Padé Exponential Poles
Match exponential series with rational functions and reveal approximation-created poles.
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Padé Exponential Poles
The Padé approximant uses a rational function—a ratio of two polynomials—to approximate a function. For the exponential function, matching its Taylor series produces rational functions with improved global behavior or specific desirable properties. However, introducing a denominator can create poles where the approximation becomes undefined, even though the original function is completely smooth and finite everywhere.
Worked Example
Let's calculate the Padé [1/1] approximation for e^x at x = 1. The formula is:
Padé [1/1](x) = (1 + x/2) / (1 - x/2)
Substituting x = 1:
- Numerator: 1 + 1/2 = 1.5
- Denominator: 1 - 1/2 = 0.5
- Approximation: 1.5 / 0.5 = 3.0
- Exact Exponential: e^1 ≈ 2.718281828459045
- Absolute Error: |3.0 - 2.718281828459045| = 0.2817181715409549
The [1/1] approximant gives a result of 3.0, which is a reasonable approximation of e ≈ 2.718.
Experiment
We can explore the poles of these approximations using the interactive module.
- Set Approximation to
Pade 1 1. - Set Probe x to
2. Observe that the denominator becomes exactly 0. The approximation is unavailable, indicating a real pole, even though e² is perfectly well-defined. - Set Probe x to
3. The approximation gives a negative value (-5.0), whereas the true exponential function is always positive. - Change Approximation to
Pade 2 2and observe how its denominator 1 - x/2 + x²/12 has no real roots, thereby avoiding real poles.
Limitations
Padé approximants are derived by matching the local behavior (Taylor series) around a point, usually x = 0. While they often provide a larger radius of convergence than Taylor polynomials of the same degree, they can introduce spurious singularities (poles) where the true function has none. Furthermore, a rational approximation matching a positive function like e^x is not guaranteed to be globally positive, as seen with the [1/1] approximant for x > 2.
References
NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods, Section 3.11: Padé Approximations. Available at https://dlmf.nist.gov/3.11
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Padé Exponential Poles module.
Try Padé Exponential Poles on Riano →