Questions tagged [convergence-acceleration]

Use this tag for questions about sequence transformations for improving the rate of convergence of a series.

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3
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1answer
155 views

Accelerated fixed-point for $x=\sin(x)$ convergence rate?

I happened to come up with an idea for accelerating the convergence of fixed-point iteration based on Aitken's delta squared acceleration method. What interests me is the case of $x=\sin(x)$, for ...
-1
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1answer
59 views

What are Padé approximants and how are they used for series acceleration?

I read an article about series acceleration on Wikipedia. I found in the non-linear acceleration section that Padé approximants can be used for series acceleration. But I am not able to understand ...
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0answers
13 views

Improvement of the convergence to the Dottie number?

Hi I study the following sequence : Let $0<a_0<1$ then the following sequence converges : $$a_{n+1}=\frac{\cos^{\alpha}(a_n)}{a_n^{\alpha-1}}$$ Where $0<\alpha<1$ And :$...
8
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1answer
221 views

$35.2850899… $ has a closed form ??

Consider the function $t(x)$ defined as : $$ x_1 = x $$ $$x_2 = x $$ $$ x_3 = 2 x^2 $$ $$ x_4 = 4 x^4 + 2 x^2 $$ and for $n > 4 $ $$ x_{n} = \frac { x_{n-1}^2 + x_{n-2}^2 + x_{n-3}^2}{x_{n-2} + ...
1
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1answer
48 views

What are the limits of series acceleration?

There are many questions on series acceleration already on this site, my question is on the limits of these type of approaches as a whole. Can I, by iteratively using acceleration techniques, make a ...
8
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0answers
308 views

Generalized limits

Cross-posted to Mathoverflow. $\DeclareMathOperator{\Lim}{Lim}$ $\DeclareMathOperator{\dom}{dom}$ $\DeclareMathOperator{\shift}{\sigma}$ $\DeclareMathOperator{\cesaro}{C}$ After reading Terry Tao's ...
1
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1answer
165 views

Evaluating a particular sum of products of exponential functions

I'm trying to get a closed-form expression for a particular type of sum, or at least a good way to approximate such sums numerically. I've tried using Mathematica, Maxima, etc, to no avail so far. ...
0
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0answers
27 views

sum up the series $f(x) = \sum_{n\geq 1 } \cos n x /\sqrt{n}$ numerically

For $x$ close to zero, the series converge slowly. Is there a way to accelerate the convergence?
2
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0answers
77 views

Convergence acceleration of successions with logarithms

I have a numerical question regarding acceleration of a succession. A preliminary: suppose that I have a succession $a_g$ that, for high $g$, asymptotically goes as $$ a_g=s_0+\frac{s_1}g+\frac{s_2}{...
7
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2answers
313 views

How to compute this constant with high precision $\sum_{n=1}^\infty \left(\frac{1}{a_n}-\frac{1}{(n+1) \ln (n+1)} \right)$

I'm interested in finding the following constant: $$b=\sum_{n=1}^\infty \left(\frac{1}{a_n}-\frac{1}{(n+1) \ln (n+1)} \right)$$ Where: $$a_1=2$$ $$a_{n+1}=a_n+\log a_n$$ This is related to my ...
2
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0answers
113 views

Sturm-Liouville Solutions and acceleration of convergence

This is my first question posted so sorry if formatting is not ideal. There are a few topics which I'm sketchy about before my applied math exam and would like additional sources for this topics. One ...
2
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2answers
176 views

Convergence acceleration of a recursively defined sequence $a_{n+1} = (1-a_n)^{\frac1p},\ p>1$

On this question of recursive sequence, I have proved its convergence. As the sequence oscillates around its limit, the convergence rate can be accelerated. Here is the definition of sequence (series) ...
6
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1answer
202 views

How to accelerate the convergence of $1 + \frac{1}{2^2} + \frac{1}{3^2} + \ldots$?

