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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18 views

How should i apply Richardson Extrapolation?

I trying to understand how the Richardson Extrapolation works, and what it is good for. The internet has lots articles about the this, but they all seem to lack in what it is useful for. I wanted ...
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0answers
20 views

approximation of function by polynomials

Given a function $f \in L^2[a,b]$, it can be written as $f(x)=\sum_{n=0}^\infty c_nL_n(x)$. where $L_n(x)$ is shifted Legendre polynomial. I am taking the finite sum to approximate. If I take some ...
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2answers
19 views

Approximation Reasoning

I can't understand one step in the following problem. We start with a function $f(x)=x^\alpha$ on the interval $(0,1)$ where $\alpha>0$ is a constant. We pick two points $x_1<x_2$ from this ...
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0answers
15 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 ...
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1answer
50 views

Prove that $y-x < \delta$

In Hardy's Pure Mathematics it says if $x^2<2, y^2>2, 2-x^2 < \delta,$ and $y^2 - 2 < \delta$, then $y-x<\delta$. I added the last two inequalities to get $(y+x)(y-x)<2\delta$. How ...
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1answer
35 views

error bound in function approximation algorithm

Suppose we have the set of floating point number with "m" bit mantissa and "e" bits for exponent. Suppose more over we want to approximate a function "f". From the theory we know that usually a ...
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1answer
35 views

approximation of $\pi$ by $\arctan$

Determinate the order n of the Maclaurin polynomial for $f(x)=4tan^{-1}x$ so that the remainader term $|R_{n}(1)|<0.000005$. Here $R_{n}(1)=\frac{f^{(n+1)}(c)}{(n+1)!}$ for some c between 0 and 1 ...
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1answer
47 views

Help me approximate this sum: $S = \sum_{j=2}^{N}{\frac{\ln \ln \ln \ j}{( \ln \ln \ j)^2}}$

I would like to figure out the asymptotic rate of growth for the sum $S = \sum_{j=2}^{N}{\frac{\ln \ln \ln j}{( \ln \ln j)^2}}$ in the limit of large $N$. Ultimately, I want to know if $S(N)$ is ...
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5answers
516 views

Why is it that this gives a good approximation of $\pi$?

At the end of a Physics examination, I decided to play around with my calculator, as I always do when I have time left over, and found that: $$\frac{1}{100} \cdot 11^{\ln(11)} \approx ...
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0answers
65 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 ...
2
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1answer
72 views

Approximation for probability of at least $t$ events

I'm reading through a paper, and they are discussing the approximate probability that $t+1$ out of $t^b$ events occur, where $b$ is a constant, and the probability of each event occurring is ...
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2answers
48 views

How can I approximate a decimal with two fractions where denominator is less or equal to $d$

I was looking for a way to approximate a decimal number with a fraction, whose denominator is less or equal to $d$. Basically, having a decimal $X$, I want to find two fractions such that ...
1
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1answer
22 views

Singular Value Decomposition for an image understanding

I'm trying to get an intuitive understanding of what an SVD decomposition does to an image. From my understanding, for an image $A \in \Bbb R^{m \times n}$, the singular values are the roots of the ...
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0answers
22 views

intersection of 2 paths

Could somebody please advise me on the best approach to take to find the intersection points of 2 paths. Each path is a data set of distance against time (I want to avoid curve fitting) and each path ...
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1answer
23 views

Approximating the circumference of given ellipse

Say we got the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{24}=1$, and the goal is to find the circumference using line integrals. So I parametrized the curve by $x=5\cos(t)$, $y=\sqrt{24}\sin(t)$. By ...
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1answer
20 views

How is optimal coordinates change chosen for Chebyshev expansion?

I'm looking into SLATEC implementation of Bessel function $J_0$ computation (readable in C in GSL), namely at its part for arguments in interval $(0,4)$. There a Chebyshev expansion is used, but the ...
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2answers
40 views

inverse complementary error function values near 0

$\newcommand{\erf}{\operatorname{erf}}\newcommand{\erfc}{\operatorname{erfc}}$Let's define for each $x>0$ $$\erf(x)=\frac {2}{\sqrt{\pi}}\int_0^xe^{-t^2} dt$$ and $$\erfc(x)=\frac ...
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0answers
32 views

Amenable groups are sofic

I am trying to understand proposition II.3.1 of http://arxiv.org/pdf/1309.2034v6.pdf, but I have some difficulties. I get the construction of the permutations $\sigma_{\gamma}$ for each $\gamma\in ...
3
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1answer
71 views

Higher Order PDE using Finite Difference

How to approximate higher-order partial differential equation using finite difference method? $$\frac{\partial^{2} y}{\partial t^{2}}+\frac{\partial^{4} y}{\partial x^{4}}=0$$
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0answers
21 views

System of Equations & Approximations

I am trying to derive Eq. (3.6) in the following thesis: http://drum.lib.umd.edu/bitstream/1903/14898/1/Khalil_umd_0117E_14726.pdf This is the equation I am trying to show: \begin{equation} ...
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1answer
21 views

Uniqueness of function approximation over three points?

