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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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Does series acceleration improve computational efficiency?

There are several methods of series acceleration. For example, Euler's transform is: $$\sum_{n=0}^\infty (-1)^na_n=\sum_{n=0}^\infty \frac{(-1)^n}{2^{n+1}}\sum_{k=0}^n (-1)^k {n \choose k} a_{n-k}$$ ...
user46190's user avatar
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Analytical transformation or an effective numerical method for calculating $\sum_{n=1}^{\infty}{K_0(\frac{n}{a})\sin(n)}$ series

I have a series for that I need to get a fast calculation: $$\sum_{n=1}^{\infty}K_0\!\left(\frac{n}{a}\right)\sin(n)$$ where $K_0$ is the $0^{\text{th}}$ modified Bessel functions of the second kind, ...
gearquicker's user avatar
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Aitken's $\Delta^2$ Method applied to the Newton's Method

Could Aitken's $\Delta^2$ Method be used to accelerate the convergence of the Newton's Method? I simplified $p_n$=$x_n$-$\frac{(x_{n+1}-x_n)^2}{x_{n+2}-2x_{n+1}+x_n}$ and obtained $p_n$=$x_n-$$\frac{(\...
J P's user avatar
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Accelerating convergence of a sequence of estimators

Let $T_n$ be a strongly consistent estimator of a parameter $\theta$. Are there algorithms for accelerating the convergence of $\lbrace T_n \rbrace$ to $\theta?$ In the case of a deterministic ...
user67724's user avatar
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Why's $\sum _{n=1}^{\infty } (-1)^n (n^{1/n}-1)$ accelerated by $\sum _{n=1}^{\infty } \frac{1}{2} (-1)^n \left(n^{1/n}-(n+1)^{\frac{1}{n+1}}\right)?$

The series I started with is defined here. The mathematical codes used below are ...
Marvin Ray Burns's user avatar
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How to accelerate the convergence of a Power series?

Is there a way to accelerate the convergence of alternating Taylor's series such as that $$ \sum_{n=0}^\infty (-1)^n a_n x^n=A $$ approaches its value with less number of terms. Since its an ...
Mihailo's user avatar
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Could someone help me accelerate the convergence of my series using Euler's transformation?

I am trying to accelerate the convergence of the series. $$ \lim_{x \to \infty} \left(\sum_{n=0}^\infty(-1)^n \frac {x^{2n+1}}{(2n+1)n!}\right) $$ For the sake of simplicity, we can disregard the ...
Mihailo's user avatar
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Could someone please help me understand Euler's series acceleration?

I am trying to accelerate a Taylor series with an alternating sign and found that Euler's acceleration is something I should look into. I found this document, https://kconrad.math.uconn.edu/blurbs/...
Mihailo's user avatar
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Are there "delta-squared processes" of higher orders than Aitken's?

Aitken's Delta-Squared Process transforms the sequence $s=(s_0,s_1,s_2,...,s_n,...)$ into the sequence $S=(S_0,S_1,S_2,...,S_n=s_{n+2}-(s_{n+2}-s_{n+1})^2/(s_{n+2}-2 s_{n+1}+s_n),...)$. If $\Delta s=(\...
RHER WOLF's user avatar
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How can I calculate at which distance a moving object stops in this physics emulation? [closed]

There is an object which has an initial velocity of $`V_1`$ and has an acceleration of $`V^2 k+d+b`$ in the opposite direction, where k,d,b are constants. What will ...
Mostofus's user avatar
-3 votes
1 answer
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Calculating time from acceleration and distance travelled [closed]

I have an interesting question that has me stumped. If an object is free falling (so accelerating at 9.8m/s^2), by gravity, and it traveled a distance of 26cm, how long did that take? The initial ...
MathsCuriosity's user avatar
1 vote
1 answer
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How to make the real and imaginary parts of the Riemann Zeta Function converge faster?

In a previous question, I derived the real and imaginary parts for $\zeta(s)$, where $s = \sigma + it$ and $\text{Re}(s) > 1$: $$\text{Re}(\zeta(s)) = \sum_{n=1}^\infty \frac{\cos(t\log n)}{n^\...
Mailbox's user avatar
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Determine the the swipe angle of a rotating circle.

