compute a Pade approximation - Maple Help

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numapprox[pade] - compute a Pade approximation

Calling Sequence

pade(f, x=a, [m, n])

pade(f, x, [m, n])

Parameters

f

-

expression representing the function to be approximated

x

-

the variable appearing in f

a

-

the point about which to expand in a series

m, n

-

desired degree of numerator and denominator, respectively

Description

• 

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 degpxm and degqxn 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 n=0 or if the third argument is simply an integer m then the Taylor (or Laurent) polynomial of 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.

Examples

withnumapprox:

padeⅇx,x,3,3

110x2+12x+1+1120x3110x212x+11120x3

(1)

pade1xsinx,x=0,4,6

75x4+5460x2+166320551x622260x4+166320x2

(2)

padeΓx,x=1,1,1

γ+12γ2+112π2x1γ+112π2+12γ2x1

(3)

padecosx,x,3,4

161150x2775x2+1+1200x4

(4)

padecosx,x,7

112x2+124x41720x6

(5)

See Also

convert[ratpoly], numapprox/hermite_pade, numapprox[chebpade], numapprox[laurent], taylor


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