Questions tagged [pade-approximation]

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.

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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 ...
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What is the Pade approximation of the matrix logarithm?

I would like to use the Pade approximation in my numerical procedure and I would like to use it to approximate the logarithm of a matrix. However, I couldn't find the correct expression for it in the ...
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Finding the nth Padé aproximant

I am trying to find the $P_N^N(\epsilon)$ and $P_{N+1}^N(\epsilon)$Padé approximant of this function $$ x(\epsilon) = \int_0^{infinity} \frac{e^{-t}}{1+\epsilon t}dt \sim \sum_{n = 0}^{infinity} n! (-...
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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 ...
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Approximating function $f(x)=\sqrt{\sqrt{e^{x}}}$

I've been tryng to find a good (accurate) low order (2-3) approximation for functions $e^x$ and $2^x$. Found out that taking square root of the function improves accuracy a bit ... and taking ...
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Derivation of Padé approximant to exponential function: unclear step in Gautschi

On pages 363 to 365 in “Numerical Analysis” by Gautschi (2012) is a derivation of the Padé approximant of the exponential function. I am stuck on a step in the beginning of the differentiation below (...
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why are there two different Pade approximation of delay

There are 2 different second order pade approximations of delay given in internet What is the difference between these two approximation? Which one is the correct Pade approximation
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Cramer's rule solution of the Padé approximant equations

Padé approximants are a particular type of rational approximation. The $L, M$ Padé approximant is denoted by $$[L/M] = P_L(x)/Q_M(x)$$ where $P_L(x)$ is a polynomial of degree less than or equal to $...
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Finding the coefficients $p_0,p_1,p_2,q_1,q_2,q_3$ of Padé approximation

Determine the Padé approximation of degree $5$ with $ n =2 $ and $ m= 3$ for $f(x) = e^{-x}$. Suppose $r$ is a rational function of degree $N$.$$ r(x) = \frac{p(x)}{q(x)} = \frac{p_0 +p_1x + \cdots + ...
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Pade approximation of $\frac{1-e^{-x}I_0(x)}{x}$

I need to expand $\frac{1-e^{-x}I_0(x)}{x}$ in Pade approximation. The answer should be $\frac{1}{1+x}$. But I'm not sure how to reach the answer. Here $I_0(x)$ is modified Bessel function of order 0 ...
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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\...
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How does the convergence sector of a continued fraction depend on the order where it is truncated?

Consider these three functions: $$ f(z) = \sqrt{bz} \cdot \coth \sqrt{bz} \\ g(z) = \sqrt{bz} \cdot \frac{\mathrm{I}_{0}( \sqrt{bz} )}{\mathrm{I}_{1}( \sqrt{bz} ) } = \sqrt{bz} \cdot \frac{\mathrm{J}...
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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 ...
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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 ...
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Padé approximation to the Poisson c.f.

Poisson variable X possess characteristic function $$ \phi_X(t) = E[e^{itX}] = e^{\lambda(z-1)} $$ where $ z = e^{it} $. If i were to obtain the Padé approximation by defining $1/e^{u} = e^{\...
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Pade approximation of a rational function

I believe I have a naive or hard question because I couldn't find any results in the Internet yet. Any help is greatly appreciated. So suppose I have two rational functions $R_1(x)$ and $R_2(x)$, i....
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Why is the Padé approximant typically written in this form?

$$R(x) = \frac{\sum_{j = 0}^m a_{j} x^j}{1+\sum_{k=1}^n b_k x^k}$$ I've started computing these to approximate my coefficients for a regression and others have been asking me how the Padé approximant ...
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Pade Approximant on Mathematica

I would like to know how to code Pade Approximation on Mathematica. And also I have solved Eigen values and have 20 coefficient values,but I am not sure how to code on mathematica.Can you please ...
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Show that $\exp(x)-\frac{1+\frac{x}{2}}{1-\frac{x}{2}} = O(x^3)$

I'm trying to show the following statement: $\exp(x)-\frac{1+\frac{x}{2}}{1-\frac{x}{2}} = O(x^3)$ I know that this is an example of Padé Approximation of the exponential. But I am not allowed to ...
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Pade Approximation of $\sin(x)$

So I am trying to get a polynomial approximation for $\sin(x)$ using Pade approximation for $n = 2$ and $m = 3$ and the Maclaurin series $(\deg 5)$ for $$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!}...
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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.
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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 $n\ge 2$? ...
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Padé approximants with $m+n$ fixed

On Wikipedia it says that the $[m/n]_f$ Padé approximant to a function $f(x)$ is the rational function $$R(x)=\frac{a_0+a_1x+\cdots+a_m x^m}{1+b_1 x+\cdots+b_n x^n} $$ such that $$R^{(k)}(0)=f^{(k)}(0)...
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How can I get those approximations?

Suppose, $u$ is the unique real solution of $x^x=\pi$ and $v$ is the unique real solution of $x\cdot e^x=\pi$ Expressed with the Lambert-w-function we have $u=e^{W(\ln(\pi))}$ and $v=W(\pi)$ Wolfram ...
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Pade approximant of infinite order

The Pade approximant states that you can approximate a function $f(x)$ by a rational function $R(x)$ of a given order. My question is, if the order of $R(x)$ goes to infinity, does $R(x)$ approach $f(...
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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 ...
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Can this approximation be made more formal?

When considering oscillating systems in physics, we end up with some response function like $$F(\omega) = \frac{\omega^2}{(\omega_0^2 - \omega^2)^2 + (\omega/\tau)^2},$$ where $\omega_0$ and $\tau$ ...
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Pade Approximation Calculate Coefficients when encountering Singular Matrix?

I will be using this post here as a reference: How to compute the pade approximation? After getting to the point where you solve a system of linear equations: $$p_0 = a_0q_0\\p_1 = a_1q_0+a_0q_1\\ \...
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Inverting the Pade approximation (going from $P_m(x)/Q_n(x)$ to $f_{m+n}(x)$)

Is there a way to easily invert the Pade approximation and get back the Taylor series it was derived from? In other words, if I have two polynomials $P_m(x)/Q_n(x)$ derived from an $(m+n)$th degree ...
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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 ...
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Looking for a Finite Difference scheme of the following form…

I'm having trouble deriving a finite difference scheme that calculates the second derivative of a function on the boundaries of a non-uniform grid and makes use of a known first derivative at the ...
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close form denominator pade approximation

How it is possible to get the denominator pade close form of the function $$\frac{1}{2} \left(\sqrt{2 \pi } e^{x/2} \sqrt{x} \text{erf}\left(\frac{\sqrt{x}}{\sqrt{2}}\right)+2\right)$$ as $$2^{-n-1} \...
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I can't solve this integral in Mathematica

I want to calculate the following integral in Mathematica 10 $$f(x) = \frac2\pi\int_0^\infty\exp\left(-\kappa_2\frac{t^2}{2!}+\kappa_4\frac{t^4}{4!}-\dots\right)\cos\left(\kappa_1t-\kappa_3\frac{t^3}{...
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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 ...
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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 ...
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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 ...