It is well known that $$ \frac{\pi^2}{6} = 1 + \frac{1}{4} +\frac{1}{9} + \frac{1}{16} + \ldots $$ I am trying to use it to calculate $\pi $. The problem is how to accelerate the convergence of ...
-1
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1answer
17k views

1. A car starts from rest and accelerates uniformly over a time of 5.21 seconds for a distance of 110 m. Determine the acceleration of the car. [closed]

A car starts from rest and accelerates uniformly over a time of 5.21 seconds for a distance of 110 m. Determine the acceleration of the car.
-1
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1answer
93 views

Does this series acceleration method have a name?

So I recently discovered a fairly simple series acceleration method. It works best for $S=\sum_{m=0}^\infty a_m$ where $a_m=(-1)^mb_m$ and $b_m$ is monotonically decreasing and positive. To start out, ...
3
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5answers
385 views

Speeding up the convergence of $\zeta(2)$

Let us denote by $S$ the sum of the series $\displaystyle\zeta(2)=1+\frac1{2^2}+\frac1{3^2}+\cdots$ Yes, I know (and you know) that $S=\frac{\pi^2}6$, but that is not relevant for the question that I ...
2
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1answer
85 views

How do mirror prox and accelerated methods differ conceptually?

In the very basic case of nice function on Hilbert space, dual space is same as primal, it seems to me that they to almost same thing: at point $x_t$ they apply a gradient obtained by looking "one ...
17
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0answers
409 views

Convergence acceleration technique for $\zeta(4)$ (or $\eta(4)$) via creative telescoping?

Question Is it already known whether the $\zeta(4):=\sum_{n=1}^{\infty}1/n^4$ accelerated convergence series $(1)$, proved for instance in [1, Corollaire 5.3], could be obtained by a similar ...
5
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1answer
116 views

Can Fourier techniques still be useful in situations where we don't have perfect periodicity (such as a missing point in a lattice)?

If we are dealing with a problem with a periodic component, for example a infinite lattice of particles in one or more dimensions and we want to calculate a solution numerically we can use Fourier ...
4
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1answer
118 views

Does the Riemann rearrangement theorem affect the Euler summation transform?

I was wondering if the Riemann rearrangement theorem has an affect on the Euler summation transform, since, when applying it to a conditionally convergent series, the rearrangement of terms may ...
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0answers
151 views

Kummer's transformation applied to a series involving the Möbius function: were rights my deductions and how get an idea of the improvement?

After I've read Kummer's recipe to get the acceleration of a series, I want to do an example related with the Möbius function $\mu(n)$. Question. I present my calculations, please A) I would ...
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0answers
107 views

Why in Jacobi eigenvalues algorithm is is important to minimize $||B-A||_F^2$?

Why in Jacobi eigenvalues algorithm is is important to minimize $||B-A||_F^2$? Where $B = J^T A J$. "Matrix Computations" by Colub and Van Loan tells it at 8.4.2. Also you can find it here Jacobi ...
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1answer
628 views

A curling stone of mass 18 kg lies on smooth horizontal ice, as shown below… [closed]

I'm working through old past papers for my current maths course. I've gotten to the following question, but nowhere in my notes does it mention acceleration. I've posted a picture of my half attempt ...
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2answers
32 views

Acceleration of a particle

I have a bidimensional circunference centered in (5,5) made of 1000 points. So the equation that describes the circunference is: $$(x-5)^2 + (y-5)^2 = 25$$ Imagine that I have a vehicle that is ...
1
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1answer
90 views

acceleration formula of parametric curve?

I have a parametric curve as follows: x(t)= 0.236t³-0.645t²+0.909t+0 y(t)= 0.189t³-0.792t²+0.603t+0 which looks like: Now, ...
0
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1answer
43 views

Numerical compuation of fast converging series

I would like to numerically compute the following series (which it related to the modified Bessel function of the second kind) up to arbitrary precision. Its definition is $$ S(x) = \sum_{k=0}^\infty \...
0
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0answers
247 views

vertical projection with gravity as acceleration

$\mathbf Question.$ A stone projected vertically upwards with initial speed of $u\ m/s$ rises $70\ m$ in the first $t$ seconds and another $50\ m$ in the next $t$ seconds. $\bullet$Find the value of $...
0
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1answer
46 views

Replacing inequality with equality.

