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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How to calculate two-point Padé approximant?

Wikipedia mentions two-point Padé approximant. I don't have access to the reference provided (Yoshiki Ueoka, Introduction to multipoints summation...). I checked also chapter 8 (The N-Point Padé ...
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Pade approximation

I am trying to model the Pade approximation of a Lorentzian graph from the taylor series. I am trying to model PA[2/2] from taylor series expansion of order N+M=4th order of derivatives taken at the ...
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Inequality of exponential function compared to its rational (Padé) approximation

In the context of studying the convexity of the real function (which is not DCP-convex but really "looks" convex) $$g(x) = \frac{1}{1-\exp(-1/x)}, \text{for } x\geq 0,$$ after some ...
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How would I generate a Pade Approximant using the coefficients of a Taylor Series?

I would like to find an effective way to make a Pade Approximant using the coefficients of a Taylor Series. I've heard of Wynn's epsilon algorithm and using the Extended Euclidean Algorithm, but what ...
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Padé approximationi of polynomial

Following wikipedia, I'm trying to compute the $[1/1]$-Padé approximation of $f(x)=x^2$. It should be of the form $\frac{a+bx}{1+cx}$, but this is either zero or a power series with a non-zero ...
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Can a homographic function be approximated by an exponential function?

Can the homographic function: $$f(x)=\frac{1+\frac{x}{a}}{1-\frac{x}{1-a}}$$ where a ∈ (0,1), be approximated by an exponential function for the interval x ∈ [0,1-a] (where the function f(x) behaves ...
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Padé approximant for non-linear least squares?

The Wikipedia article on Padé approximant makes it sound like "Padé series" is capable of approximating a function better than Taylor series can. Can Padé series be treated as a drop-in ...
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Pade approximation for a function of 2 matrices

Is there any successful examples of using Pade approximation to calculate a function of 2 matrices, such as computing $C(A, B)$ such that $e^C = e^A e^B$? I know that Pade approximation has been ...
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Compute a (3,4)-Padé approximant for sin(x)

The task is to compute a (3,4)-Padé approximant for $sin(x)$, that is, two rationals polynomials $r,t$ with $\deg r < 3, \deg t \leq 4$ such that t has a nonzero constant coefficient and $r/t \...
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Resources for Two-point Padé approximation

I am a 12th grade high school who is doing his Extended Essay from maths. I wish to learn two-point padé approximation as it would help me with my EE. If anyone knows any helpful source on the topic ...
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Numerically guessing a polynomial rational function from its power series expansion

Suppose I have a function $f \in \mathbb{Q}(x, y)$ which does not have a pole at the origin (in particular I can write it as the ratio of two polynomials with integer coefficients in $\mathbb{Z}[x, y]$...
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How do I wield function approximations (Taylor series, Pade approximants) effectively?

I'm reading a chapter in "Economic and Financial Decisions under Risk" which quickly covers the foundations for things like "constant relative risk aversion." At one point, this ...
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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: $$h\left(a\right)\equiv-\int_{-\infty}^{\infty}p\left(x,a\right)\ln\left(p\left(x,a\right)\right)dx$$ where $$p\left(x,a\right)...
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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 ...
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Homographic invariance of value transformations

Following is a theorem from Baker Graves Morris book on Padé Approximants : The proof is not clear. I have two questions : 1- How there are polynomials $p_M(z)$ and $q_M(z)$ of degree at most $M$ ...
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Pade approximation of $1-\frac{(1-x^2)\sin^2(\theta)\sin^2(\theta-y)}{(1-(1-x^2)^{1/2}\cos(\theta)\cos(\theta-y))^2}$ upto $2^{nd}$ order

I am very new to Pade' approximation concept, so some detailed derivation for the approximate result of the following function would be very helpful. The function that I wish to approximate in the ...
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In what sense is Pade aproximation "best"

The wikipedia article on Pade approximations says that its the "best" rational approximation of a given function but doesn't elaborate further. I've seen some form of this claim posted ...
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Is it possible to find the Padé approximation for $\ln{(1+x)}$? [duplicate]

On the Padé approximant wiki, people present various form of padre approximant for $\sin{(x)}, \exp{(x)}$, erf$(x)$ but does not provide instruction on how to create them. What would be the correct ...
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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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How is this method of fitting a Padé approximant to data called?

This website shows an implementation in R that fits a Padé approximant: $$ R(x)= \frac{\sum_{j=0}^m a_j x^j}{1+\sum_{k=1}^n b_k x^k}=\frac{a_0+a_1x+a_2x^2+\cdots+a_mx^m}{1+b_1 x+b_2x^2+\cdots+b_nx^n} ...
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How are these equations derived?

In the below snippet, the equations are listed that need to be used to solve for the Padé approximant. How are these equations derived? I get they want the derivatives to be equal at zero, but I don't ...
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fitting a 1d function with an asymptote

I am trying to fit a positive function $f(x)$ , $0 < x\leq 1$, with the following properties: $$f(0) =\infty$$ $$f(1) = 1$$ $$\frac{df}{dx}(1) > 0$$ $$f(x) > 0,\text{ }\forall\text{ ...
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Padé Approximation of $\sin(x)$ using Matlab

How to determine Padé approximation of degree $6$ for $f(x)=\sin x$, and compare the results at $x_i = 0.2i$ for $i = 0,1,2,3,4,5$, with $f(x)$ and with its sixth Maclaurin polynomial with $n=2$ and $...
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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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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 and the Padé approximant

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 $L$...
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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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