Given a function $f(x)$, we want to approximate $f$ using $P(x)$, such that: $P(x_0) = f(x_0)$, $P(x_2) = f(x_2)$, $P'(x_1) = f'(x_1)$. Prove that such a $P$ is unique $\iff$ $x_1 \neq ...
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1answer
38 views

What can be said about $f''$ if the trapezoidal approximation is always an overestimate?

For any $a$ and $b$ the Trapezoidal approximation of the integral $\int_a^b f(x)\,dx$ is an overestimate. What can you conclude about the second derivative of $f$? I think it might mean that the ...
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1answer
39 views

Under which condition is $\hat\Sigma\approx\frac{1}{T-1}(X'X)$

Let $X$ be $T\times N$ random matrix. We are interested in the sample variance covariance matrix of $X$. It holds that \begin{align} ...
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2answers
36 views

Approximate summation of flooring function

I have this summation: $\sum_{k=0}^x \lfloor{\frac{k}{c}}\rfloor$ Do you have any ideas on any general expressions that can approximate this? P.S I know I can approximate it with a Fourier ...
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0answers
14 views

selecting points on a domain which represent the derivative of a function

I'm working on some algorithm part of which entails me to subdivide a domain based on the derivative of a function. Let's just consider the 1D case with a closed and bounded domain $[a,b]$ for a ...
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0answers
17 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 ...
4
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1answer
70 views

Is there an alternative to Taylor expansion of functions with more control over the error distribution?

As a physics student I see the Taylor series being (ab)used very often, mostly for the expansion of $\exp(x)$. The usual line of though is that the error (remainder) term of a high order is likely ...
3
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2answers
49 views

Asymmetric second difference quotient?

I need to find (approximate) the second derivative of a discrete function. Usually I would approximate the second derivative with $$f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2}\tag{1}$$ In my case, ...
4
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1answer
38 views

Order of Growth of a Sum

Let $n>k$ and $r$ be arbitrary positive integers. Define $q=k/n$. I want to show that $$ \sum_{i=0}^{rk} \binom{rn}{i}q^i(1-q)^{rn-i}(rk-i)=\Theta(\sqrt{r}) $$ as $r\rightarrow \infty$. I've ...
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3answers
105 views

Approximating $e^{x}/(e^{x} - 1)$

Is it correct to tell that we can approximate \begin{equation*} \frac{e^{x}}{(e^{x} - 1)} \end{equation*} by: \begin{equation*} \frac{1}{x} \end{equation*}
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0answers
26 views

How to approximate (with minimal error) a 16-dimensional linear equation system with inaccurate solutions?

I have a system of linear equations with 16 coefficients that I'm trying to solve. I have an effectively unlimited number of equations using those coefficients (i.e., ax1+bx2+...+p*x16=y). I know how ...
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3answers
33 views

Maclurin Series. (Approximation)

Given that $y=\ln \cos x$, show that the first non-zero terms of Maclurin's series for $y=-\frac{x^2}{2}-\frac{x^4}{12}$. Use this series to find the approximation in terms of $\pi$ for $\ln 2$. My ...
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1answer
47 views

Sofic groups alternative definition

I am trying to solve exercise two from the following document : http://mtm.ufsc.br/~daemi/soficworkshop/Course%20notes/Lupini%20Lecture%202.pdf I suspect there is an error in the exercise, but I'm ...
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0answers
38 views

Why do I get a big error when I compute this integral with Gauss-Legendre Quadrature?

