Documentation |
Pade approximation
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 = x_{0} is a rational expression
approximating f. The parameters p and a_{0} are given by the leading order term f = a_{0} (x - x_{0})^{p} + O((x - x_{0})^{p + 1}) of the series expansion of f around x = x_{0}. The parameters a_{1}, …, b_{n} are chosen such that the series expansion of the Pade approximant coincides with the series expansion of f to the maximal possible order.
The expansion points infinity, -infinity, and complexInfinity are not allowed.
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 - x_{0} 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])
f |
An arithmetical expression or a series of domain type Series::Puiseux generated by the function series |
x |
An identifier |
x0 |
An arithmetical expression. If x0 is not specified, then x0 = 0 is assumed. |
[m, n] |
A list of nonnegative integers specifying the order of the approximation. The default values are [3, 3]. |