All Questions
Tagged with recursive-algorithms numerical-methods
23
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How to best approach a numerical computational solution towards matching $e^{-r}$ and $k_2\sin(k_1\,r)$ as well as their derivatives $\frac{d}{d\,r}$?
In my question, "Why does it seem like two parameters $k_1$ and $k_2$ are needed to match $e^{-r}$ and $k_2\sin(k_1\,r)$ as well as their derivatives $\frac{d}{d\,r}$?", it was identified ...
- 951
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Recursive approximations of inverse square law
I have a toy electrostatics simulation that consists of some number of 2D point particles that each have a real-valued "charge" $q_i$, which then exert forces on each other proportional to $...
- 640
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Recursive algorithm for numeric integration with singularities
As part of my research, I have to integrate Green's Function to get charge density but Green's function has integrable-singularity near band-edge and integration has been a headache with normal ...
2
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1
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Rules for Choosing Bounds and Initial Conditions when Using 2nd Order Runge Kutta Methods
I have a question regarding 2nd order Runge-Kutta methods, specifically where it regards the bounds of the solution.
Let's say I have to solve a 1st order ODE $\frac{dy}{dx}=f(x,y)$ numerically using ...
- 433
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why is this iterative algorithm for calculating the Chebyshev polynomials stable?
It is known that $\cos n \theta $ is a polynomial of $\cos \theta $. This polynomial is the so called Chebyshev polynomial of order $n$, often denoted as $T_n(x)$.
We have the recursion relation
$$T_{...
- 1,015
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39
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How to derive a recursive formula from the following formula
How to derive a recursive formula from the following formula,
$$
u_{n}=a_{n-1}u_{0}+\sum_{k=1}^{n-1}(a_{n-1-k}-a_{n-k})u_{k}+\Gamma(2-\alpha)h^{\alpha}f(t_{n},u_{n})?
$$
P.S.:
Consider the following ...
- 25
1
vote
2
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107
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Tail recursive formulation of the Legendre polynomial relation
The recursive formula for Legendre polynomials is widely known:
$(n + 1) P_{n+1}(x)
= (2n + 1) x P_{n}(x) - n P_{n-1}(x).$
Let us rewrite the above as follows for convenience:
$P_{n}(x)
= \frac{2n - ...
1
vote
1
answer
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How to calculate the following variance in a recursive way
Suppose we need to divide people into two groups A and B, the first person will be assigned to either of the group with probability $0.5$, from the second person, the assignment will be done based on ...
- 21
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is there a faster method to calculate $1/x$ ($x$ an integer) than this?
I gave this stackexchange a second go. Is there a faster way to calculate $1/x$ than the following:
Calculate $100/x$ (.or other arbitrary positive power of $10$) with remainder
Write multiplier in ...
user645636
6
votes
1
answer
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Why does calculating $\exp z$ using $\ln z$ via newton-raphson method fail to converge?
I am trying to calculate $\exp z$ using $\ln z$ via Newton-Raphson method $$x_{n+1} = x_n-\frac{f(x_n)}{f^{'}(x_n)}$$and got the formula $$x_{n+1}=x_n-\frac{\ln x_n-z}{\frac{1}{x_n}}$$ where $z = a + ...
- 101
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1
answer
513
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*Numerical* Convergence of the Babylonian Method?
I understand the sequence $x_{n+1} = \frac12\left(x_n + \frac2 {x_n}\right) $ converges to $ \sqrt2 $ algebraically.
That is proved by means of fixed-point method or monotone convergence theorem and ...
- 23
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Let $X$ be a random variable, $\frac{\mathbb{E}[e^{sX}-e^{tX}]}{s-t}$ for $s \approx t$. As
Question
Let $X$ be a random variable for which we only have the value of its Moment Generating Function $M_X$ on a discrete set of points, I am looking for a stable method to compute:
$$\frac{M_X(s) ...
- 757
1
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77
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How do I fight loss of significance and/or improve convergence for this recursive algorithm?
While trying to answer this question I used the series approach and obtained a recursive algorithm. While checking it numerically, I found it suffering from "catastrophic cancellation", i.e. loss of ...
- 31.7k
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2
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Numerical stability of a geometric series
Suppose I want to compute $S_n = \sum_{i = 0}^n \beta^i$. We do have an exact formula for the answer: $S_n = \frac{1 - \beta^{n + 1}}{1 - \beta}$. Let's suppose though that instead of using this ...
- 1,303
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Prob. 24, Chap. 5 in Baby Rudin: For $\alpha>1$, let $f(x) = (x+\alpha/x)/2$, $g(x) = (\alpha+x)/(1+x)$ have $\sqrt{\alpha}$ as their only fixed point
Here is Prob. 24, Chap. 5 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
The process described in part (c) of Exercise 22 can of course also be applied to functions ...
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