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
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
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
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
593 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
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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
votes
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
294 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
132 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
485 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
370 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
575 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
261 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
votes
0answers
476 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
347 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
793 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 ...
3
votes
0answers
136 views

Why is a Complementary Filter a Good Approximation?

Can someone help me understand what the full expression for the original complementary filter should be and why the one proposed by Colton is a good approximation? Context: Having found some ...
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
161 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
190 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
264 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
108 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
57 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
118 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 ...
3
votes
1answer
673 views

Upper and lower bounds of a ratio involving vector norms

I'm working on a signal processing problem and need to analyze the following expression $$ G = \frac{n}{\sum\limits_{i=1}^n |w_i|} \frac{ \sum\limits_{i=1}^n g_i w_i^2}{\sum\limits_{i=1}^n g_i |w_i|} ...
5
votes
3answers
283 views

“$O$” notation in Stirling approximation

In the Stirling approximation the formula as typically used in applications is $$\ln n! = n\ln n - n +O(\ln(n))$$ I'm confused about the last term "$O$" . What does it mean actually, and when do we ...
5
votes
0answers
155 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
1answer
233 views

Approximating a weird sum

How can I approximate the sum$$\sum_{k=1}^n \left(\frac{2k}{n} \left\lceil \frac{n}{k} \right\rceil \left\{ \frac{n}{k} \right\}-1\right)$$ where $\{x\}$ is the fractional part function, and $\lceil ...
5
votes
5answers
7k views

Software to find a function for data approximation

I've got some y(x) 2D data set. I would like to find a function fitting this data: Is there any open source or free software to find a function to approximate a data sequence like the above? Here ...
2
votes
1answer
54 views

Prove that s is finite and find an n so large that $S_n$ approximate s to three decimal places.

$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that ...
0
votes
1answer
115 views

Integral approximation.

Can you help me to show that $$\int^{\pi/2}_{0}{d\theta\over (1-m^2\cos^2\theta)^2} \approx {(2-m^2)\pi\over4(1-m^2)^{3/2}}$$ to first order, such that $0 \lt m \lt 1$
1
vote
1answer
269 views

Knot placement for a natural cubic spline

I am trying to approximate a function via a natural cubic spline. Suppose I sample the function on a grid i.e. I know the value of the function at a fixed number of equidistant points, say on 200 ...
3
votes
0answers
66 views

When can the commutator of two matrices be neglected in series expansions?

Under what conditions can the higher order commutators in the Baker–Campbell–Hausdorff formula be neglected when the commutators does not vanish exactly and there is no small parameter in the ...
3
votes
1answer
153 views

Iterative model fitting

I have a sequence of points $\{(x_k,y_k,z_k)\}$ and I need to fit some $2D$ model $P(x,y)$ that approximates $z$ in some sense. The $z_k$$'s$ are noisy samples of some $2D$ function $z_k = f(x,y) + ...
4
votes
5answers
554 views

Improving Newton's iteration where the derivative is near zero?

I'm implementing a root-solver for finding x coordinates of a function f(x), after I have an y-coordinate. The function is periodic, roughly sinusoidal with constant amplitude but non-linearly ...
8
votes
1answer
181 views

Is this sequence convergent?

I heard this question from a professor a couple years ago. I still think about it... Does the sequence $(a_n)_{n\in \mathbb N}$ with $$a_n=\sqrt[n]{|\sin(n)|}$$ converges ( to $1$ ) ? I believe ...
2
votes
1answer
130 views

approximate a square function with a linear one

I have to code a function in matlab (F1) whose values range from 0.740261423849103 to some where around 0.95. Then there is another function (F2) which is usually the square of F1. Is there any way I ...
15
votes
1answer
420 views

A cute approximation for $\cot(2\pi x)$(!?)

Numerical calculations and some theory leads to the suggestion that $$\cot(2\pi x) \rightarrow\frac{1}{2\pi}\sum_r \frac{1}{x-r}$$ where $r$ ranges over all the roots of $B_{2n+1}$ (Bernoulli ...
0
votes
1answer
41 views

approximation on a graph

I expect my question to sound very naive, please excuse for this. I have a set of $(x, y)$ points on a graph, which are the measurements of a real-life process. Now I want to draw an "approximation ...
3
votes
1answer
1k views

Method of dominant balance

Find the leading asymptotic behaviour as $x \rightarrow \infty$ of $$x^2y'' + (1 + 3x)y' + y = 0 $$ Can someone kindly explain me how to solve this problem? Im learning asymptotic analysis, and I ...
1
vote
1answer
129 views

approximating function $a+b\cdot 2^x$

Hi I need to implement a function $a+b\cdot 2^x$ in a highly resource constrained device. a and b are constants and x is a variable. How do I go about finding a simplified version of this expression ...
3
votes
1answer
315 views

Show that if m/n is a good approximation of $\sqrt{2}$ than $(m+2n)/(m+n)$ is better

Claim: If $m/n$ is a good approximation of $\sqrt{2}$ then $(m+2n)/(m+n)$ is better. My attempt at the proof: Let d be the distance between $\sqrt{2}$ and some estimate, s. So we have ...
3
votes
1answer
92 views

If $f \in\operatorname{Lip}_K[a, b]$, show that $f$ can be uniformly approximated by polynomials in $\operatorname{Lip}_K[ a, b]$.

If $f \in\operatorname{Lip}_K[a, b]$, show that $f$ can be uniformly approximated by polynomials in $\operatorname{Lip}_K[ a, b]$. Context: $f \in \operatorname{Lip}_K[a,b]$ then it is ...
2
votes
1answer
540 views

Rounding .5 - why isn't rounding away from zero the 'right' answer?

I am familiar with the issue of 'how should one roung .5?', and I am familiar with the conventional solutions, but I don't understand why there isn't a correct answer. When you're formulating a ...
2
votes
1answer
178 views

approximate error between integral an sum

I am new here. My problem: There is an integral $I:=\int_0^1 f(x)\,dx$ for $f\colon [0,1]\to\mathbb{R}$ and I want to compute it by ...