For questions related to approximations
0
votes
0answers
15 views
Approximate size of mesh
this is programming/mathematic question and i am not sure how to properly call this problem so excuse me if this is well-known problem already.
I have an unknown-sized mesh consisting of nodes. Every ...
1
vote
0answers
61 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 ...
0
votes
1answer
69 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
52 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 ...
0
votes
1answer
14 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
53 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
52 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
48 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
29 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
63 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 ...
5
votes
0answers
275 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
$$
...
12
votes
1answer
197 views
Why is integer approximation of a function interesting?
I have recently learnt the following result:
Let $g \in \mathbb{R}[x]$ be a polynomial with $g(0) = 0$. Then, for any $\varepsilon > 0$, the set of positive integers $n$, such that $g(n)$ is ...
0
votes
0answers
9 views
Haar and Walsh systems
Does anybody know what is inner product between Haar and Walsh functions? For example if $H_{n,k}$ is Haar function and $ W_{l}$ Walsh function, what is $<f,g>$?
Thank you
0
votes
1answer
35 views
Compute approximation difference
How can one compute
$$e^{\frac{-j(j+1)}{2n}} -\prod_{i=1}^{j}\left(1-\frac{i}{n} \right)$$
assuming $j,n \geq 2$? I am interested in understanding how quickly this number tends towards zero as $n$ ...
4
votes
0answers
53 views
Approximation of Semicontinuous Functions
Assume that $k \in \mathbb{N}$ and $f : \mathbb{R}^d \rightarrow [0,\infty)$ is lower semicontinuous, i.e. $f(x) \leq \liminf_{y \rightarrow x} f(y)$ for all $x \in \mathbb{R}^d$.
Does there exist a ...
2
votes
0answers
80 views
Low-rank approximation to the Graph Laplacian matrix of a regular grid.
As mentioned in the title, does anybody know any methods of efficient low-rank approximation $LL^T$ to the Graph Laplacian matrix $A$ corresponding to a square lattice? (except PCA)
0
votes
1answer
24 views
Approximations with differentials
My book states that for large n and small a, $$\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$$
and so the ...
1
vote
1answer
105 views
Integration over a combination of confluent hypergeometric, power, and exponential functions
I am trying to work out this integral. If there is no closed form, can you think of any approximations to it?
$$\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$$
...
3
votes
0answers
98 views
Runge-Kutta Error Analysis
Could anyone explain to me how to reduce the error propagated by using Runge-Kutta of order 4? Or can anyone give me a nice reference to it.
1
vote
1answer
101 views
Finding approximate eigenvalues of perturbed matrix
Assume I have some constant matrix $A$ to which I add a perturbation, resulting in $M(\epsilon )=A+\epsilon B$ the perturbed matrix ($B$ is constant as well), and that I can easily find the ...
4
votes
2answers
128 views
Finding good approximation for $x^{1/2.4}$
I would like to a good (8 bits accuracy) approximation for $x^{1/2.4}$ in the range $[0, 1]$. This transform is used for converting linear intensities to SRGB compressed values, so it's important that ...
2
votes
2answers
90 views
Approximating the error function erf by analytical functions
The Error function
$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt$
shows up in many contexts, but can't be represented using elementary functions.
I compared it with another function ...
2
votes
0answers
30 views
Approximate Differential Equation?
Let $x \in \mathbb{R}$ be a variable and $c\in\mathbb{R}$ a parameter. Also, let $f(x,c)$ be a function dependent on $x$ and $c$. Furthermore, define a Differential Equation which is solved by the ...
3
votes
0answers
37 views
Bound on Permutations [duplicate]
I am trying to prove the following inequality,
$$n^{(l)} = n(n-1)\cdots(n-l+1) \geq \frac{n^l}{e}\quad\text{ for }\quad 2 \leq l \leq \sqrt{n}\;.$$ So my approach is to observe that $n^{(l)} = ...
0
votes
0answers
23 views
Anyone tried B-F-S in training a neural network?
Anyone tried B-F-S (Broyden-Fletcher-Shanno) in training a neural network?
I know SCG or LM are more popular, but I have already written an optimized library that have BFS method, I really dont want ...
1
vote
0answers
36 views
alternating series estimation with integral?
We know that there are some approximation like Abel's identity. If $\lambda_n$ is increasing and
$$
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 ...
0
votes
0answers
45 views
How can weakly/strongly decreasing or increasing approximate sums be explained?
I'm reading around big O to get some concept about performance for data structures. The mathematics book recommended by the open book ~ maths for computer science
In the book (pg 456) part of the ...
3
votes
2answers
50 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, ...
0
votes
0answers
49 views
Reciprocal uncertainty
In the equation $y = \dfrac{1}{L + 0.4 d}, L$ has an uncertainty of $± \,0.025$ and $d$ has an uncertainty of $±\,0.01$.
I understand that the uncertainty of $L + 0.4 d$ is $0.025 + 0.4 \times 0.01 ...
4
votes
1answer
47 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
125 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
108 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 ...
2
votes
1answer
39 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
47 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
51 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 ...
2
votes
0answers
20 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
98 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}}$$
If I want to ...
1
vote
3answers
47 views
Find approximation of $y= {x^2}$
I have a function $\large{f(x)=\sqrt{x} \space \space \space x \forall \geq 0}$
I am looking for a quadratic approx. to $f(x)$ at $x=9$.
So far, I know that the quadratic approx. at $x = x_0$ ...
1
vote
0answers
24 views
$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$
Here $i$ is complex number, $n$ is positive integer. Show that
$$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$$
This question appears from Stein's ...
1
vote
1answer
103 views
Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes
I have to numerically calculate the following definite integral
$$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$
for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
0
votes
0answers
22 views
Multivariate linear regression/aproximation
I'm solving a problem which features a function $f: \mathbb{R}^4 \rightarrow \mathbb{R}^4.$ I don't know the function, but I assume it's linear and it can be expressed as $ \mathbf{y} = \mathbf{A} ...
5
votes
5answers
227 views
What exactly is “approximation”?
There are a lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$
$$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what ...
0
votes
0answers
134 views
Approximate a complicated mystery function
Let there exist a mystery function ƒ.
ƒ accepts exactly 2 arguments, A & B.
As B approaches A, ƒ approaches A, at a simple exponential growth rate E.
As B approaches 0, ƒ approaches the mean ...
0
votes
0answers
105 views
Please help me find this limit and inverse Laplace transform
I need help solving this (I suggest something hereafter but I am not sure if it's ok):
I would like to find an approximate solution of the function $\bar{f}(s)$ defined in the Laplace space. At long ...
2
votes
1answer
36 views
Approximate N order polynomial as a weighted sum of lower order polynomials
I want to represent a polynomial such as $x^5$ with a sum of weighted polynomials so that
$$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
83 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
vote
0answers
64 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
70 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
2answers
59 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
1answer
61 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 ...
