kernel

Usage

kernel L
kernel A

Parameter	Type	Description
L	${\mathbb{Q}}[n,E]$	A difference operator
A	${\mathbb{Q}}(n)^{m,m}$	A matrix of fractions

Description

kernel L returns a basis for $\mbox{Ker}~L \cap {\mathbb{Q}}(n)$, while kernel A returns a basis for $\mbox{Ker}(Y(n+1) = A Y(n)) \cap {\mathbb{Q}}(x)^m$.

Example

The equation

$$L = (n+2)(n+4)y(n+2)-(2(n+1)(n+3)+1)y(n+1)+(n+1)^2 y(n) = 0$$

has the following rational solutions:

1 --> L := (n+2)*(n+4)*E^2-(2*(n+1)*(n+3)+1)*E+(n+1)^2;
2 --> K := kernel(L);
3 --> tex(K);

$$[{{1} \over {n^{2}+3\,n+2}}]$$

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 := (n+2)*(n+4)*E^2-(2*(n+1)*(n+3)+1)*E+(n+1)^2;
> rationalKernel(L,E,n);

$$\left[{{1} \over {n^{2}+3\,n+2}}\right]$$

See Also

polynomialSolution, rationalSolution