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RationalCanonicalForm

  

construct two differential rational canonical forms of a rational function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

RationalCanonicalForm[1](F, x)

RationalCanonicalForm[2](F, x)

Parameters

F

-

rational function of x

x

-

variable

Description

• 

Let F be a rational function of x over a field K of characteristic 0. The RationalCanonicalForm[i](F,x) calling sequence constructs the ith differential rational canonical forms for F, i=1,2.

  

If the RationalCanonicalForm command is called without an index, the first differential rational canonical form is constructed.

• 

The output is a sequence of 2 elements R,V, called RationalCanonicalForm(F), where R,V are rational functions over K such that

1. 

F=R+xVV.

2. 

 gcddenomR,numerRiⅆⅆxdenomR=1for allintegersi. 

• 

If the third optional argument, which is the name 'polyform', is given, the output is a sequence of 4 elements a,b,c,d, where a,b,c,d are polynomials over K, b,c,d monic such that R=ab, V=cd.

• 

The use of RationalCanonicalForm[1] is for testing similarity of two given hyperexponential functions. For RationalCanonicalForm[2], the polynomials b,c,d are also pairwise relatively prime. RationalCanonicalForm[2] is used in a reduction algorithm for hyperexponential functions.

Examples

withDEtools:

F4x2+4x+13x+129x129x2+12x3+4x2+1x3+4x22

F:=4x2+4x+13x+129x129x2+12x3+4x2+1x3+4x22

(1)

R1,V1RationalCanonicalForm[1]F,x

R1,V1:=12x812x7108x648x5239x4+48x350x2+144x47x+12x12x3+4x22,x+14x24x3+4x23

(2)

R2,V2RationalCanonicalForm[2]F,x

R2,V2:=5x916x814x7134x6+39x5331x4+96x3+32x2+16x7x+12x12x3+4x22,x24

(3)

a1,b1,c1,d1RationalCanonicalForm[1]F,x,'polyform'

a1,b1,c1,d1:=12x812x7108x648x5239x4+48x350x2+144x47,x+12x12x3+4x22,x+14x24,x3+4x23

(4)

References

  

Geddes, Keith; Le, Ha; and Li, Ziming. "Differential rational canonical forms and a reduction algorithm for hyperexponential functions." Proceedings of ISSAC 2004. ACM Press, (2004): 183-190.

See Also

DEtools[AreSimilar]

DEtools[MultiplicativeDecomposition]

DEtools[PolynomialNormalForm]

DEtools[ReduceHyperexp]

SumTools[Hypergeometric][RationalCanonicalForm]

 


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