Approximation Theory – How to Decide on the Numerator/Denominator Ratio in a Pade Approximation?

approximation-theorypade approximationsoft-question

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 question:

If I take $M+N$ as fixed, how do I make a decision on the ratio between $M$ and $N$? I expect this will depend on the function at hand, or perhaps on the computing system we are designing the approximant for, but in general what are the rules of thumb for this? I have tried to research this but have seen no way to make the decision, yet I feel the decision must be important somehow.

Best Answer

I give you my personal solution.

Suppose that you expand your function $f(x)$ around a given point $x=a$ and that the first term corresponds to a degree $p$. Then I choose the $[p+n,n]$ Padé approximant. This exactly coincides with @SV-97 comment.

If you look at this question of mine, which never recieved any answer, you will see one example and a similar concern.

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Is this “A unified theory of logic”

Here's an excerpt:

Definition 4. The set of all objects to which a statement refers once properly interpreted will henceforth be referred to as the semantic set of the statement. The semantic set captures as many objects as necessary to capture the full meaning of the statement, including not just physical objects but also any conceptual objects and relationships that are relevant. The semantic set of a statement and the meaning of a statement are synonymous. The semantic set fully expresses a statement's entire meaning. Just like the symbolic set, it is simply an inert object. Once constructed it is no longer open to reinterpretation or parsing.

(The author does not define what a statement is.) So, no: This is not math.

For comparison, here's a similarly early definition from a random introductory book on mathematical logic on my shelf:

Definition 4.1. A deduction of a formula $P$ from a set of formulas ${\mathcal E}$ (in a language $L$ in $\mathfrak{L}_1$) is a finite sequence of formulas $P_1, \dots, P_n = P$ with the property that for each $i = 1, \dots, n$, at least one of the following alternatives holds:

(a) $P_i\in {\mathcal E}$;

(b) $\exists j < i$ such that $P_i$ is a direct consequence of $P_j$ using Gen;

(c) $\exists j, k < i$ such that $P_i$ is a direct consequence of $P_j$ and $P_k$ using MP.

The Pade approximation of the matrix logarithm

I think you are making some mistake in computing. The following PARI/GP code (with a $1/2$ factor included)

default(realprecision, 7);     
X = matdiagonal([1.002, 1.01, 1.1]);
A = X~*X - matid(3);
L = 1/2 * (A+A^2/2) / (1+A+A^2/6);
print(L);

outputs

[0.001998003, 0, 0; 0, 0.009950331, 0; 0, 0, 0.09530948]

while the code

log([1.002, 1.01, 1.1])

outputs [0.001998003, 0.009950331, 0.09531018] and the two answers are close. Also note that the code

X = 1 + x; A = X*X - 1;
L = 1/2 * (A+A^2/2) / (1+A+A^2/6);
print(L + O(x^5));

outputs x - 1/2*x^2 + 1/3*x^3 - 1/4*x^4 + O(x^5) which is the first four terms of the power series expansion of $\,\log(1+x).$

Best Answer

Related Solutions

Is this “A unified theory of logic”

The Pade approximation of the matrix logarithm

Related Question