DEtools - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Differential Equations : DEtools : Differential Operators : DEtools/eigenring

DEtools

  

eigenring

  

compute the endomorphisms of the solution space

  

endomorphism_charpoly

  

give the characteristic polynomial of an endomorphism

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

eigenring(L, domain)

endomorphism_charpoly(L, r, domain)

Parameters

L

-

differential operator

r

-

differential operator in the output of eigenring

domain

-

list containing two names

Description

• 

The input L is a differential operator. Denote V(L) as the solution space of L. eigenring computes a basis (a vector space) of the set of all operators r for which r(V(L)) is a subset of V(L). So r is an endomorphism of the solution space V(L). The characteristic polynomial of this map can be computed by the command endomorphism_charpoly(L,r).

• 

For endomorphisms r, the product of L and r is divisible on the right by L. If the optional third argument is the equation verify=true then eigenring checks if the output satisfies this condition. This should not be necessary though.

• 

The argument domain describes the differential algebra. If this argument is the list Dt,t then the differential operators are notated with the symbols Dt and t. They are viewed as elements of the differential algebra C(t)[Dt] where C is the field of constants.

• 

If the argument domain is omitted then the differential specified by the environment variable _Envdiffopdomain will be used. If this environment variable is not set, then the argument domain may not be omitted.

• 

These functions are part of the DEtools package, and so they can be used in the form eigenring(..) and endomorphism_charpoly(..) only after executing the command with(DEtools).  However, they can always be accessed through the long form of the command by using DEtools[eigenring](..) or DEtools[endomorphism_charpoly](..).

Examples

withDEtools:

Take the differential ring C(x)[Dx]:

ADx,x

ADx,x

(1)

LDx4+2xDx2+2Dx+x24

LDx4+2Dx2x+x2+2Dx4

(2)

Compute a basis v for the endomorphismsr:VLVL. Compute an eigenvalue e of r. Then compute the greatest common right divisor G. Then the solution space VG is the kernel ofre:VLVL.

veigenringL,A

v1,Dx2+x

(3)

forrinvdoprint'r'=r;Ffactorendomorphism_charpolyL,r,A;foreinsolveF,xdoprintG=GCRDre,L,Aend doend do

r=1

G=Dx4+2xDx2+2Dx+x24

r=Dx2+x

G=Dx2+x+2

G=Dx2+x2

(4)

References

  

For a description of the method used, see:

  

van der Put, M., and Singer, M. F. Galois Theory of Linear Differential Equations, Vol. 328. Springer: 2003. An electronic version of this book is available at http://www4.ncsu.edu/~singer/ms_papers.html.

  

van Hoeij, M. "Rational Solutions of the Mixed Differential Equation and its Application to Factorization of Differential Operators." ISSAC '96 Proceedings. (1996): 219-225.

See Also

DEtools[DFactor]

DEtools[GCRD]

DEtools[Homomorphisms]

diffop