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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58 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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0answers
74 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
17 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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40 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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0answers
23 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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59 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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39 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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98 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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56 views

How do we know that the first few digits of an approximation for $\pi$ are correct?

For Gregory–Leibniz series, wikipedia has - "after 500,000 terms, it produces only five correct decimal digits of π.". But how do you know that those five decimal values are correct when you reach ...
2
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45 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. ...
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48 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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142 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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55 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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98 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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203 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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35 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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38 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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79 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
32 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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54 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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244 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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60 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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63 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
122 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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286 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
121 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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90 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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50 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
132 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
49 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 ...
2
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0answers
570 views

Approximation of integral using series expansion of the integrand.

I have a smooth function $x \rightarrow f_\epsilon (x)$ on $x\in[-1\ldots 1]$ (dependent on the continuous parameter $\epsilon$) and I want to approximate the integral $$ I=\int_{-1}^1 f_\epsilon ...
2
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0answers
441 views

Polynomial Interpolation and Error Bound

Problem: Use the Lagrange interpolating polynomial of degree three or less and four digit chopping arithmetic to approximate cos(.750) using the following values. Find an error bound for the ...
2
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0answers
145 views

Generating all positive integers from three operations

This question arose on sci.math, but since almost all the competent mathematicians there have migrated here, I thought I'd give the question a wider audience. Starting with an integer $t>2$, ...
2
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114 views

explicit error bounds for Multivariate interpolation

I want to interpolate a function of $d$ variables over a Cartesian grid, using multivariate interpolation, while characterizing interpolation error in terms of bounds on partial derivatives of the ...
2
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0answers
74 views

On bounding the average cost of top-down merge sort

Let $A_n$ be the average number of comparisons to sort $n$ keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that $$ A_0 = A_1 = 0;\quad A_n = ...
2
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0answers
73 views

Algorithm to predict next 3D points

For example, having this data: year x/y/z 2007 10/20/70 2008 20/10/70 2009 30/10/60 2010 40/10/50 2011 40/15/45 We want to predict what will be the x/y/z in ...
2
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0answers
146 views

approximation of the sum

I have difficulties to find an approximation formula (or bound from the below) for the following sum: $$ \sum_{k=1}^n\left( \frac{1}{35}\right)^{k-1}(n-k)!\left(k-\frac 32\right)!. $$
2
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116 views

Approximating sums like $\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$

Can anyone tell me how to approximate the following functions? $f_3(n) = \displaystyle\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$ $f_4(n) = ...
2
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0answers
180 views

calculate the rate of change

I am trying to calculate the change frequency for a set of data. Each bit of data has the date-time it was created. I would like to say for a specific set of data the change frequency is hourly, ...
2
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0answers
219 views

When is it valid to convert a function inside a probability integral to the indicator function?

I am faced with an approximation that replaces a probability density function with the indicator function and I am at a loss as to why this is valid. We want to model the lifetime $T$ of a website ...
2
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0answers
284 views

Multivariate B-Spline Derivatives

To construct a multivariate B-spline, we simply take the Kronecker tensor product between the univariate basis functions constructed for each individual dimension. What I'd like to know is how do you ...
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18 views

Trapezoidal Rule yielding the exact value of the integral

It is clear that if a function $f(x)$ is linear over the domain $a \leq x \leq b$, then one application of the trapezoidal rule, over the same domain, will yield the exact value of ...
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21 views

Please check this perturbation solution of polynomial root and truncation order.

I have a quintic polynomial where the coefficients depends on a parameter $c$, i.e. $$ a_0(c)+a_1(c)x+a_2(c)x^2+a_3(c)x^3+a_4(c)x^4+x^5 $$ I know that the roots of the polynomial are real and ...
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0answers
18 views

How to compare experimental data with teorethical prediction

I would like to know, what is the method to approximate experimental data to teorethical one. I have heard about polynomial regression. After calculating particular matrices and solving set of ...
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0answers
53 views

Is there an analytic approximation to integral of this form?

Started working on trying to find an analytical approximation to this integral and not getting very far. Any assistance or direction is greatly appreciated! Thanks Vince $$\int_{0}^{t} ...
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0answers
76 views

Derivation of approximation of Error function

In Abramowitz and Stegun there are some formulas for approximation of Error funtion. I am intrested in the formulas $7.1.25$ to $7.1.28$, here is one of them ($7.1.26$). ...
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40 views

Iterative algorithms to approximate unknown multivariate convex real-valued functions by a set of linear upper/lower bounds

I am looking for iterative algorithms that can approximate a multivariate convex real-valued function $f(\vec{x})=y,\, \Bbb{R}^n\rightarrow \Bbb{R}$. The function is not known beforehand, but it is ...
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68 views

Interesting approximation of distribution of numbers in a Farey sequence

I was investigating the distribution of the numbers in a Farey sequence and found some pattern. It is known that the number of elements in Farey sequence can be found using Euler totient function. So ...
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0answers
18 views

How do we derive efficiency from robustness in the virtual ant solution to the traveling salesman problem?

Using virtual ants/swarm intelligence to solve the Traveling Salesman Problem is an example of using a robust system to solve an efficiency problem. We normally think of robustness and efficiency as ...