I'm using Gauss-Legendre Quadrature to solve the following integral: $\int_0^{1}x^xdx$ After I've compared the result with the MatLab vpa(int(...)) of the same integral I've noticed that the ...
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2answers
61 views

find which two points an arbitrary point is nearest to

I have a line segment of connected points (a path in 2D), and a point $P$ that is not calculated based on this segment, although I can guarantee that the point will be placed along the path. Based on ...
3
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1answer
49 views

Alpha max plus beta min algorithm for three numbers

There exists fast algorithm to approximate length of 2D vector - Alpha max plus beta min algorithm. It says that $\alpha\cdot\max(x,y)+\beta\cdot\min(x,y)\approx\sqrt{x^2+y^2}$ for some constants ...
2
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2answers
43 views

$\int_2^x\frac{dt}{\log^kt}=O\left(\frac{x}{\log^kx}\right)$

I seek to prove the identity $$\int_2^x\frac{dt}{\log^kt}=O\left(\frac{x}{\log^kx}\right)$$ I was given the following hint: Split the integral into $\int_2^{f(x)}+\int_{f(x)}^x$ for a ...
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0answers
12 views

Approximating a grid-valued signed distance function with a continuous function

I want to solve a continous optimization problem using IPOPT. My optimization involves a signed distance function whose values are defined on a 2D grid. Since IPOPT can't handle piecewise functions, I ...
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1answer
37 views

Using SVD to approximate matrix-vector multiplication?

Given some matrix A, is it possible to use Singular Value Decomposition to approximate Ax for some vector x within some error bound? According to Efficient low rank matrix-vector multiplication, it ...
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1answer
53 views

Expectation of trigonometric functions involving random variables.

This is more a formulation question. I need help making a sales pitch (lol). I am working on an practical engineering problem where I encounter functions of the form: $\cos(\phi + d_\phi)$, $ ...
1
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0answers
24 views

how to approximate compute p(a|b, c) , using p(b|c), p(a|*, c), p(a), p(b), p(c)

how to approximate compute p(a|b, c) , using p(b|c), p(a|*, c), p(a), p(b), p(c) p(a|*, c), * means anythins. p(a|*, c) = $\sum_i p(a|i, c) $ error = | p(a|b, c) - f(p(b|c), p(a|*, c), p(a), p(b), ...
3
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1answer
30 views

Trapezoidal Rule in Stirling's Approximation

https://en.wikipedia.org/wiki/Stirling's_approximation Here $$\sum \limits_{i=1}^n log(i) $$ is approximated as $$\int^n_1log(x) dx\ +\ 1/2\ log(n)$$ but I would approximate it as ...
4
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3answers
117 views

Calculate fractional part of square root without taking square root

Let's say I have a number $x>0$ and I need to calculate the fractional part of its square root: $$f(x) = \sqrt x-\lfloor\sqrt x\rfloor$$ If I have $\lfloor\sqrt x\rfloor$ available, is there a ...
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0answers
7 views

Mapping Function from Generated Data

Assuming solution space data is 1:1 mapped to domain space, is there a good/well known approach to flushing out a mapping function/approximation having access to lots of mapped data? I imagine I am ...
0
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1answer
31 views

Finding the integer parts of irrationals

When working with continued fraction expansions, I sometimes have to calculate the integer part of irrationals quickly without a calculator, what would be an effective way to do this? For example, ...
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1answer
22 views

Approximation Error of Stirlings Formula

Stirlings Approximation : $n! \approx \sqrt{2\pi}n^{n+\frac{1}{2}}e^{-n}$. So $100!$ has an approximate percentage error of about $\frac{100}{12n} = \frac{1}{12}$. Using this information, how does ...
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2answers
59 views

what is the best approximation for sine?

can you tell me which is the best approximation for cosine/sine functions. It should also reduce the computational complexity. I've already tried the Bhaskara-1 approximation. Can you suggest me ...
2
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1answer
37 views

Approximating non-rational roots by a rational roots for a quadratic equation

Let $a,b,c$ be integers and suppose the equation $f(x)=ax^2+bx+c=0$ has an irrational root $r$. Let $u=\frac p q$ be any rational number such that $|u-r|<1$. Prove that $\frac 1 {q^2} \leq |f(u)| ...
1
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1answer
14 views

Linearization of an implicitly defined function

$f(x,y,z)=e^{xz}y^2+\sin(y)\cos(z)+x^2 z$ Find equation of tangent plane at $(0,\pi,0)$ and use it to approximate $f(0.1,\pi,0.1)$. Find equation of normal to tangent plane. My attempt: I found that ...
7
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1answer
160 views

Accelerating approximations for arccos

I have recently built a method to accelerate drastically the accuracy of the following approximation of $\arccos(x)$ : $f_n(x)=2^n\sqrt{2-2g^{n-1}(x)}$ where $g(x)=\frac{1}2\sqrt{2+2x}$ and ...