Usage
kernel L
kernel(L, p)
kernel A
Parameter | Type | Description |
---|---|---|
L | A differential operator | |
A | A matrix of fractions | |
p | A polynomial |
Description
kernel L returns a basis for , while kernel A returns a basis for . If a second argument is present, then a basis for the subspace of all solutions whose denominator has all its roots among the roots of . For example, kernel(L, 1) returns a basis for , the space of all the polynomial solutions of .
Example
The operator
has the following rational kernel:
1 --> L := D^3+(5*x^4-8/15*x^2-1/5)/(x^5-14/15*x^3-1/15*x)*D^2 +(-26*x^2-22/15)/(x^4-14/15 *x^2-1/15)*D +(-30*x^2-6)/(x^5-14/15*x^3-1/15*x); 2 --> K := kernel(L); 3 --> tex(K);
To compute its polynomial kernel:
4 --> P := kernel(L, 1); 5 --> tex(P);
Finally for the elements of the kernel whose denominator is a power of (although the result is the same as the rational kernel above, this computation is about 3 times faster):
6 --> K1 := kernel(L, x); 7 --> tex(K1);
Usage within MAPLE
> L := D^3+(5*x^4-8/15*x^2-1/5)/(x^5-14/15*x^3-1/15*x)*D^2 +(-26*x^2-22/15)/(x^4-14/15 *x^2-1/15)*D +(-30*x^2-6)/(x^5-14/15*x^3-1/15*x); > rationalKernel(L,D,x);
> polynomialKernel(L,D,x);
See Also
polynomialSolution, rationalSolution