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numapprox

  

hermite_pade

  

compute a Hermite-Pade approximation

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

hermite_pade([f1, f2,..., fn], x, N)

hermite_pade([f1, f2,..., fn], x, [d1, d2,..., dn])

hermite_pade([f1, f2,..., fn], x=a, N)

hermite_pade([f1, f2,..., fn], x=a, [d1, d2,..., dn])

Parameters

f1, ..., fn

-

expressions representing the functions to be approximated

x

-

the variable appearing in the f_i's

a

-

the point about which to expand in a series

N

-

non-negative integer

d1, ..., dn

-

degree bounds

Description

• 

The function hermite_pade computes a Hermite-Pade approximation of degree (d1,..., dn) for the functions f1,..., fn with respect to the variable x. When the degrees are not specified, but rather the order N is given, then an approximation of minimal degree is computed.

• 

Specifically, f1,..., fn are expanded in Taylor series about the point x=a (if a is not specified then the expansion is about the point x=0), to order d1+...+dn+n1, and then the Hermite-Pade rational approximation is computed.

• 

The (d1,..., dn) Hermite-Pade approximation is defined to be the list of polynomials p1x,...,pnx with degp[i]xdi such that the Taylor series expansion of p1xf1x+...+pnxfnx has maximal valuation at x=a.

• 

Various levels of user information will be displayed during the computation if infolevel[hermite_pade] is assigned values between 1 and 3.

• 

This code is based on a procedure by H. Derksen in previous versions of the share library.

• 

The command with(numapprox,hermite_pade) allows the use of the abbreviated form of this command.

Examples

withnumapprox:

hermite_padesinx,cosx,ⅇx,x=0,3,2,5

5x3+45x2255x1275,75x2+495x+120,x520x4+160x3600x2+900x120

(1)

hermite_padesinx,cosx,x=π,7

6xπ2+15,xπ315x+15π

(2)

ffcos2xx+1+3,cosx2+xcosx+1,cos2x+1,cosx:

gghermite_padeff,x=0,20

gg:=2,2x4,3x,2x2+4x

(3)

simplifyff1gg1+ff2gg2+ff3gg3+ff4gg4

0

(4)

References

  

Beckermann, B., and Labahn, G. "A uniform approach for Hermite Pade and simultaneous Pade approximants and their matrix-type generalizations." Numerical Algorithms, Vol. 3, (1992): 45-54.

  

Beckermann, B., and Labahn, G. "A uniform approach for the fast computation of matrix-type Pade approximants." SIAM Journal on Matrix Analysis and Applications, Vol. 15, No. 3, (1994): 804-823.

  

Derksen,H. An algorithm to compute generalized Pade-Hermite forms. 1994. Available at http://www.math.lsa.umich.edu/~hderksen/preprints/pade.dvi.

See Also

numapprox[pade]

series

 


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