For questions related to approximations

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38 views

Expanding the Exponential at Infinity

Can anyone give me a good approximation to $e^{-ax}$ for large $a$? This is based on trying to find a good approximation to $$ \int\frac{dx}{\sqrt{x}}e^{-\left(x^{2}+a x\right)} $$ when $a$ is ...
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0answers
26 views

Approximation of sum with binomial summands

I am new here, so hopefully my question will be understood correctly. I have a function (originating from expected untility theory in economics) that looks the following: ...
2
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1answer
61 views

Approximating hypergeometric function F(1,1+a,2+a,z) for z->1

in my studies a normalization constant for a pmf includes the hypergeometric function ${}_2F_1(1,1+a,2+a,z)$ The parameters are in the range $0.99<z<1$ and $0<a<5$. I have tried some ...
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0answers
28 views

Approximation of eigenvalue

I'm having trouble with the following problem: Let $A\in M_n(\mathbb{C})$ be a Hermitian matrix, $\mu\in\mathbb{C}$, $\epsilon>0$ and $x\in\mathbb{C}^n$ such that $\mid\mid x\mid\mid=1$ and ...
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3answers
33 views

An approximation: 2*number of events = twice the probability

My statistical mechanics textbook uses an approximation to derive a well-known result. The approximation is: Suppose for an event, the probability of an outcome is P. For n events, the probability of ...
6
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2answers
175 views

Find the integral part of $\sum_{i=2}^{10000}\frac1{\sqrt{i}}$

$$A = \frac1{\sqrt{2}}+\frac1{\sqrt{3}}+\cdots+\frac{1}{\sqrt{10000}}$$ Find $\lfloor A\rfloor$ where $\lfloor x\rfloor$ is the greatest integer less than, or equal to $x$ I got stuck on this, so ...
2
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1answer
46 views

Approximation of a factorial

In books I can find a lot of approximations of factorial, which doesn't require the a lot of multiplications, like for example Stirling's approximation. Very popular is also the Euler's solution which ...
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1answer
41 views

Forcing coefficient of best fit quadratic

So I have a question regarding best-fit polynomials. Currently, using matrices and then Gaussian elimination, I come up with f(x) = Ax^2 + Bx + C and use the derivative of that fit to come up with the ...
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0answers
27 views

Matching First and Second Derivatives: Taylor Series

I have $$ f(x) = \sqrt{\ln\left(a\cosh^{2}(mx)(1+bx^{2})\right)} $$ If I expand this as a series I should get something of the form $$ \sqrt{\ln a}+gx^{2}+\mathcal{O}(x^{4}) $$ but I'm having ...
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0answers
40 views

Approximation for the logarithm of a summatory

I would like to find an approximation for: $$ \log \left(\sum_{i=1}^{N} a_i\exp(-b_i^2)\exp(-c_i^2)\right) $$ with $$ a_i = \frac{1}{\sqrt{(e^2 + e_i^2)(g^2 + g_i^2)}} \\ b_i = \frac{b-d_i}{2(e^2 + ...
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0answers
44 views

Does a complex polynomial can be approximated by a Euler-type exponential function?

Can one uniformly approximate a complex polynomial with a Euler-type exponential function? I mean: let $E\subset\Bbb C$ be a compact set with the connected complement $\Omega:= \bar {\Bbb C}\setminus ...
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1answer
33 views

MatLab: estimate number of iterations

I want automatically estimate iterations number in matlab. Suppose we have for(int i = 1; i < N; i++). It's clear that for-loop prodices ...
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0answers
17 views

approximation by superpositions of a sigmoidal function

I have a question about superposition of sigmoidal function with limited bound. Let $f$ be a signmoidal function with amplitude $\alpha$. For example, let's assume we have tanh(): $f_i=\alpha \cdot ...
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0answers
10 views

Correlation 4-point

I need to calculate $\langle x_{i}x_{j}x_{k}x_{l}\rangle $, where $$ \langle f(x) \rangle = \int e^{-\frac{1}{2}A_{ij}x^{i}x^{j} - \frac{\lambda }{4!}\sum_{i}x_{i}^{4}} f(x)d^{n}\mathbf x , $$ for ...
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0answers
21 views

Copying the Curvature of One Function onto Another: Approximation

I have a polar function $$ r(\theta)=\left(r+\epsilon\right)\cos(\theta)-\sqrt{r^{2}-\left(r+\epsilon\right)^{2}\sin^{2}(\theta)} $$ Is it possible to methodically conjure another polar function ...
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0answers
30 views

Using Polars to Approximate a Cartesian line: Approximating an Integral

I have the equation of the lower semicircle of radius $r$ centred at a distance $a+r$ above the x-axis $$ f(x)=r+a-\sqrt{r^{2}-x^{2}} $$ which I can approximate (for small $x$) as $$ f(x)\approx ...
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2answers
18 views

Divide $n$ seats by a list of $\mathbf{w}$ weights proportionally.

