numapprox
pade
compute a Pade approximation
Calling Sequence
Parameters
Description
Examples
pade(f, x, [m, n])
pade(f, x=a, [m, n])
pade(f, x, m)
pade(f, x=a, m)
f
-
expression representing the function to be approximated
x
the variable appearing in f
a
the finite point about which to expand in a series
m, n
desired degree of numerator and denominator, respectively
The function pade computes a Pade approximation of degree m,n for the function f with respect to the variable x.
Specifically, f is expanded in a Taylor (or Laurent) series about the point x=a (if a is not specified then the expansion is about the point x=0), to order m+n+1, and then the Pade rational approximation is computed.
The m,n Pade approximation is defined to be the rational function pxqx with degpx≤m and degqx≤n such that the Taylor (or Laurent) series expansion of pxqx has maximal initial agreement with the series expansion of f. In normal cases, the series expansion agrees through the term of degree m+n.
If the order of the lowest order term in the Laurent series is a negative integer v and n+v<0, then no rational approximation with a denominator of degree at most n can exist, and an error is raised. If v>m≥0, the return value is 0.
If the third argument is simply an integer m, then the Taylor (or Laurent) polynomial of (relative) degree m is computed.
Various levels of user information will be displayed during the computation if infolevel[pade] is assigned values between 1 and 3.
The command with(numapprox,pade) allows the use of the abbreviated form of this command.
withnumapprox:
padeexpx,x,3,3
110x2+12x+1+1120x3110x2−12x+1−1120x3
pade1xsinx,x=0,4,6
1+13396x2+511088x4551166320x6−53396x4+x2
padeΓx,x=1,1,1
γ+−γ22+π212x−1γ+π212+γ22x−1
padecosx,x,3,4
1−61x2150775x2+1+1200x4
padecosx,x,7
1−12x2+124x4−1720x6
padeexpxx3,x,4,0
Error, (in `convert/ratpoly`) no rational approximation with denominator degree <= 0
padeexpxx3,x,4
1x3+1x2+12x+16+x24
padeexpx3−1,x,2,5
0
See Also
convert[ratpoly]
numapprox/hermite_pade
numapprox[chebpade]
numapprox[laurent]
taylor
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