Rational approximation and Padé approximants

Let f be analytic near zero and choose nonnegative integers m and n. A normalized type-(m,n) Padé representation seeks polynomials p and q of degrees at most m and n, with q(0)=1, for which qf−p vanishes through degree m+n. Because q is nonzero in some neighborhood of zero, the quotient then matches the Taylor expansion of f to that order. The type gives degree upper bounds, so a successful representation can have smaller effective degrees.

The normalization is convenient but must not be assumed feasible for every index and every Taylor series. Some matching systems are defective or require a denominator vanishing at the expansion point before cancellation; we only call the displayed normalized construction successful when its equations have a solution. Common factors can make coefficient representations nonunique, even when the resulting rational function is unique. Cross-multiplying two successful normalized quotients shows their difference numerator has degree at most m+n but vanishes to higher order, so it is identically zero.

Write the Taylor coefficients of f and the unknown coefficients of q. In the product qf, coefficients above degree m through degree m+n must vanish because p cannot contain them. These equations determine possible denominator coefficients; the remaining low-degree coefficients define p. The system is linear in the unknown coefficients even though the quotient is nonlinear as a function of its denominator. This is a useful reduction, not a guarantee that the linear system is well conditioned.

Exact coefficient matching can be sensitive when the data nearly admit a lower-degree rational representation. A nearly canceling numerator zero and denominator zero may appear as a spurious pole-zero pair, often called a Froissart doublet. Rounding or noisy Taylor coefficients can move such features dramatically. Rank-revealing or regularized computations may prefer a lower effective degree, but that is an additional algorithmic choice. A tiny series residual alone does not certify a meaningful pole or accurate values near that pole.

A high-order zero of the local error describes behavior as the argument approaches the expansion point. It does not bound the quotient on a distant interval, especially near denominator zeros. A rational approximation optimized uniformly over a prescribed interval solves a different problem, with restrictions that exclude poles on that interval. Rational interpolation at distinct sample points is another different construction. Sharing a numerator/denominator representation does not make their objectives or convergence theorems interchangeable.

Before using a rational approximant, state the region of interest, inspect the denominator there, and evaluate an actual error or residual bound suited to the application. If the target has known singularities, they can guide the approximation space, but fitted poles should still be interpreted cautiously. Padé approximants can exactly recover a sufficiently low-degree rational target from its series; that favorable case does not establish convergence for every analytic function or every sequence of degree pairs.