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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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 ...
3
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64 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
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304 views

Minimizing maximum absolute column sum norm of the residual between a matrix and its $k$-rank approximation

Let $X \in \mathbb{R}^{m\times n}$ be a matrix with rank $r$. How can we find the optimal $\tilde{X} \in \mathbb{R}^{m\times n}$ whose rank is $k$ where $k\leq r$ and the reconstruction error in ...
3
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83 views

Is that observation really a property of the log of coefficients of continued fractions (example: cf(log(3)/log(2))

I'm again looking at the problem of approximation of perfect powers of 2 to that of 3 (I assume $\small q_N = 2^S / 3^N \gt 1 $) and specifically using the continued fraction representation of $\small ...
3
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282 views

Approximating a system of differential equations as a Bézier curve

I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve. Sorry if my ...
3
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277 views

Approximation of a real number as a linear combination of two reals with coprime integral coefficients

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
2
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56 views

Integral of a Gaussian with Trigonometric functions Involved

I am having a difficult time evaluating an integral unlike any integral I have seen before. To get right into things here is the integral: $$\frac{A}{\sigma_o\sqrt{2\pi}}\int_{-\infty}^\infty ...
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184 views

Why is $e$ close to $H_8$, closer to $H_8\left(1+\frac{1}{80^2}\right)$ and even closer to $\gamma+log\left(\frac{17}{2}\right) +\frac{1}{10^3}$?

The eighth harmonic number happens to be close to $e$. $$e\approx2.71(8)$$ $$H_8=\sum_{k=1}^8 \frac{1}{k}=\frac{761}{280}\approx2.71(7)$$ This leads to the almost-integer ...
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29 views

Meaure-theoretic induction: Why dyadic approximation?

In measure-theoretic induction proofs we always use the dyadic approximation of a non-negative measurable function $Y$ as $$Y_n = \sum_{k=0}^{n2^n-1} k/2^n 1\left(\frac{k}{2^n} \leq Y < ...
2
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27 views

Approximation of integrable function by polynomials

Assume $f\in \mathscr{R}(\alpha)$ on $[a,b]$, and prove that there are polynomials $P_n$ such that $$\lim \limits_{n\to \infty}\int_{a}^{b}|f-P_n|^2d\alpha=0.$$ Proof: Let $\varepsilon>0$ be given ...
2
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22 views

For $\sum_{n=1}^\infty (-1)^n\frac{(x-3)^n}n$ approximate when x = 4, with an error that does not exceed .01

For $\sum_{n=1}^\infty (-1)^n\frac{(x-3)^n}n$ approximate with specific details the series when x = 4, but with an error that does not exceed .01. That is, find a value of n so that the nth partial ...
2
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35 views

Textbook on Approximation

Does anyone know of a good book on approximations methods? In particular finding approximations to sums, integrals, etc via upper/ lower bounds, power series approximations, etc and bounding the ...
2
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24 views

Approximation of vector-valued Lipschitz functions

I am looking for methods of approximating vector-valued Lipschitz functions up to arbitrary precision, by convolution with heat kernels in particular. Here is what I am interested in. Let $f$ be a ...
2
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35 views

Approximate algorithms for integer linear programming (for optimal subset selection)

I'm trying to select an optimal subset of some items. I've tried 2 optimal approaches (branch-and-bound and integer programming) but both proved impossible for the size of the problem. I'm wondering ...
2
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58 views

Does Stone-Weierstrass apply in $\mathbb{R}$?

Does the Stone-Weierstrass theorem (Weierstrass approximation theorem) apply in $\mathbb{R}$. The Stone-Weierstrass theorem in $\mathbb{R}$ would be: Given continuous ...
2
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32 views

How are Floating Point approximations done by integer operations? (Source Wikipedia)

Please help me understand the mathematics involved in Wikipedia page of Floating point, section of Piecewise Linear approximation to exponential and logarithm. Following is the link Piecewise linear ...
2
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15 views

Problems with Exchange Procedure in Remez Algorithm

So first off: *** This code is not being used in production software. It is a personal project of mine, trying to understand approximation theory and advanced curve fitting. In other words, I'm ...
2
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55 views

Asymptotic Approximation

After analyzing performance of a cooperative system, I get the following expression for the system outage probability: $P = 1 - \frac{{{e^{ - 2\mu /{\beta _M}}}}}{{\Gamma \left( {{\alpha _3}} ...
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57 views

Approximate an integral with Bessel functions

Given $r,a,\lambda\in\mathbb{R}$, $r<a$, how can I find an approximate solution for the following definite integral? $$ \int_0^\infty J_0 (\lambda r)J_1(\lambda a)\frac{1}{\sqrt{n+\lambda^2 ...
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63 views

Is there a closed-form solution (even approximated) to this inequality?

I have the following function: $f(x, \theta) = (1-\theta)(x+1)^{-\theta}\left[ \frac{2-2\theta}{1- 2\theta} (N^{1-2\theta} - (x+1)^{1-2\theta}) - (x+1)^{-\theta}(N^{1-\theta} - (x+1)^{1-\theta}) ...
2
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81 views

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ...
2
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0answers
23 views

Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is ...
2
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56 views

Can trigonometric functions for double precision be implemented in terms of those for single precision?

In some program environments like GLSL there is support for single and double precision numbers for arithmetic and square roots computation, but only single precision trigonometric functions are ...
2
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30 views

Extension of laplace method's

It is well known that the integral $\int_a^b e^{-n \cdot f(x)}dx$ can be approximate by $\sqrt{\frac{2\pi}{n|f''(x_0)|}}e^{-n\cdot f(x_0)}$ at $x_0$ the maximum of $f(x)$ in $(a,b)$ (for large $n$..). ...
2
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0answers
65 views

Is there any way to justify this approximation of the solution of a rather simple equation?

