Linear differential systems
function:eigenring
FUNCTION: eigenring - compute the eigenring of a system of linear differential
equations
CALLING SEQUENCE:
eigenring(A, x)
PARAMETERS:
A - a square rational function matrix with coefficients in an algebraic
extension of the rationals
x - a name
SYNOPSIS:
The eigenring function computes a matrix representation of the eigenring of
a system of linear differential equations.
The returned matrix is a general rational solution of the system
T' = AT - TA.
EXAMPLES:
> A := array(1 .. 3, 1 .. 3,[(2, 3)=(-2+4*x)/x/(x^2+2),(1, 2)=-4/x,
(2, 1)=0,(1,3)=0,(2, 2)=1,(1, 1)=1,(3, 3)=1,(3, 1)=0,(3, 2)=9/(x^2+2)]);
[1 - 4/x 0 ]
[ ]
[ -2 + 4 x ]
[0 1 ----------]
[ 2 ]
A := [ x (x + 2)]
[ ]
[ 9 ]
[0 ------ 1 ]
[ 2 ]
[ x + 2 ]
> eigenring(A, x);
[_C1 0 0 ]
[ ]
[ 0 _C1 0 ]
[ ]
[ 0 0 _C1]
> C :=
array(1 .. 4, 1 .. 4,[(3, 4)=1,(1, 2)=1,(2, 2)=0,(2, 1)=0,(1, 3)=0,(3, 3)=
0,(4, 4)=-6/x,(3, 2)=0,(2, 3)=1,(3, 1)=0,(2, 4)=0,(4, 1)=-1/x^8,(4, 3)=-2*(x^2-
1)/x^4,(1, 1)=0,(4, 2)=2*(3*x^2-1)/x^5,(1, 4)=0]);
[ 0 1 0 0 ]
[ ]
[ 0 0 1 0 ]
[ ]
[ 0 0 0 1 ]
C := [ ]
[ 2 2 ]
[ 1 3 x - 1 x - 1 ]
[- ---- 2 -------- -2 ------ - 6/x]
[ 8 5 4 ]
[ x x x ]
> eigenring(C,x);
[ 3 4 5]
[ 7 _C1 + _C2 -2 _C1 x - _C1 x _C1 x _C1 x ]
[ ]
[ _C1 ]
[ - --- 4 _C1 + _C2 _C1 x 0 ]
[ 3 ]
[ x ]
[ ]
[ _C1 _C1 ]
[ 3 --- - --- 5 _C1 + _C2 _C1 x ]
[ 4 3 ]
[ x x ]
[ ]
[ 2 2 2 ]
[ _C1 (12 x + 1) _C1 (3 x + 2) _C1 (2 x - 1) ]
[- --------------- 2 -------------- - -------------- _C2 ]
[ 7 4 3 ]
[ x x x ]