### Pade Approximants: Factors to Consider?

Posted:

**Thu Jul 27, 2017 2:18 am UTC**I have never used Pade Approximants before but am about to take the first step.

I have a few questions and am looking for advice.

The data to be approximated is a non-linear angular quantity, θ, which is a function of parameter k and has been computed numerically. I am hoping to find a relatively simple approximant for θ, sin(θ), or cos(θ) as a function of k. If I know a function for one of these quantities, the other two quantities are easy to compute.

Plots for sin(θ) and cos(θ) have been made, as well as for their first and second derivatives: d/dk{sin(θ)], d/dk[cos(θ)], d

Do these factors play any role in choosing which function for which a Pade approximant should be determined?

i.e., does the accuracy of the approximant change depending upon whether the Pade approximant is for

(i) θ,

(ii) sin(θ), whose concavity changes over the range of investigation, or,

(iii) cos(θ), whose concavity does NOT change over the range of investigation?

Another question about Pade approximations in general:

Does a Pade approximant reveal whether the underlying function is a regular, monomial, polynomial?

For example, say I generate data for a known quadratic function: y = k*x

I pretend the function is unknown and create a Pade approximant for it.

Would this Pade approximant reveal to me that original true quadratic?

I have a few questions and am looking for advice.

The data to be approximated is a non-linear angular quantity, θ, which is a function of parameter k and has been computed numerically. I am hoping to find a relatively simple approximant for θ, sin(θ), or cos(θ) as a function of k. If I know a function for one of these quantities, the other two quantities are easy to compute.

Plots for sin(θ) and cos(θ) have been made, as well as for their first and second derivatives: d/dk{sin(θ)], d/dk[cos(θ)], d

^{2}/dk^{2}[sin(θ)], and d^{2}/dk^{2}[cos(θ)]. A plot of d/dk[sin(θ)] changed concavity over the range of investigation from concave down to concave up. A plot of d/dk[cos(θ)] is concave up over the entire range of investigation.Do these factors play any role in choosing which function for which a Pade approximant should be determined?

i.e., does the accuracy of the approximant change depending upon whether the Pade approximant is for

(i) θ,

(ii) sin(θ), whose concavity changes over the range of investigation, or,

(iii) cos(θ), whose concavity does NOT change over the range of investigation?

Another question about Pade approximations in general:

Does a Pade approximant reveal whether the underlying function is a regular, monomial, polynomial?

For example, say I generate data for a known quadratic function: y = k*x

^{2}.I pretend the function is unknown and create a Pade approximant for it.

Would this Pade approximant reveal to me that original true quadratic?