This functionality does not run in MATLAB.
pade(f, x, <[m, n]>) pade(f, x = x0, <[m, n]>)
pade(f, ...) computes a Pade approximant of the expression f.
The Pade approximant of order [m, n] around x = x0 is a rational expression
approximating f. The parameters p and a0 are given by the leading order term f = a0 (x - x0)p + O((x - x0)p + 1) of the series expansion of f around x = x0. The parameters a1, …, bn are chosen such that the series expansion of the Pade approximant coincides with the series expansion of f to the maximal possible order.
If no series expansion of f can be computed, then FAIL is returned. Note that series must be able to produce a Taylor series or a Laurent series of f, i.e., an expansion in terms of integer powers of x - x0 must exist.
The Pade approximant is a rational approximation of a series expansion:
f := cos(x)/(1 + x): P := pade(f, x, [2, 2])
For most expressions of leading order 0, the series expansion of the Pade approximant coincides with the series expansion of the expression through order m + n:
S := series(f, x, 6)
This differs from the expansion of the Pade approximant at order 5:
series(P, x, 6)
The series expansion can be used directly as input topade:
pade(S, x, [2, 3]), pade(S, x, [3, 2])
Both Pade approximants approximate f through order m + n = 5:
map([%], series, x)
delete f, P, S:
The following expression does not have a Laurent expansion around x = 0:
series(x^(1/3)/(1 - x), x)
Consequently, pade fails:
pade(x^(1/3)/(1 - x), x, [3, 2])
Note that the specified orders [m, n] do not necessarily coincide with the orders of the numerator and the denominator if the series expansion does not start with a constant term:
pade(x^10*exp(x), x, [2, 2]), pade(x^(-10)*exp(x), x, [2, 2])
An arithmetical expression. If x0 is not specified, then x0 = 0 is assumed.
A list of nonnegative integers specifying the order of the approximation. The default values are [3, 3].
Arithmetical expression or FAIL.