Wronskian Matrix

Let \(R\) be a differential ring and let \(y_1,y_2,\ldots,y_n\) be elements of \(R\). The wronskian matrix of \(y_1,y_2,\ldots,y_n\) is defined as the \(n\times n\) matrix

\[\begin{split} W(y_1,y_2,\ldots,y_n)=\begin{pmatrix}y_1&y_2&\ldots&y_n\\ \delta_R(y_1)&\delta_R(y_2)&\ldots&\delta_R(y_n)\\ \vdots&\vdots&\ddots&\vdots\\ \delta_R^{n-1}(y_1)&\delta_R^{n-1}(y_2)&\ldots&\delta_R^{n-1}(y_n)\end{pmatrix}\end{split}\]

The wronskian determinant, or simply the wronskian, of \(y_1,y_2,\ldots,y_n\) is the determinant of the wronskian matrix \(W(y_1,y_2,\ldots,y_n)\).

WronskianMatrix(L): [RngDiffElt] -> AlgMatElt

Given a sequence of differential ring elements \(L\), return the Wronskian matrix of \(L\) whose entries are elements of the universe of \(L\).

WronskianDeterminant(L): [RngDiffElt] -> RngDiffElt, AlgMatElt

Given a sequence of differential ring elements \(L\), return the determinant of the Wronskian matrix of \(L\) as well as the matrix itself.

Example: Wronskian Matrix and Determinant (ex-95dc85)

> F<z> := RationalDifferentialField(Rationals());
> WronskianMatrix([1,z,z^2]);
[1 z z^2]
[0 1 2*z]
[0 0 2]
> WronskianDeterminant([1,z^2,1/z]);
6/z
[z z^2 1/z]
[1 2*z -1/z^2]
[0 2 2/z^3]

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