this is just an approximation - that it won't give the exact value of $f''(x)$ and therefore there is an error term. However, I ...
2
votes
4answers
428 views

Analytical approximation of an integral

I think there is no analytical solution for $$ \int_{K}^{\infty} \frac{exp(-x)}{x} dx $$ where $K > 0$. Instead, is there an analytical approximation?
2
votes
1answer
67 views

Approximation of a function

Let $f:[0, 1] \rightarrow [0, \infty]$ be a function of $x$, with a parameter $\theta > 0$, such that $f$ is continuous $f$ is strictly decreasing $f(0) = \infty$ $f(1) = 0$ For example, $f(x) ...
1
vote
2answers
99 views

Asymptotic expansion for $\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du$?

Is there an asymptotic expansion for the function: \begin{equation} g(x)=\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du, \end{equation} over the domain $x\in [0,\infty)$ in terms of ...
1
vote
0answers
84 views

Expansion in powers

Let $n=2k, k \in Z_+$. Let $$P_k\left(\frac{t}{\sqrt n}\right)=n!\sum_{\begin{smallmatrix} n_1+\ldots+n_k=n \\ j=n_2+2n_3+\ldots+(k-1)n_k\end{smallmatrix}}\left(\frac{-t^2}{n}\right)\frac{1}{\prod_{i=...
0
votes
3answers
894 views

Approximating the integral $\int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$

How would I go about evaluating $I = \int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$ to four decimal accuracy?
2
votes
0answers
483 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 ...
3
votes
2answers
143 views

Bounds on $ \sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)…\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}$

I have a confluent hypergeometric function as $ _{1}{{F}_{1}}\left( a,a+b,z \right)$ where $z<0$ and $a,b>0$ and integer. I am interested to find the bounds on the value it can take or an ...
1
vote
1answer
289 views

Find taylor polynomial that approximates e^x with accuracy at least 1.

Find Taylor polynomial at $x=0$ which approximates $e^x$ with accuracy at least $1$ for each $x \in [-2,2]$. I dont undestand these questions that involve the $n^{th}$ remainder. I know I need to ...
1
vote
1answer
2k views

What is a fitting parameter in least-squares?

I was doing some least-squares homework when I saw this term "fitting parameters". I was asked to implement the leas-squares fit using polynomials of $p-th$ degree to a generic dataset. This is done. ...
1
vote
2answers
8k views

Power series for $\ln(1+x^2)$

In the problem I am asked to use a power series representation of $\ln(1+x)$ to approximate the integral from $0$ to $0.5$ of $\ln(1+x^2)$ to within 4 decimal places. So far I have found a series for ...
0
votes
1answer
30 views

Approximation related to resonance

Can someone help me with this problem. We have $$x(t)=N \sin (w_{0} t)+\frac{w_0}{w_1}e^{\frac{-t}{T}}\sin (w_{1}t)$$ and $w_1=(1+\frac{\delta_1}{N^2})w_0$ for some $|\delta_1|\leq 1$. I need to ...
2
votes
2answers
51 views

Formula for the pseudofrequency using approximations

We know that $w_1=\frac{\sqrt{4w_{0}^2-b^2}}{2}$ and $(1+u)^{\alpha}=1+\alpha u + \mathrm{O}\left(u^2\right)$ I need to show that $\frac{w_1}{w_0}=1+\frac{\lambda}{N^2}+\mathrm{O}\left(\frac{1}{N^3}\...
1
vote
0answers
186 views

Approximate CDF of the sum of a gaussian and a truncated gaussian

I am looking for a quick-to-compute approximation of the CDF of $X+Y$, where $X \sim N(0,\sigma_1^2)$ and $Y$ is a truncated gaussian, more specifically, a gaussian with mean $0$, standard deviation ...
1
vote
1answer
326 views

