A note on compound matrices
The aim of this note is to show the advantages of the so called compound matrix method over standard boundary value solvers to compute the eigenvalues of linear eigenvalue problems. In the first section the authors give a brief description of the method. If the eigenvalue problem is written in the form: \(y'(z)=A(\lambda,z)y(z)\), \(z\in[0,1]\), \(By(0)=Cy(1)=0\) where \(y\in \mathbb{C}^ n\) and \(B\) and \(C\) constant matrices with \(\text{rank }B=n-p\geq p=\text{rank }C\), there exist \(p\) linearly independent solutions \(y_ j(\lambda;z)\in \mathbb{C}^ n\), \(j=1,\dots,p\) of the differential equation satisfying the boundary conditions at \(z=0\). In the compound matrix method all possible minors of order \(p\) of such a set of independent solutions are taken as new independent variables. In this way the eigenvalue problem is described by means of a new set of \({n \choose p}\) differential equations together with suitable boundary conditions which is solved with a shooting type method. Finally, to illustrate the performance of the compound matrix method the authors present some numerical examples arising in physical problems together with some comments on the advantages and disadvantages of this technique.
scientific article; zbMATH DE number 149168