I am considering the problem of finding the root of equation $$f(x)=-x+\sum_{i=m}^{i=n} \sqrt{x+i}=0$$ where $m,n$ can both be from very small to very large integers. Since, in a single calculation, ...
2
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43 views

Continued fraction approximation to a function and its derivative

I am recently working on fitting a model with incomplete beta function. In order to put it into my optimization algorithm, I must find out the derivatives of the incomplete beta function $B_p(x,y)$ ...
2
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0answers
101 views

Lower bound on a polynomial far from its zeros

Let $p(x) = \sum_{i=0}^{d}c_{i}x^{i} \in \mathbb{R}[x]$ and assume that all its zeros are real and in $[-1,1]$. I am interested in lower bounding the value of $|p(a)|$ in case $a \in [-1,1]$ is far ...
2
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0answers
55 views

Multivariate Appoximation

I have a mathematical model for a complex system which I would like to approximate it. My idea is to run this complex model once and produce some outputs, and then fit a polynomial for these outputs. ...
2
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50 views

How good an approximation to the derivative is an arc-length based approximation?

Note - my original definition below was wrong. I hope this replacement is better. The usual approximation to $f'(x)$ with step size $h$ is $D_h(f, x) = \frac{f(x+h)-f(x)}{h} $. This has so many nice ...
2
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0answers
185 views

How to find the approximate basic frequency or GCD of a list of numbers?

I could't actually summarize the question in the title, so I'll explain my situation. I want to tell the integer numbers which act as the best approximate basic frequencies of a list of real numbers: ...
2
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0answers
60 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
2
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0answers
105 views

Application of Weierstrass approximation theorem

How to approximate a continuous function to a desired accuracy using a polynomial? Theorem: For any $\varepsilon > 0$ and $f \in C([a,b])$, there exists a polynomial $p$ such that $\sup_{x \in ...
2
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0answers
338 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 ...
2
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0answers
36 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 ...
2
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41 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 ...
2
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0answers
94 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 ...
2
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0answers
34 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) ...
2
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60 views

Approximating sums

I got a general question, that is motivated by a recent problem. So let me first describe the problem and then add the general part: I got a rather simple (using only basic elements) equation, which ...
2
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0answers
308 views

Real approximation to the maximum using Laplace's method integral

The Laplace's Method states that under some conditions, it holds that: $ \sqrt{\frac{2\pi}{M(-g''(x_0))}} h(x_0) e^{M g(x_0)} \approx \int_a^b\! h(x) e^{M g(x)}\, dx \text { as } M\to\infty$ Where ...
2
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0answers
51 views

Is there a known algorithm for approximating all the real and imaginary zeros of any well behaved equation of a single variable?

Does there currently exist a general algorithm (or set of algorithms used together) that will approximate all the zeros of any well behaved non-differential equation of a single variable which has a ...
2
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0answers
62 views

Approximating a function using its integral

Question: Let $f:\Bbb R \to \Bbb R \in C^{1}, \forall \delta>0:$ $$F_\delta = \frac 1{2\delta}\int^{x+\delta}_{x-\delta} f(t) \, d(t)$$ in $[a,b]$ prove that $\forall \varepsilon>0 \exists ...
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0answers
69 views

Extending $W^{2,n}(\Omega)\cap C^2(\Omega)$-functions to $W^{2,n}(\Omega)$

I have the following theorem: Let $u\in C^2(\Omega)\cap W^{2,n}(\Omega)$ satisfy $Lu\geq f$ for $f\in L^n(\Omega)$. Then for any ball $B_{2R} = B_{2R}(y)\subset\Omega$ and $p>0$ where $u$ is ...
2
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0answers
131 views

Good upper bound on a binomial sum

What is a good upper bound on the following binomial sum: $$\sum_{i,j< \frac{m}{n}} {m \choose i}{m-i \choose j} z^{i+j}$$ where $z = \frac{1}{n-2}$?
2
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324 views

numerical approximation to logarithm

we know that $$ \ln(x)=\frac{1}{x} \int_{0}^{1}dt \frac{1}{1+xt} $$ then given a cuadrature formula inside $(0,1)$ is that true $$ \ln(x)= \frac{1}{x}\sum_{i}\frac{w_{i}}{1+xt_{i}} $$ wht other ...
2
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0answers
132 views

Weighted sum of ratio of partial sum of binomial coefficients

I would like to approximate the following sum when $n \rightarrow \infty$ and $n \gg k$, $$\sum_{x = k}^n \sum_{y > x}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m}}{\sum_{m = 0}^k {y - 1 \choose ...
2
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0answers
95 views

$\epsilon$-net of a $n$-dimensional $\ell_2$-ball

Let $B$ be an $\ell_2$-ball of radius $r$ in $\mathbb{R}^n$. I want to find the cardinal of a (not too big) $\epsilon$-net of $B$, that is the cardinal of a finite set $V\subset B$ such that $\forall ...
2
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0answers
432 views

Approximations of the incomplete elliptic integral of the second kind

For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a ...
2
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0answers
55 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 ...
2
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0answers
135 views

Approximation of complementary filter: Why is it a good approximation?

Having found some unofficial sources on Sensor Fusion (Thousand Thoughts Sensor Fusion and The Balance Filter by Shane Colton) I'm struck by how the approximation of the complementary filter that is ...
2
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0answers
54 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 ...