For questions related to approximations

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1answer
32 views

How to Split a 2D Gaussian pdf into a Grid of Equally Sized Volumes

Let $f(x,y)$ be a Gaussian pdf for some known mean and covariance. Given $(x_0, y_0)$ and $(x_N, y_M)$ such that $$\int_{x_0}^{x_N} \int_{y_0}^{y_M} f(x,y) dy dx \approx 1$$ I would like to split ...
0
votes
1answer
70 views

Picard Approximation

Consider the initial value problem $$\frac{dy}{dx} = y^2 + 3x^2 - 1, \\ y(1) = 1$$ on D = {|x-1| <= 1, |y-1| <= 1} Find the second approximation to the solution and estimate the error term ...
2
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1answer
79 views

Minimum Argument Difference to Make the Lower Bound > the Upper Bound

Assume $g$ is a function that grows asymptotically as $$ g(n) \in\frac n {log(n)} + O(\sqrt n),\,n \in \Bbb N\tag1 $$ I wish to find $h(n)$ such that $$ g(n) \le g(n+h(n)). $$ i.e. Given the bounds ...
0
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1answer
34 views

Sequence of piecewise constant functions converging to any $L^2$ function

Let $\{P_i\}_{i=1}^\infty$ be a sequence of partitions of the interval $[0,1]$ with a vanishing mesh. Additionally $H_i$ be the space of piecewise constant functions (step functions) with pieces ...
4
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1answer
68 views

Applications of the Exponential Integral?

this is my first time asking a question on here so please forgive me if I have made any formatting mistakes. I have the integral $f(x) = \int_0^\infty \frac{e^{-t}}{x + t} \; dt$ and I have shown the ...
1
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0answers
10 views

Approximation method for combinations

I have the expression: $$\frac{{D+\frac{k}{\theta}}\choose D-z}{{1+m}\choose{D-z}}$$ In the above expression, $m$, $k$, $\theta$, and $D$ are constants. What is the approximation of the above ...
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2answers
45 views

Can a function be approximated by finite number of Taylor expansion terms outside of disk of convergence?

Suppose we have a finite number of terms for Taylor expansion of a conditionally convergent function. For example, $f=\frac1{1-x}$ with expansion $f=\sum_{n=0}^\infty x^n$. This expansion diverges ...
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0answers
9 views

Simplifcation of the Function Required

I want to simplify the following expression $$ \sum_{x=0}^{D+m} \frac {x!} {(n+1)_x} (\frac {\theta_{eff}} {\mu})^x {D+m \choose x} $$ The parameters $D, m, n, \theta_{eff}, \mu, $ are constants. ...
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0answers
27 views

Proving Chebyshev's Theorem (polynomials)

A polynomial $P_n(x)$ of degree $n$ is a polynomial of best approximation to the function $f \in C[a,b]$ if and only if there are at least $n+2$ Chebyshev alternant points on the closed interval ...
0
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1answer
40 views

Solving an ODE in Maple via Picard approximations

I am trying to use Maple to apply Picard approximations to the following ODE: $$ 2xyy' + x^2 - y^2 = 0,\ y(0) = 1 $$ I transformed it into the form $y' = f(x, y)$: $$ y' = \frac {y^2 - x^2} {2xy} ...
2
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2answers
50 views

Approximation of difference of harmonic numbers

Harmonic number $H_n$ is equal to $$H_n = \sum_{i=1}^n \frac{1}{i}$$ Asymptotic expansion of harmonic humber is $$(1) H_n = \ln n + \gamma + \frac{1}{2n} - O\left(\frac{1}{n^2}\right)$$. Very popular ...
3
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1answer
74 views

Approximating $\sqrt{101}$ using Taylor series methods

I'm trying to approximate $\sqrt{101}$ using the Taylor series for the function $f(x)=\sqrt{x}$ centered at the point $x=100$. I need to obtain an approximation that is within $0.01$ of the correct ...
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3answers
37 views

Having trouble calculating approximations using Taylor polynomials

I have a problem to approximate $\sqrt{1.06}$ using a third degree Taylor polynomial. The way I learned was to pick a center that we would know the answer to that is close to the value we're trying ...
1
vote
1answer
82 views

If $\int_0^1 f(x) e^{nx} dx = 0$ for every n, then f=0

$f$ be a continuous function [0,1] to $R$. $\int_0^1 f(x)e^{nx} dx = 0$ for all $n \in N\cup\{0\}$ how to prove $f(x)= 0$ in $[0,1]$ for all $x\in[0,1]$? I solved "$\int_0^1 f(x)x^n dx = 0$ for all ...
3
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1answer
32 views

Asymptotic approximation of a certain sum

During calculations of an expectation of some random variable, I have encountered the following sum: \begin{equation} \sum_{t=2}^{n+1} \frac{t(t-1) \cdot n!}{(n-t+1)!\cdot n^t} \end{equation} I ...
2
votes
1answer
80 views

Using newthon method to find nth root - did wolframalpha get it wrong?