I have $n$ number of seats and I have list of weights $\mathbf{w} \in \mathbb{R}^{k}$ which is a probability distribution with $k$ possible values, $\sum_{i=1}^k{w_i}=1$. I want to divide the $n$ ...
0
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2answers
38 views

How to use $t(29/\sqrt{2})<0$ where $t(x)=x^2-41x+420$ to prove that $41/29<\sqrt{2}<42/29$??

So I was investigating different ways to approximate $\sqrt{2}$. Here's my latest: $$Let:t(x)=x^2-41x+420$$ then the roots of $t(x)$ are $20$ and $21$. I showed that then $t(x)=(x-20)(x-21)$ and ...
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2answers
170 views

Evaluate $\sum_0^\infty \frac{1}{n^n}$

Courtesy of this xkcd comic I now know that $$ \sum_{n=1}^\infty \frac{1}{n^n} \approx \ln^e(3) $$ Echoing the views of the comic itself, if I ever find myself taking $\ln^e(x)$ then something has ...
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0answers
29 views

How to evaluate the derivate of a hypergeometric function w.r.t. one of its parameters?

I have to numerically evaluate the derivative of the hypergeometric function w.r.t. its first and second parameters $\large\frac{\partial}{\partial a}{_2F_1}\left(a , b ,c;z\right)$ and ...
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1answer
33 views

Approximations to the Roots of a Function

I want to find approximations to the root of a function in two variables using the Newton-Raphson method. I can use the method on a function in a single variable but I'm lost as to how you can use it ...
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1answer
94 views

Weierstrass Approximation Theorem for $\Bbb C$

The Weierstrass approximation theorem states that any continuous function $ f : I \rightarrow \Bbb R $ on a closed, bounded, connected subset $ I \subseteq \Bbb R $ can be uniformly approximated by ...
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3answers
465 views

Geometric explanation of $\sqrt 2 + \sqrt 3 \approx \pi$

Just curious, is there a geometry picture explanation to show that $\sqrt 2 + \sqrt 3 $ is close to $ \pi $?
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4answers
120 views

Approximation to $\sqrt{\cos(\theta)}$?

I have this formula, (it is just the law of cosines angle formula): $$ d = \sqrt{a^2 + b^2 - 2ab \ cos(\theta)} $$ Here is my issue. I am wondering if there is a way to 'extract' the $cos$ term. My ...
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0answers
32 views

Is there a way to expand Re(Li(a^z)) in series?

I'm searching a way to expand $ f(z) = Re(Li(a^z)), a \in R, z \in C $ in series. The computer-friendly, quickly convergent series is a huge plus. For being 'computer-friendly' I mean a relatively ...
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2answers
35 views

Approximating 'big' ratio with 'small' ratio

Given a ratio $ \frac{m}{n}, p \in N, q \in N $ where either $m$ or $n$ (or both) is a very big number, how can we find a ratio $ \frac{p}{q}, p \in N, q \in N $ which estimates $ \frac{m}{n} $ up to ...
0
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0answers
49 views

Minimax approximation

Problem: Give an example to show that the minimax approximation operator $X(f)$ is not linear. So, if $f$ is any function, I have tried to find $g$ such that $X(f+g)=X(f)$ but $Xg$ is not the zero ...
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1answer
64 views

Why is Simpson's rule exact for cubics?

I can't understand: Why is Simpson's rule exact for cubic polynomials?
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1answer
100 views

Approximating Logs and Antilogs by hand

I have read through questions like Calculate logarithms by hand and and a section of the Feynman Lecture series which talks about calculation of logarithms. I have recognized neither of them useful ...
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0answers
28 views

Time Discretization

I wonder why we work with constant discretization in Time Discretization of numerical approximation for numerical scheme if we take not necessarily constant Discretization Is the numerical scheme ...
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0answers
28 views

Counterexample to smooth approximation of sobolev function on closure of set without $C^1$ boundary

I'm working through the following problem, and I just need a hint to finish it I think. Consider the set $\Omega = B(0,1) \backslash \left\{x\in \mathbb{R}^N : x_N = 0 \right\}$. We are given the ...
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0answers
43 views

Proof for ACYCLIC PARTITION being an NP-complete problem

I'm new to this site, so please pardon me for any mistakes and please feel free to edit the question to help get better answers. I'm interested in reading any proof of ACYCLIC PARTITION (Garey and ...
0
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1answer
29 views

Absolute error computation

Given is a function $f(x)=4x^{2}$, which we want to evaluate for $x\in \left [ 1,2 \right ]$, $\widetilde{x}\in \left [ 1,2 \right ]$ is the approximation of $x$. What can be the value of the ...
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0answers
25 views

Approximately minimising a transcendental function.

