kernel

Usage

kernel L
kernel(L, p)
kernel A

Parameter	Type	Description
L	$\mathbbm{Q}[x,\frac d{dx}]$	A differential operator
A	$\mathbbm{Q}(x)^{n,n}$	A matrix of fractions
p	$\mathbbm{Q}[x]$	A polynomial

Description

kernel L returns a basis for $\mbox{Ker}~L \cap \mathbbm{Q}(x)$, while kernel A returns a basis for $\mbox{Ker}(Y' = A Y) \cap \mathbbm{Q}(x)^n$. If a second argument $p$ is present, then a basis for the subspace of all solutions whose denominator $b \in \mathbbm{Q}[x]$ has all its roots among the roots of $p$. For example, kernel(L, 1) returns a basis for $\mbox{Ker}~L \cap \mathbbm{Q}[x]$, the space of all the polynomial solutions of $L y = 0$.

Example

The operator

$$L = \frac{d^3}{dx^3}+{{5\,x^{4}-{{8} \over {15}}\,x^{2}-{{1}... ...6} \over {x^{5}-{{14} \over {15}}\,x^{3}-{{1} \over {15}}\,x}}$$

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);

$$[{{1} \over {x}}, x^{5}-{{10} \over {9}}\,x^{3}+{{5} \over {21}}\,x]$$

To compute its polynomial kernel:

4 --> P := kernel(L, 1);
5 --> tex(P);

$$[ x^{5}-{{10} \over {9}}\,x^{3}+{{5} \over {21}}\,x ]$$

Finally for the elements of the kernel whose denominator is a power of $x$ (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);

$$[{{1} \over {x}}, x^{5}-{{10} \over {9}}\,x^{3}+{{5} \over {21}}\,x]$$

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);
  

  
  
  $$\left[{{1} \over {x}}, x^{5}-{{10} \over {9}}\,x^{3}+{{5} \over {21}}\,x\right]$$
  
  

  > polynomialKernel(L,D,x);
  

  
  
  $$\left[ x^{5}-{{10} \over {9}}\,x^{3}+{{5} \over {21}}\,x \right]$$
  
  

  
  See Also

  polynomialSolution, rationalSolution