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### kernel

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

• In order not to conflict with the linalg[kernel] function in MAPLE, kernel is available under MAPLE under the names polynomialKernel and rationalKernel. So the above examples in MAPLE would be:
> 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);