Pade (version 1.0.7)

Pade: Padé Approximant Coefficients

Description

Given Taylor series coefficients \(a_n\) from \(n = 0\) up to \(n = T\), the function will calculate the Padé \(\left[L / M\right]\) approximant coefficients so long as \(L + M \leq T\).

Usage

Pade(L, M, A)

Value

Pade returns a list with two entries:

Px

Coefficients of the numerator polynomial starting at \(x^0\).

Qx

Coefficients of the denominator polynomial starting at \(x^0\).

Arguments

L

Order of Padé numerator

M

Order of Padé denominator

A

vector of Taylor series coefficients, starting at \(x^0\)

Author

Avraham Adler

Details

As the Taylor series expansion is the “best” polynomial approximation to a function, the Padé approximants are the “best” rational function approximations to the original function. The Padé approximant often has a wider radius of convergence than the corresponding Taylor series, and can even converge where the Taylor series does not. This makes it very suitable for computer-based numerical analysis.

The \(\left[L / M\right]\) Padé approximant to a Taylor series \(A(x)\) is the quotient

$$\frac{P_L(x)}{Q_M(x)}$$

where \(P_L(x)\) is of order \(L\) and \(Q_M(x)\) is of order \(M\). In this case:

$$A(x) - \frac{P_L(x)}{Q_M(x)} = \mathcal{O}\left(x^{L + M + 1}\right)$$

When \(q_0\) is defined to be \(1\), there is a unique solution to the system of linear equations which can be used to calculate the coefficients.

The function accepts a vector A of length T + 1, composed of the \(a_n\) of the of truncated Taylor series

$$A(x) = \sum_{j=0}^T a_j x^j$$

and returns a list of two elements, Px and Qx, the Padé numerator and denominator coefficients respectively, as long as \(L + M \leq T\).

References

Baker, George Allen (1975) Essentials of Padé Approximants Academic Press. ISBN 978-0-120-74855-6

See Also

This package provides similar functionality to the pade function in the pracma package. However, it does not allow computation of coefficients beyond the supplied Taylor coefficients and it expects its input and provides its output in ascending---instead of descending---order.

See the minimaxApprox package for polynomial and rational minimax approximations to functions.

Examples

A <- 1 / factorial(0:10) ## Taylor sequence for e^x up to x^{10} around x_0 = 0
Z <- Pade(5, 5, A)
print(Z)                             ## Padé approximant of order [5 / 5]
x <- -.01                            ## Test value
Actual <- exp(x)                     ## Proper value
print(Actual, digits = 16)
Estimate <- sum(Z[[1L]] * x ^ (seq_along(Z[[1L]]) - 1)) /
    sum(Z[[2L]] * x ^ (seq_along(Z[[2L]]) - 1))
print(Estimate, digits = 16)         ## Approximant value
all.equal(Actual, Estimate)