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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0
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1answer
292 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 ...
7
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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 $$ ...
12
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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 $n$ ...
13
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1answer
612 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
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0answers
274 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
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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
296 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
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0answers
232 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
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1answer
913 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
909 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
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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
56 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
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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)} = ...
1
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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
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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
158 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
511 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 ...
7
votes
3answers
467 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
379 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
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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
60 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
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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
589 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
44 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
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5answers
1k 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 ...
2
votes
1answer
105 views

Approximate N order polynomial as a weighted sum of lower order polynomials

I want to represent a polynomial such as $x^5$ with a sum of weighted polynomials so that $$x^5 - (ax^4 + bx^3 + cx^2 + dx + e) = \epsilon$$ My aim is to pick these weights $(a,b,c,d)$ assuming that ...
3
votes
2answers
285 views

Approximation by Taylor polynomial

Let $f(x) = (1 − x)^{-1}$ and $x_{0} = 0.$ (a) Find the nth Taylor polynomial $P_{n}(x)$ for $f(x)$ about $x_{0}$. (b) Find the smallest value of $n$ necessary for $P_{n}(x)$ to approximate $f(x)$ ...
1
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0answers
130 views

Approximation of SDE

I have been struggling with the following problem: If you want to find a numerical result by simulating the paths of a stochastic differential equation, in particular a geometric brownian motion I ...
3
votes
0answers
85 views

Question about linearization

Given a data matrix $D\in\mathbb{R}^{N \times N}$ Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$, if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j ...
1
vote
1answer
480 views

Ei[x] Approximation

I'm working with the function $F(x)=e^{-k(x+1)}\int_1^x\frac{N^2}{t(N-t)}e^{kt}dt$. Breaking it down into into single fractions helps a little, yielding: $F(x)=Ne^{-k(x+1)} \int_1^x [\frac{1}{t} ...
2
votes
2answers
364 views

Approximate continous function with linear growth condition by Lipschitz function

Suppose a continuous function $f(u)<K(1+|u|)$ for some positive number $K$. How can we find a sequence of Lipschitz functions $f_{n}$ that converge to $f$ uniformly on $\mathbb{R}$. If we require ...
0
votes
1answer
560 views

Least-squares approximation polynomial

Consider the function $\displaystyle f(x) = \frac{1}{\alpha (x-\beta)^2 + 1}$ in the interval $I = [-1,1]$. Set $\beta = 0$. How do I get the expression for the least-squares polynomial, say $\tilde ...
0
votes
3answers
254 views

Finding an approximate diagonal in a grid

Imagine a 2 dimensional grid, with a variable size of $ x*y $. For this example of figure 1, let $ x=14; y=5 $. Now one may position "pixels" in this gird. They can only be placed on the grid's points ...
10
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0answers
473 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} ...
2
votes
1answer
100 views

Comparing two ratios with tolerances

I'm trying to figure out what the most robust way to compare two ratios is. What I have is a set of proportions between two or more quantities. For example, it might be 0.5:0.5, or 0.45:0.2:0.35. Then ...
2
votes
1answer
338 views

approximating a discrete function with a continuous one

Let $f:[0,1]\rightarrow \mathbb{R}$ be a continuously differentiable function that reaches a global maximum at $x^*\in(0,1)$. Now, consider its 'discrete' counterpart. That is, consider the collection ...
3
votes
1answer
761 views

Rayleigh-Ritz method for an extremum problem

I am trying to use the Rayleigh-Ritz method to calculate an approximate solution to the extremum problem with the following functional: $$ L[y]=\int\int_D (u_x^2+u_y^2+u^2-2xyu)\,dx\,dy, $$ $D$ is the ...
1
vote
0answers
66 views

Approximation of factorial - Stirling formula [duplicate]

Possible Duplicate: Elementary central binomial coefficient estimates How can I prove that $$ \binom{n}{n/2} = \Theta\left(\frac{2^n}{\sqrt n}\right) $$ I tried with Stirlings ...
2
votes
0answers
135 views

Approximation of complementary filter: Why is it a good approximation?

Having found some unofficial sources on Sensor Fusion (Thousand Thoughts Sensor Fusion and The Balance Filter by Shane Colton) I'm struck by how the approximation of the complementary filter that is ...
2
votes
0answers
56 views

convex optimization with inconsistent constraints

If you have a problem in convex optimization where all $N$ constraints ($N >> 0$) yield no possible solution but you are able to rank, or weight the constraint in terms of their importance are ...
1
vote
1answer
160 views

How to approximate $\text{li}(z)$ numerically?

I'm trying to implement a function to calculate $\pi(x)$ via Riemann's formula: $$ \pi(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan ...
3
votes
1answer
189 views

Brownian motion and hitting frequency

Suppose we have a Brownian motion $B_t$ with $B_0 = 0$ and $B_t - B_s \sim N(0,t-s)$. Every time $B_t$ hits $\pm h$, where $h$ is some "barrier" $>0$, I pay someone £1 and the brownian motion ...
1
vote
1answer
262 views

How do I determine appropriate rational approximations to a sum of square roots in order to bound the error accumulation?

I have two numbers, $A$ and $B$, that are sums of integer multiples of a set of square roots of small primes (and 1) and their products: $A = a_0 + a_1\sqrt 2 + a_2\sqrt 3 + a_3\sqrt 5 + a_4\sqrt 6 + ...
1
vote
0answers
107 views

Intuition for approximating Ei(x)

I'm working with a function that is defined to be the sum of a series, and I'd like to either find its values, or approximate them analytically: $F(x) = \frac{1}{w} ...
-1
votes
2answers
56 views

Aprroximate graph to function

there is a set of points which set a graph that is not linear. Is there any method to approximate a function that is close enough to this graph? I've read some articles and got to know approximation ...
1
vote
1answer
117 views

Need numerical approximation for Fourier{max(0,f(x,y))} given Fourier{f(x,y)}

Given $\mathscr{F}\{f(x,y)\}$ is there a way to numerically approximate $\mathscr{F}\{max(0,f(x,y))\}$ ? I am not necessarily looking for a closed formula. Even some iterative method would be ...