I currently have a closed form solution for the error probability of a certain type of wireless channel. By letting all $S_i$ terms denote constants, using $U(\cdot,\cdot,\cdot)$ to denote the ...
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1answer
45 views

What is a good approximation of $(1-p)(1-q)$ as $(1-x)^2$, for $p,q \in (0,1)$?

I'm doing some scientific modeling, and I want to use $(1-x)^2$ to approximate $(1-p)(1-q)$, with $p, q \in (0,1)$. $p$ and $q$ are probabilities, and are not near zero. My intuition is that since ...
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0answers
3 views

Multidimentional Scaling with Pairwise distance “vectors”

Consider a random variable $z$ with a Gaussian distribution : $$ \mathbf{z} \sim \mathcal{N} ( \mathbf{m}, \mathbf{V} ) $$ Where $\mathbf{m}$ and $ \mathbf{V}$ are mean and variance parameters. ...
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2answers
223 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
2
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1answer
77 views

Approximation of indicator function of an open set by continuous functions

Let $(X,\mathcal{T})$ be a locally compact separable Hausdorff space and $A \in \mathcal{T}$ open. Does there exist a sequence $(f_n)_{n \in \mathbb{N}}$ of (bounded) continuous functions such that ...
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1answer
53 views

Approximating Trig Functions with Polynomials

I was thinking about the graphs of different trig functions and noticed that most of them are of a similar shape to some different types of polynomials. For example: Higher degree polynomials create ...
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4answers
62 views

Approximating $\ln{\frac{x-y}{x+y}}$

For $x>>y$, $$\ln{\frac{x-y}{x+y}} = \ln{\left[ x\left( 1-y/x \right) \right]} -\ln{\left[ x\left( 1+y/x \right) \right]} \approx -2\frac{y}{x}$$ However, the following does not work: ...
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1answer
36 views

A power approximation function

I am trying to construct a function that would approximate $a^b$ using Maclaurin series. Here are my reasoning: Since $$a^b=e^{b\ln a}$$ and $$e^x=\sum^{\infty}_{k=0} \frac{x^k}{k!}$$ it should ...
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1answer
32 views

Show $\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx<\frac{1}{aM^{a+b}e^{M^a}}$?

is there a way to show the inequality \begin{equation}\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx<\frac{1}{aM^{a+b}e^{M^a}}\end{equation} for positive constants $M$ and $C$ ...
3
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0answers
297 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ ...
2
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1answer
127 views

Stone-Weierstrass theorem with $p(1/x)$

I am trying to prove that for a continuous $f\colon[1,\infty)\rightarrow\mathbb R$ and $f(x)\to a$ as $x\to\infty$ it could be approximated by $g(x)=p(1/x)$ where $p$ is a polynomial.
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0answers
79 views

Sharpness of the upper bound $(1-x)^n \leq 1 + \frac{nx}{2}$

Here is a known inequality: $$(1-x)^n\leq 1+\frac{nx}{2}\qquad \text{for} \, \frac 1n\leq x\leq 1 $$ I am wondering if there is a better upper bound than this? Thank you.
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0answers
41 views

Efficient approximation of derivatives of an integral

Suppose $ \phi(z) $ is the probit function (http://en.wikipedia.org/wiki/Probit). And $$ Z = \int \phi(\mathbf{w}^\top \mathbf{x}) \mathcal{N}(\mathbf{w}; \mathbf{\mu}, \mathbf{\Sigma}) d\mathbf{w} ...
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149 views

Are there known pairs of simple numbers equal to huge precision, but not equal strictly?

Are there known pairs of numbers $a$ and $b$, which at first look at them seemed likely to be equal, and after checking up to $10^n$ decimal places appeared to agree, but suddenly for some $n$ they ...
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0answers
45 views

Approximating an IVP

I wish to solve the IVP: \begin{align} x(0) =& -1 \\ x' =& 1 + x^2 - t^3 \end{align} With a fourth order taylor series method, I solved the ODE on the interval [0, 2] and then made the ...
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0answers
23 views

How to find a function that can approximate another blackbox function programmaticly?

This question has been posted on http://stackoverflow.com/questions/21758016/how-to-find-a-function-that-can-approximate-another-blackbox-function-programmat I was suggested to post it here. I ...
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1answer
51 views

How to approximate L^1[0,1] functions?

Do functions on a uniform grid with n points in the interval $[0,1]$ approximate $L^1[0,1]$ functions, as $n \to \infty$? I want to sample functions in $L^1[0,1]$ space numerically and I want to be ...