For questions related to approximations

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1answer
30 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 ...
2
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1answer
68 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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4answers
392 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
105 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
31 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
33 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 ...
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0answers
40 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 ...
3
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1answer
55 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
57 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
20 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
26 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
40 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
21 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 ...
0
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0answers
18 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 ...
0
votes
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
2 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
111 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
58 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 ...
1
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1answer
36 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 ...
2
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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: ...
0
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1answer
69 views

ln-exp approximation

I am looking for an approximation to the following expression: $\ln (1+e^{-x})$ If $x$ is small then it is not a problem. However, is there a (polynomial, rational) approximation for relatively ...
0
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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 ...
0
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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
270 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
100 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
67 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
38 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} ...
9
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0answers
136 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 ...
1
vote
1answer
42 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
17 views

Help find a name of used method

Consider some function $f(x)$ is provided as a formula or 2D plot. In my software application I virtually split this function to fixed-sized ($\Delta t$) intervals, so I get a sequence of $(f_i, ...
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0answers
19 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 ...
0
votes
1answer
48 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 ...
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0answers
26 views

Find allowed error of an argument regarding the allowed error of a function.

To what precision can $x$ be obtained with logarithmic table (with $5$ digit table) if $x$ lies between $300$ and $400$? Any ideas?
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0answers
41 views

Probability of a sample from a random variable with Gaussian distribution

I am studying a paper [1] which states that, as far as I understand, the probability of a single sample $x$ taken from a random variable $X$ with Gaussian distribution equals the Gaussian distribution ...
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0answers
38 views

Approximating the probability that a range bounds a given number, with very large numbers

Let $m$ numbers be chosen uniformly from $0,\dots,n-1$ without replacement and then sorted in ascending order as $\ell_0,\dots,\ell_{m-1}$. Let there be $b,e,x$ such that $0 \le b \le e \le m$ and $0 ...
4
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0answers
54 views

Can we use a sum of residues to develop an asymptotic expansion for this unknown function?

In the course of solving a particular physical problem, I have derived a relationship between two unknown functions: $$ f(s) = \frac{s \sinh{\frac{\pi s}{2}}}{2 \pi i \beta} \int_{-c- i ...
1
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1answer
35 views

How to approximate a smooth function

Now I have a target smooth function f which is infinitely differentiable over $R^d$, $f \in C^{\inf}(R^d)$. $f = \Sigma c_ig_i(x)$, where $c_i$s are unknown coefficients and $g_i(x)$ is a smooth ...
4
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4answers
86 views

$\sin x$ approximates $x$ for small angles

In physics, particularly in waves, we make use of the fact that for small angles (less than $\pi/12$-ish), the sine function value of an angle is pretty close to the value of the angle itself (in ...
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2answers
48 views

Why sum of sigmoids is a good approximation of softplus function?

According to this paper: Rectified Linear Units Improve Restricted Boltzmann Machines, $\sum_{i=1}^N \sigma(x-i+\tfrac{1}{2}) \approx \log(1+e^x)$ (equation 7) where $\sigma(z) = \frac{1}{1+e^{-x}}$ ...
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0answers
29 views

exponential approximation

I am reading about AdaBoost algorithm, I cannot understand (7). How can it be like this? If you want full document, you can get it at http://cseweb.ucsd.edu/classes/fa01/cse291/AdaBoost.pdf Thank ...
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0answers
21 views

approximating Gegenbauer polynomials (or ultraspherical or Jacobi)

Looking for hardcore orthogonal polynomial people here... If we hold the degree $\ell$ constant and take the order $\alpha$ to infinity, the Gegenbauer polynomial $G_\ell^{(\alpha)}$ approaches the ...
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1answer
70 views

From 'The Joy of x' book: John Hubbard and problems with multiple roots

My math skills are super rusty. In an effort to get some vigor back I started some reading and picked up The Joy of x based on its rave reviews.. I just couldn't make any sense out of the following ...
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0answers
59 views

Why some curious almost-identities

I read somewhere that $$e^{\pi\sqrt{163}}$$ is almost an integer and strangely enough this isn't just a random coincidence but rather there exists some general theory ...
0
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2answers
97 views

What is the advantage of using Fourier Series representation rather than the function itself?

Suppose we have a function $f$ defined over $[a,b]$ to the real numbers, i.e. $f: [a, b] \to \mathbb R.$. We can approximate this function as Fourier Series. Suppose $a_n, b_n$ is the Fourier series ...
2
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0answers
25 views

Non Approximation result

Say we have a constant approximation algorithm for the following objective: $$\min_x f(x) \;\;\;\;\;\; (1)$$ Now, we want to solve the following objective: $$ \max_x (N - f(x)) \;\;\;\;\;\; (2) ...
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0answers
55 views

Calculate a 5x5 Vandermonde system for a 5 point mesh

This is problem 1.2 in Randall J Leveque's textbook, "Finite Di fference Methods for Ordinary and Partial Di fferential Equations". I'm struggling with how to actually do the computation, I'm not so ...
0
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1answer
23 views

How to find sequence from set of values

I have set of probability values arranged in ascending order, p1<p2<p3<...<pM. Now I want to assign set of numbers in the same manner in which probabilities are increasing. It means I ...
0
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0answers
28 views

any approximation for $\sum_{i_3=3}^{n-k+3}\sum_{i_4=i_3+1}^{n-k+4}\sum_{i_5=i_4+1}^{n-k+5}\cdots\sum_{i_k=i_{k-1}+1}^{n} 1, (n \gg k)$?

Is there any approximation for $$\sum\limits_{i_3=3}^{n-k+3}\sum\limits_{i_4=i_3+1}^{n-k+4}\sum\limits_{i_5=i_4+1}^{n-k+5}\cdots\sum\limits_{i_k=i_{k-1}+1}^{n} 1, \quad (n \gg k)$$ ? We know that ...
1
vote
5answers
73 views

Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$

Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$ I know that $\prod_{i=3}^k (n-i) < \prod_{i=3}^k n = n^{k-2}$ Also a tighter upper bound is appreciated.
1
vote
2answers
38 views

Approximate minimum of function

Let: $$f(\Delta)=\sqrt{b \log ^n\left(\frac{E}{\Delta }\right)+4\frac{\Delta ^2}{E^2}},\ \ n=4,6$$ $$E,b,\Delta>0,\ \ \Delta \ll E$$ The function has a single minimum, but the explicit ...