A Padé approximation is the use of a ratio of polynomials to approximate a function. This can be seen as a generalization of the Taylor series which can better account for singularities in the function.
Questions tagged [pade-approximation]
58 questions
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Using Padé approximants for the quaternion exponential
On a lark, I wanted to know if one can use Padé approximants to compute the exponential $\exp(z)$ of a quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$. Since Mathematica has a package meant for manipulating quaternions, as well as computing the…
J. M. ain't a mathematician
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13
votes
5 answers
Approximating $\log x$ with roots
The following is a surprisingly good (and simple!) approximation for $\log x+1$ in the region $(-1,1)$:
$$\log (x+1) \approx \frac{x}{\sqrt{x+1}}$$
Three questions:
Is there a good reason why this would be the case?
How does one go about…
Nathaniel Bubis
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How does one decide on the numerator/denominator ratio in a Pade approximation?
I appreciate that Pade approximants are often nicer than Taylor series; I know that if you take a Pade approximant of order $M/N$ it corresponds loosely to a Taylor approximation of order $M+N$.
Soft question, and apologies if this is deemed a bad…
FShrike
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9
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2 answers
How to approximate $\sin(x)$ using Padé approximation?
I need to write a function for $\sin(x)$ using Padé approximation.
I found here a formula, but I don't know how to achive this.
Thanks in advance.
9
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1 answer
Is anything known about $ \small{b_0+\tfrac{a_1}{\left(b_1+\tfrac{a_2}{\left(b_2+...\right)^n}\right)^n}} $?
What is known about this generalized "continued fraction"
$$
b_0+\frac{a_1}{\left(b_1+\frac{a_2}{\left(b_2+\frac{a_3}{\left(b_3+\dotsb\right)^n}\right)^n}\right)^n}
$$
when the integer $n\ge 2$?
Wikipedia and wolfram articles on generalized…
Cave Johnson
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Bennett's Inequality to Bernstein's Inequality
Bennett's Inequality is stated with a rather unintuitive function,
$$
h(u) = (1+u) \log(1+u) - u
$$
See here. I have seen in multiple places that Bernstein's Inequality, while slightly weaker, can be obtained by bounding $h(u)$ from below,
$$
h(u)…
duckworthd
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votes
2 answers
Wrong stability results when using Padé approximation
The Padé approximation of the exponential function, $F(s) = e^{-\tau s}$, is used often in control theory. I wonder whether its use can lead to erroneous results regarding the stability properties of the closed-loop system.
Let's say we have a…
7
votes
2 answers
Rigorous rationale for the Pade Approximant?
I recently asked a Question for which the only Answer I got was a recommendation to punt and use the Pade Approximant.
This is the first time I recalling seeing this, and intuitively it seems like a good approximation. I have seen and generated…
Jerry Guern
- 2,764
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votes
3 answers
Bounding error of Padé approximation
I'm trying to understand how one would understand the error of a given Padé approximation for a function.
For instance, the $[2,1]$ approximant for $\log(1+x)$ is $\frac{x(6+x)}{6+4x}$. Is there a reasonable way to understand how good this…
davidlowryduda
- 94,345
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1 answer
Continuation of functions beyond natural boundaries
The article Continuation of functions beyond natural boundaries by John L. Gammel states
I am particularly interested in the convergence of the $[N/N+1]$ Padé approximants beyond the natural boundary, since, as is well known, Borel [2] has shown…
user76284
- 6,408
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votes
0 answers
roots of Padé approximating polynomials to the exponential function
I need to obtain (numerically) the roots of the denominator in the Padé approximation to the exponential function $e^{-x}$, in Python. I can calculate its coefficients in closed form (see below). But the numpy.roots() function to find roots of a…
Jason S
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votes
1 answer
What algorithm is used in Matlab's pade function?
I try to find out for quite some time now, how Matlab implements the calculation of Padé Approximants using its symbolic pade function. (the code of is buried in a compiled mex-file)
I compared its output with the "direct calculation" via…
Robert Seifert
- 285
4
votes
2 answers
Implementation help for Extended Euclidean Algorithm
I'm not sure if this question is entirely on-topic here, please notify if not. I feel it is more a math related problem, than a programming problem.
Following the advice in this answer I'm trying to implement the Extended Euclidean Algorithm. The…
Robert Seifert
- 285
3
votes
3 answers
How to create a Pade approximation for a difficult function with a divergent Taylor series?
I've been trying to create a good approximation for this function:
$$h\left(a\right)\equiv-\int_{-\infty}^{\infty}p\left(x,a\right)\ln\left(p\left(x,a\right)\right)dx$$ …
Jerry Guern
- 2,764
3
votes
1 answer
How to prove the following application of the Stiltjes series expansion
We begin with a density given by
$$
\tag 1
K(\xi)=\sum_{k=1}^K p_k\delta\left(\xi -\xi_k\right)
$$
The question is how to prove the following
$$
\tag 2
\int_0^{max(\xi)}K(\xi)\frac{(z\xi)^{1-K}}{1-z\xi}d\xi=\sum_{k=1-K}^{K}z^kM_k
$$
where the…
Alexander Cska
- 434