I'm trying to implement the n-th root algorithm as outlined here: http://en.wikipedia.org/wiki/Nth_root My code however, takes a lot of iterations (more than 100) to converge. I tried to check with ...
1
vote
1answer
28 views

Why does the asymptotic equation of the modified Bessel of the second kind (Iv) have an imaginary part?

This is a follow up to this question. How does one arrive at the asymptotic expressions for the bessel functions? After looking at: G. N. Watson, "A Treatise on the Theory of Bessel Functions", 2nd ...
0
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1answer
36 views

Understanding graph layering approximation algorithms

I've been trying to understand the Graph Layering Approximation Algorithms for both Set Cover and Feedback Vertex Set problems. I am using Vijay V. Vazirani's "Approximation Algorithms" book. So let ...
2
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0answers
42 views

Finding the cubic near minimax approximation for sin(x) on (0,pi/2)

I am really stuck here. Here is the question that I have. Find the cubic near minimax approximation for $f(x)=\sin(x)$ on $(0,\pi/2)$. So I defined $h(x)=ax^3 + bx^2 + cx + d - sin(x)$ The max ...
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0answers
23 views

Approximation of a quadratic form

Let $\mathbf{x}=(x_1,\cdots,x_n)^T\in\mathbb{R}^n$ and $A\in\mathbb{S}_{++}^n$ be a symmetric positive definite matrix. Also, let $Q\colon\mathbb{R}^n\to\mathbb{R}$ be the quadratic form given by $$ ...
1
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1answer
92 views

Asymptotics of sum of Binomial Coefficients (Binomial distribution) - Poisson approximation?

Let $$f(n):=\sum_{i=k}^n {n \choose i } p^i (1-p)^{n-i}$$ where $k\geq 2$ is a fixed Parameter and $p=p(n) \in (0,1]$ depends on $n$ where $np\leq 1$. We consider $n \rightarrow \infty$. I've found ...
0
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0answers
34 views

If there exists a polynomial of best approximation of degree n, there also exists a polynomial of best approximation of degree n+1.

First I'd like to say that although this question was asked before (here) and is from the same text, the answer used methods that were not introduced in the text. Let $P_n(x)$ be a polynomial of ...
1
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2answers
49 views

Is $n\binom{\epsilon n}{t}>t\binom{n}{t}$ for large $n$ and fixed $\epsilon$ and $t$

Let $\epsilon$ and $t$ be fixed numbers with $t$ and integer. I came across the following inequality in a counting problem. $$n\binom{\epsilon n}{t}>t\binom{n}{t}.$$ I want to show that for $n$ ...
1
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2answers
68 views

How does one arrive at the asymptotic expressions for the bessel functions?

It is known that Bessel functions for large arguments will behave as exp or cos/sin however I was wondering how does one arrive at those results. The motivation being that I would like to use these ...
0
votes
1answer
37 views

Finding/approximating 2 unknowns using one equation

I’m doing experimental data in a chemistry lab and I have faced this mathematical problem at a point of my work. Hope you guys can help me with that. What would be the best way to find two constants m ...
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0answers
11 views

Best multivariate approximation scheme for scattered data approximation

what is the best multivariate approximation scheme for scattered data approximation on a set E of points in $R^d$?
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3answers
51 views

Where did the linear approximation/linearization formula come from?

Where did the linear approximation/linearization formula come from? I understand that it takes root in the point-slope form and slope intercept form of a linear equation, but I don't understand where ...
0
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0answers
40 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 ...
10
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2answers
440 views

How prove this sequence $a_{n}=\sqrt{n}+\frac{1}{2}-\frac{1}{8\sqrt{n}}+o\left(\frac{1}{\sqrt{n}}\right)?$

let sequence $\{a_{n}\}$ such $$a_{1}=1,a_{n+1}=1+\dfrac{n}{a_{n}}$$ show that: $$a_{n}=\sqrt{n}+\dfrac{1}{2}-\dfrac{1}{8\sqrt{n}}+o\left(\dfrac{1}{\sqrt{n}}\right)?$$ This result is china student ...
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0answers
48 views

Approximate an integral

In a physics textbook, I came across the integral $$I(r_1,r_0)=\int_{r_0}^{r_1}\frac{1}{1-2m/r}\left[1-\frac{r_0^2(1-2m/r)}{r^2(1-2m/r_0)}\right]^{-1/2}dr$$ The author said that the integrand can be ...
0
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2answers
30 views

Understanding an approximation equation

In this biology textbook I found the following approximation: $$\frac{1}{2N}\left( 1-\frac{1}{2N} \right)^t ≈ \frac{1}{2N}e^{\frac{-t}{2N}}$$ Can you help me to understand this approximation and ...
0
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2answers
36 views

Product approximation

In this biology textbok I found the following approximation: $$\prod_{i=1}^{k-1}1-\frac{i}{2N} ≈ 1-\frac{{k \choose 2}}{2N} $$ Can you help me to understand this approximation and help me to ...
1
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2answers
36 views