B-Spline Interpolation/Approximation

I've got a couple of probably very simple questions, yet some googling didn't bring up what I was looking for. First what I want to do: I have a grid, and the gridpoints are function values. I want to ...
3
votes
0answers
84 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 ...
1
vote
1answer
61 views

approximation of law sines from spherical case to planar case

we know for plane triangle cosine rule is $\cos C=\frac{a^+b^2-c^2}{2ab}$ and on spherical triangle is $ \cos C=\frac{\cos c - \cos a \cos b} {\sin a\sin b}$ suppose $a,b,c<\epsilon$ which are ...
0
votes
2answers
145 views

Aproximate calculation in decimals

I am trying to refresh on precision of calculations. If we have the decimal fractions: $.234673$, $.322135$, $.114342$, $.563217$ each known to be correct to six figures why are each of the decimals ...
0
votes
1answer
463 views

Need some help understanding notation for composite gauss quadrature formula

Reading through some notes on 2-point gauss quadrature, I came across the following general formula. I'm currently doing an assignment with 3-point quadrature, and have gotten to a similar formula, ...
3
votes
0answers
65 views

Root calculation by hand (division-like algorithm)

I remember from my highschool days a division-like algorithm for calculating square, even cubic roots. I know the continued fraction method, some series and Newton's method. I have checked similar ...
0
votes
2answers
59 views

Using approximation to find a value of theta

So I have one vector of alpha, one vector of beta, and I am trying to find a theta for when the sum of all the estimates (for alpha's i to n and beta's i to n) equals 60. Basically what I did is ...
0
votes
1answer
303 views

Approximating probability of success of Bernoulli trials using Kullback–Leibler divergence

In "Probabilistic Graphical Models" book by Daphne Koller and Nir Friedman they have the following approximation of probability of r successful outcomes of N Bernoulli trials: $P(S_N=r)\approx \exp(-...
7
votes
0answers
340 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^{...
12
votes
1answer
238 views

Why is integer approximation of a function interesting?

I have recently learnt the following result: Let $g \in \mathbb{R}[x]$ be a polynomial with $g(0) = 0$. Then, for any $\varepsilon > 0$, the set of positive integers $n$, such that $g(n)$ is ...
0
votes
1answer
36 views

Compute approximation difference

How can one compute $$e^{\frac{-j(j+1)}{2n}} -\prod_{i=1}^{j}\left(1-\frac{i}{n} \right)$$ assuming $j,n \geq 2$? I am interested in understanding how quickly this number tends towards zero as $n$ ...
14
votes
1answer
633 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 ...
4
votes
0answers
282 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)
0
votes
1answer
62 views

Approximations with differentials

My book states that for large n and small a, $$\dfrac{1}{(n+a)^2} - \dfrac{1}{(n)^2} \approx -\dfrac{2a}{n^3}$$ Let $f(x) = \dfrac{1}{x^2}$, $$ df = -\dfrac{2}{n^3}dx$$ with $$dx = a$$ and so the ...
3
votes
2answers
304 views

Integration over a combination of confluent hypergeometric, power, and exponential functions

I am trying to work out this integral. If there is no closed form, can you think of any approximations to it? $$\int_0^T e^{a (T-x)} (T-x)^{1+m+n} x^k \, _1F_1\Big(1+n;2+m+n;a (x-T) \Big) \, dx$$ ...
3
votes
0answers
245 views

Runge-Kutta Error Analysis

Could anyone explain to me how to reduce the error propagated by using Runge-Kutta of order 4? Or can anyone give me a nice reference to it.
2
votes
1answer
932 views

Finding approximate eigenvalues of perturbed matrix

Assume I have some constant matrix $A$ to which I add a perturbation, resulting in $M(\epsilon )=A+\epsilon B$ the perturbed matrix ($B$ is constant as well), and that I can easily find the ...
7
votes
3answers
939 views

Finding good approximation for $x^{1/2.4}$

I would like to a good (8 bits accuracy) approximation for $x^{1/2.4}$ in the range $[0, 1]$. This transform is used for converting linear intensities to SRGB compressed values, so it's important that ...
11
votes
3answers
6k views

Approximating the error function erf by analytical functions

The Error function $\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt$ shows up in many contexts, but can't be represented using elementary functions. I compared it with another function $...
2
votes
0answers
57 views

Approximate Differential Equation?