Rate of Convergence: Definition Let $\{x_n\}_{n\ge 0}$ is a sequence that converges to a number $x*$. Suppose that $\{a_n\}_{n\ge 0}$ is another sequence known to converge to $0$. We say $\{x_n\}$...
0
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2answers
48 views

uniform acceleration of two bodies

X and Y are 400m apart. X is travelling towards Y with initial speed 3m/s and acceleration 4m/s^2. At the same time Y is travelling toeards X with initial speed 7m/s and acceleration 2m/s^2. After ...
1
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3answers
192 views

What is the computational benefit of Aitken's $\Delta^2$ process?

Let $(x_n)$ be a linearly convergent sequence. Then $$y_n := x_n - \frac{(x_{n+1}-x_n)^2}{x_{n-2} - 2x_{n+1} + x_n}$$ is called Aitken's $\Delta^2$ process. Remarkably, $(y_n)$ converges faster than $(...
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0answers
20 views

Augmenting an algorithm with a monotonic step

Suppose algorithm $A$ is known to converge to the global maximum from any feasible point. I believe it is the case that if I augment $A$ with a monotonically increasing step $B$, then the new ...
0
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1answer
49 views

Deceleration of an aircraft while braking.

I'm trying to determine the distance an aircraft traveled during a period of brake application using the following information: Initial speed (v0) = 92 nautical miles per hour, Final speed (v1) = 67 ...
1
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0answers
133 views

Extension of Chebyshev's inequality

Consider about the sum of iid random variables $$ X_t = \sum_{i=1}^{t}x_i $$ $x_t$ are of zero mean and finite variance $\sigma^2$. Let $M,m>0$. By Chebyshev's inequality, it is easy to show that $...
0
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2answers
485 views

Calculate acceleration knowing initial speed, required speed, and distance to cover

How to get the acceleration in m/s by knowing the initial speed, required speed and the distance by which this required speed is to be achieved. Basically I am developing a car simulation program, ...
3
votes
3answers
301 views

Series acceleration with Fourier-Bessel series coefficients

I was investigating methods for series acceleration and I found this identity: $$e=\sum _{n=0}^{\infty } \left(\sqrt{\frac{\pi }{2}} (2 n+1)\right) I_{n+\frac{1}{2}}(1)$$ where $I$ is the modified ...
1
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0answers
257 views

Wynn-epsilon convergence

How could I use the Wynn-epsilon alghoritm in Matlab to accelerate the convergence of a Maclaurin series? I want to extimate the first derivative of $f(x)$, so $$f'(x)= \sum_{k=0}^\infty {ka_kx^{k-1}...
7
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3answers
291 views

What is the fastest way to $\pi$?

There are many known sequences convergent to $\pi$ with different convergence accelerations. For example both of the following sequences are convergent to $\pi$ when $n$ goes to $\infty$: (a) $a_n=2^{...
2
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2answers
407 views

Pade Approximations convergence acceleration

Why Pade Approximatoins accelerate the convergence of series? Generally speaking, what is there an advantage in the sence of convergence acceleration using rational interpolation? Thanks much in ...
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0answers
59 views

Convergence acceleration using rational approximation

How Pade Approximations accelerate the convergence of series??? Here I don't mean only the Pade Approximations, just in general how do the rational approximations contribute to series convergence ...
6
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2answers
414 views

Value of $\sum_{n=0}^{\infty} \frac{(-1)^n}{\ln(n+2)}$

While testing implementations of Wynn's $\epsilon$-algorithm and Levin's u-transformation I need the value of $$\sum_{n=0}^{\infty} \frac{(-1)^n}{\ln(n+2)} \cdot$$ The results of my algorithms are in ...