Padé Approximant Calculator - Rational Function Approximation

Overview of Padé Approximants

The Padé approximant is a rational function that matches a given Taylor series to as high an order as possible. While a Taylor polynomial uses only a finite number of derivatives to form a polynomial, a Padé approximant uses those derivatives to construct a ratio of two polynomials. This often produces a more accurate approximation on a wider interval, especially near singularities where a pure polynomial might diverge. Mathematically, for nonnegative integers $m$ and $n$ , the Padé approximant of order $m, n$ is the rational function $\frac{P}{Q}$ where $P$ has degree at most $m$ and $Q$ has degree at most $n$ , with $Q (0) = 1$ for normalization. The coefficients are chosen so that the Maclaurin series of $\frac{P}{Q}$ agrees with the series of the original function through order $m + n$ .

Why Rational Approximations?

Polynomials provide good local approximations, but they can behave poorly farther from the expansion point. Rational functions can capture asymptotic behavior more faithfully. For example, the Padé approximant of $\frac{1}{1}$ reproduces the function exactly with only a first-order denominator. In many contexts—including control theory, quantum mechanics, and numerical integration—Padé approximants serve as a bridge between a power series and a more globally accurate representation.

Computing the Coefficients

To build the approximant, we first compute the Taylor series coefficients $a_{k}$ of the function up to order $m + n$ . Then we solve a linear system for the denominator coefficients $q_{j}$ (with $q_{0} = 1$ ) so that the power series of $Q$ times the original series agrees with a polynomial of degree $m$ . Finally, we compute the numerator coefficients $p_{k}$ by convolving the $q_{j}$ with the $a_{k}$ . Although these steps may sound complex, they rely on straightforward linear algebra that can be coded concisely.

Benefits of Padé Approximants

One of the key advantages is improved accuracy away from the expansion point. Consider approximating $e^{x}$ . The Taylor series truncated at order four provides modest accuracy near zero, but diverges quickly for large $x$ . A Padé approximant of order $2, 2$ can capture the function well across a broader interval because the denominator models the exponential growth. In numerical computations where both accuracy and efficiency matter, Padé approximants offer a valuable compromise.

Using the Calculator

Enter a function recognized by math.js, such as sin(x) or exp(x). Choose non-negative integer orders $m$ and $n$ for the numerator and denominator. When you press Compute, the calculator differentiates the function repeatedly at $x = 0$ to obtain series coefficients. It then solves for the denominator coefficients with a small system of linear equations and displays the resulting rational function. Because symbolic differentiation happens directly in your browser, complex expressions or very high orders may take a moment to process.

Worked Example

Suppose we want a $1, 1$ Padé approximant of $e^{x}$ . The first two derivatives at zero are all $1$ . Solving the equations yields $P (x) = 1 + \frac{1}{2} x$ and $Q (x) = 1 - \frac{1}{2} x$ . The ratio $\frac{P}{Q}$ matches the Taylor series of $e^{x}$ through $x^{2}$ and offers better accuracy for larger $x$ than the quadratic Taylor polynomial alone.

Limitations

Padé approximants are powerful, but they require care. The underlying linear system might be singular for certain order choices or functions, which means a Padé approximant does not exist in those cases. Additionally, while the approximant often extends the range of accuracy, it can introduce spurious poles—points where the denominator vanishes though the original function remains finite. Checking that these poles do not fall within the interval of interest is a prudent step when using the approximant in practical computations.

Historical Context

Henri Padé introduced his approximation scheme in the late nineteenth century as an improvement over Taylor polynomials. Since then, the method has appeared in diverse areas, from quantum field theory to digital signal processing. Its power lies in capturing more global information with only local derivatives. Modern symbolic algebra systems automate the construction of Padé approximants, but understanding the underlying theory grants insight into why they work so well.

Beyond the Basics

Advanced applications extend Padé approximants to series around points other than zero, or to multivariate functions. They also form the basis for analytic continuation, where a series that converges only near the origin is turned into a rational function valid over a larger domain. In numerical analysis, Padé approximants underpin some exponential integrators and continued fraction expansions. Experimenting with the calculator exposes you to these possibilities and encourages further exploration.

Tips for Accurate Computation

Stick to modest orders when first learning. High orders require many derivatives and can suffer from numerical instability. If you encounter an error, try reducing $m$ or $n$ . Remember that the input function must be analytic at the expansion point; discontinuities or absolute values lead to divergent series and unusable approximants. When the approximant exists, compare it to the original function by plotting or evaluating at sample points to verify accuracy.