Let $x \in \mathbb{R}$ be a variable and $c\in\mathbb{R}$ a parameter. Also, let $f(x,c)$ be a function dependent on $x$ and $c$. Furthermore, define a Differential Equation which is solved by the ...
3
votes
0answers
42 views

Bound on Permutations [duplicate]

I am trying to prove the following inequality, $$n^{(l)} = n(n-1)\cdots(n-l+1) \geq \frac{n^l}{e}\quad\text{ for }\quad 2 \leq l \leq \sqrt{n}\;.$$ So my approach is to observe that $n^{(l)} = \dfrac{...
1
vote
0answers
59 views

alternating series estimation with integral?

We know that there are some approximation like Abel's identity. If $\lambda_n$ is increasing and $$ C(x)=\sum_{\lambda_n\le x}c_n,\qquad(c_n\in\mathbb{C}) $$ Then if $X\ge\lambda_1$ and $\phi(x)$ has ...
3
votes
2answers
112 views

Discrete approximations of $\nabla^2{\bf v}$

I am writing a Navier Stokes solver. The vector field is represented as a grid with integer coordinates I am looking at other people's computer code. I don't entirely understand the vector calculus, ...
4
votes
1answer
165 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 ...
0
votes
1answer
520 views

Discrete approximation - exponential function and integrals

Let $f$ be a complex-valued continuous function on $\mathbb{R}_+$ with compact support and let $g, h$ be two complex-valued continuous functions on $\mathbb{R}_+$ such that $g$ is bounded and $|h(t)|$...
7
votes
3answers
471 views

Even integer approximations to multiples of pi

I admit that I'm probably out of my depth with this question, but I can't help but feel curious. I wanted to show that, in the sequence $\{\sin(n)\}$, there is never a largest term (the sequence ...
4
votes
1answer
403 views

What is the meaning of “mean-field”?

In lots of Bayesian papers, people use variational approximation. In lots of them they call it "mean-field variational approximation". Does anyone know what is the meaning of mean-field in this ...
1
vote
2answers
71 views

Roots of the equation $I_1(b x) - x I_0(b x) = 0$

I'm interested in the roots of the equation: $I_1(bx) - x I_0(bx) = 0$ Where $I_n(x)$ is the modified Bessel function of the first kind and $b$ is real positive constant. More specifically, I'm ...
0
votes
1answer
121 views

Efficient method of approximating a distribution with Gaussian

Given a univariate uni-modal density function $f(x)$ (very hard to compute its cumulative distribution function (CDF) $F(x)$, not to mention its inverse CDF $F^{-1}(x)$), how to find the best ...
3
votes
0answers
36 views

Characterize a large class of shapes using a finite number of parameters

I am doing some numerical computations searching for an optimal shape for a certain functional. In my particular case, the shape $\Omega$ is a 2 dimensional star shaped domain by the origin, which ...
6
votes
2answers
148 views

Get minimum N in approximation of pi

The following expression is an approximation of PI, where N determines the precision. $$\pi (N) = \frac{4}{N} \sum_{i=1}^{N}\frac{1}{1 +\left ( \frac{i -\frac{1}{2}}{N} \right )^{2}}$$ If I want to ...
1
vote
3answers
61 views

Find approximation of $y= {x^2}$

I have a function $\large{f(x)=\sqrt{x} \space \space \space x \forall \geq 0}$ I am looking for a quadratic approx. to $f(x)$ at $x=9$. So far, I know that the quadratic approx. at $x = x_0$ ...
1
vote
0answers
27 views

$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$

Here $i$ is complex number, $n$ is positive integer. Show that $$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$$ This question appears from Stein's ...
2
votes
2answers
604 views

Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes

I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
0
votes
0answers
45 views

Multivariate linear regression/aproximation

I'm solving a problem which features a function $f: \mathbb{R}^4 \rightarrow \mathbb{R}^4.$ I don't know the function, but I assume it's linear and it can be expressed as $ \mathbf{y} = \mathbf{A} \...
6
votes
5answers
2k views

What exactly is “approximation”?

There are a lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$ $$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what ...