Approximation of $x\log(x/a)$ for $x$ near a

I'm trying to see where the approximation $$(x-a) + ((x-a)^2)/2a$$ of $x\log(x/a)$ comes from (for x near a). Might be missing something very trivial but I've already tried the usual expansions ...
0
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0answers
9 views

Approximate the local behavior of an unknown distribution with uniform distribution.

consider an arbitrarily smooth distribution function over support $S$. I am only interested in local behavior that happens in a very small area. To what extent can I approximate the local distribution ...
1
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1answer
36 views

division by sum of exponentials of large negative numbers

I need to evaluate the following numerically: $$ f = \frac{\exp(a)}{\exp(a)+\exp(b)+\exp(c) + \exp(d)} $$ $a,b,c$ and $d$ are large negative numbers, they are smaller than -1000. Numerically ...
0
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0answers
17 views

approximating a probability density

Let $f(x)$ be the probability density of a random variable $X$. Let the support of $f(x)$ be positive reals. If $f(x)$ is sufficiently smooth then one can approximate it with its Taylor series cut off ...
1
vote
1answer
28 views

Use tangent line to find approximation

$f(x) = x^2 - 3x + 5$, the tangent line to the graph of $f$ at $x = 3$ is used to approximate values of $f(x)$. Which of the following values $3.4$ $3.5$ $3.6$ $3.7$ $3.8$ is the greatest value of x ...
0
votes
1answer
50 views

Approximation of $\frac{1}{2^{N}} \binom{N}{\frac{1}{2}(N + s)} $ for big N

I have the following function that I have to approximate for big N and I really have no clue what to do - I hope that anyone can help me out: $$P_N(s) = \begin{cases} \frac{1}{2^{N}} ...
2
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2answers
72 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
5
votes
1answer
97 views

How to show this: $\sum_{k=2}^{n}\frac{\ln{k}}{k^2}\approx \ln{n}\cdot\left(\zeta_{n}{(2)}-\frac{\pi^2}{6}\right)+C$

Show that: $$\sum_{k=2}^{n}\dfrac{\ln{k}}{k^2}\approx \ln{n}\cdot\left(\zeta_{n}{(2)}-\dfrac{\pi^2}{6}\right)+C,n\to\infty$$ where $$\zeta_{n}{(k)}=\sum_{j=1}^{n}\dfrac{1}{j^k}$$and $C$ is ...
2
votes
1answer
22 views

How many bits of difference in a relative error?

I would like to know if there is a formula or any other way to find out how many bits of difference between two values given the relative error. For instance: $$\epsilon_{\text{rel}} = \frac{V - ...
1
vote
4answers
91 views

Demonstrate that $\displaystyle \frac{(2n - 2)!!}{(2n - 3)!!} \simeq 1.7 \sqrt{n}$

As in the title, I know that $\displaystyle \frac{(2n - 2)!!}{(2n - 3)!!} = \frac{(2n - 2)(2n - 4)\cdots 4 \cdot 2}{(2n - 3)(2n - 5) \cdots 3 \cdot 1} \simeq 1.7 \sqrt{n}$ Could you give some hint ...
1
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1answer
61 views

Approximations. Newton's method - composite Simpson's rule

Can you help me please to solve this problems and if you can give me some helpful information. Thanks!
0
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0answers
49 views

Approximating piecewise linear function

I'm trying to derive an analytic approximation to the following piecewise linear function: $$ f(x) = \left\{ \begin{eqnarray} \frac{x}{x_s} & & \text{if} & x < x_s \\ ...
0
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0answers
35 views

Approximating $C^2$ functions with compactly supported $C^2$ functions

Let $C^2$ be the space of twice-continuously differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$ and $C^2_K$ be the subset of functions in $C^2$ with compact support (that are zero outside ...
0
votes
1answer
154 views

Approximation to the square root

I was reading an article that approximated a square root operator as follows $\sqrt{1+x+y} \cong \sqrt{1+x} + \frac{1}{2}y + O(xy,y^2) $ At first glance that looks like a Taylor series expansion, ...
5
votes
1answer
176 views

Subdifferential boundary conditions: Testing with $L^2$ or $H^{1/2}$ functions

My question was essentially this: Does it make a difference if I test subdifferential boundary conditions with functions from $L^2(\Gamma)$ or $H^{1/2}(\Gamma)$? In the following, I will phrase the ...
3
votes
2answers
111 views

Approximate $|x|$ with a smooth function

I am trying to get the derivative of $|x|$, and I want that derivative function, say $g(x)$, to be a function of x. So it really needs the |x| to be smooth (ex. $x^2$); I am wondering what is the ...
10
votes
5answers
339 views

Why is $e^\pi - \pi$ so close to $20$?

$e^\pi-\pi\approx 19.99909998$ Why is this so close to $20$?
0
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0answers
32 views

Padé approximant of transfer function with gain and time delay.

$$ H(\omega) = A e^{-j \omega \tau} $$ I'm trying to use Padé approximation to generate a numerator and denominator polynomial for the above transfer function but genuinely struggling with how to ...