# Interpolation by rational functions reference

I have been hearing a lot about a theory of interpolation using rational function, parallel to that of polynomial interpolation.

I'm looking for a book chapter, or even short lecture notes, that will give me the basic results and tools of this field.

• Possibly this? en.wikipedia.org/wiki/Padé_approximant – Noam D. Elkies Dec 19 '16 at 21:33
• Thanks, I didn't know this is its name. However, I'm looking for something with more details and proofs. – Amir Sagiv Dec 19 '16 at 21:35
• Approximation is different from interpolation. Padé is the former, but the OP asked for the latter. Now maybe approximation is what he really needs, but still... – Robert Israel Dec 19 '16 at 22:46
• Wikipedia is not usually the place to go for lots of details and proofs. It does, however, often give good pointers to details and proofs. Also, once you know the term for it you can search for other sources. (And interpolation is just solving linear equations for the coefficients of the interpolating rational function, though there may still be nontrivial issues of numerical stability and computational efficiency.) – Noam D. Elkies Dec 19 '16 at 22:50
• @RobertIsrael , I didn't notice it in first glance, but I am looking for an interpolation method. – Amir Sagiv Dec 20 '16 at 6:18

Here are four references on the subject (the main ones as far as I know) :

Baker, George A.; Graves-Morris, Peter, Pad\'e approximants. Second edition. Encyclopedia of Mathematics and its Applications, 59. Cambridge University Press, Cambridge, 1996.

Nikishin, E. M.; Sorokin, V. N. Rational approximations and orthogonality. Translations of Mathematical Monographs, 92. American Mathematical Society, Providence, RI, 1991.

Petrushev, P. P.; Popov, V. A., Rational approximation of real functions. Encyclopedia of Mathematics and its Applications, 28. Cambridge University Press, Cambridge, 1987.

Walsh, J. L. Interpolation and approximation by rational functions in the complex domain. Fourth edition. American Mathematical Society Colloquium Publications, Vol. XX American Mathematical Society, Providence, R.I. 1965.

There is also a short chapter on rational interpolation in :

Rivlin, T.J. An introduction to the approximation of functions. Dover Books on Advanced Mathematics. Dover Publications, Inc., New York, 1981.

• I would particularly recommend the first and last one. The OP might be interested in the phenomenon of inaccessible support points, amply demonstrated by trying to find a rational interpolant to $\{(0,1),(1,2),(2,2)\}$, among many other delicate issues. I'd also add as a reference chapter 3 of Cuyt and Wuytack's Nonlinear Methods in Numerical Analysis. – J. M. is not a mathematician Dec 20 '16 at 5:51

It seems that "rational interpolation" has been around for quite some time. It is often a better choice, especially if the function to be approximated or interpolated has a pole in or near the domain of interest.

Here is a paper by Walsh from the 1930's.

Here is a presentation slide.

For a recent reference that includes efficient computational techniques developed in the last few years, see Chapters 26-27 of L. N. Trefethen's Approximation Theory and Approximation Practice. You can get the full Latex source of the chapter for free, including MATLAB code and exercises, here.

Let me quote the chapter summary (from Trefethen):

Generically, there exists a unique type $(m,n)$ rational interpolant through $m+n+1$ data points, but such interpolants do not always exist, depend discontinuously on the data, and exhibit spurious pole-zero pairs both in exact arithmetic and even more commonly in floating point. They can be computed by solving a matrix problem involving a Toeplitz matrix of discrete Fourier coefficients. Uniqueness, continuous dependence, and avoidance of spurious poles can be achieved by reducing $m$ and $n$ when the minimal singular value of this matrix is multiple. It may also be helpful to oversample and solve a least-squares problem.

As mentioned therein, I believe the history of this problem starts with Cauchy's 1821 note V in this book (in French).