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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

\begin{displaymath}
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}}
\end{displaymath}

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);

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

To compute its polynomial kernel:
4 --> P := kernel(L, 1);
5 --> tex(P);

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

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);

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


Usage within MAPLE


next up previous contents index
Next: leftGcd Up: Supported functions Previous: factor   Contents   Index
Manuel Bronstein 2002-09-04