I have a circle rotating with the acceleration of $5000^{\circ}/s^{2}$ for time $0:T/2$; then $-5000^{\circ}/s^{2}$ for time $T/2:T$, and the velocity is $20^{\circ}/s$. How can I determine the swipe ...
MKVENG's user avatar
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Equation to accelerate an object along a known curve until it reaches a target distance

This problem is deceptively simple, but it's been driving me mad: I have a function, $f(x)$. At $f(0)$, it returns $0$ -- at $f(1)$, it returns $MaxSpeed$. I have an object at rest and a target point ...
Jay2645's user avatar
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Moving Average for Speed and Acceleration

I am tracking a person using a set of ultrasound tracking device. The ultrasound system provides (x,y) location data at 8Hz frequency, and there are 131 data points in total. When calculating the ...
olliver_27's user avatar
1 vote
1 answer
175 views

Why is this wrong? Calculate 3D position from acceleration

I would like to ask a question related to math/physics since I'm just a developer and I don't have enough physics knowledge to solve this problem alone. I'm trying to calculate a 3d position having ...
Eloh's user avatar
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3 votes
1 answer
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Accelerate convergence of sequence

Is it always possible to extract a subsequence from my generic sequence $(q_n)$, such that the convergence of the subsequence to the same limit $r$ is faster then the original?
Crash Bandicoot's user avatar
3 votes
1 answer
344 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 ...
Simply Beautiful Art's user avatar
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1 answer
334 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 ...
Dhaval Bothra's user avatar
7 votes
1 answer
258 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} + ...
mick's user avatar
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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 ...
user2944352's user avatar
10 votes
0 answers
709 views

Generalized limits

Cross-posted to Mathoverflow. $\DeclareMathOperator{\Lim}{Lim}$ $\DeclareMathOperator{\dom}{dom}$ $\DeclareMathOperator{\shift}{\sigma}$ $\DeclareMathOperator{\cesaro}{C}$ After reading Terry Tao's ...
user76284's user avatar
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2 votes
1 answer
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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. The ...
Mike Battaglia's user avatar
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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?
S. Kohn's user avatar
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2 votes
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101 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}{...
Salvatore Baldino's user avatar
8 votes
2 answers
375 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 ...
Yuriy S's user avatar
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2 votes
0 answers
158 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 ...
Cooked's user avatar
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2 votes
2 answers
229 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) ...
Hans's user avatar
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6 votes
1 answer
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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 ...
poisson's user avatar
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-1 votes
1 answer
28k 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.
Katie's user avatar
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-1 votes
1 answer
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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, ...
Simply Beautiful Art's user avatar
5 votes
4 answers
671 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 ...
José Carlos Santos's user avatar
2 votes
1 answer
167 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 ...
Ben Usman's user avatar
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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 ...
Américo Tavares's user avatar
6 votes
1 answer
175 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 ...
sonicboom's user avatar
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4 votes
1 answer
159 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 ...
Simply Beautiful Art's user avatar
1 vote
0 answers
198 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 ...
user avatar
0 votes
0 answers
115 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 ...
Yola's user avatar
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-1 votes
1 answer
1k 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 ...
Michael Williams's user avatar
0 votes
2 answers
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 ...
Unnamed's user avatar
  • 449
1 vote
1 answer
206 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, ...
ksmith's user avatar
  • 111
0 votes
1 answer
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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 \...
User133713's user avatar
0 votes
0 answers
253 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 $...
sushi181's user avatar
0 votes
1 answer
54 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 $\{x_n\}$...
Sohail Ahmed's user avatar
0 votes
2 answers
107 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 ...
sushi181's user avatar
3 votes
3 answers
529 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 $(...
Leo's user avatar
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0 votes
0 answers
21 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 ...
Cliff AB's user avatar
  • 239
0 votes
1 answer
70 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 ...
InvertedPlane's user avatar
1 vote
0 answers
172 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 $...
Andy Xu's user avatar
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0 votes
2 answers
614 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, ...
Fabian's user avatar
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