Schur decomposition of a real upper Hessenberg matrix

The Schur decomposition is a representation of matrix A as A = S^T T S, where S is an orthogonal matrix (matrix of Schur vectors), T - quasi-upper triangular matrix, i.e. triangular matrix with blocks of sizes 1x1 and 2x2 on the main diagonal.

This algorithm performs the Schur decomposition of the matrix in upper Hessenberg form by using a QR algorithm with multiple shifts. This algorithm is a block-matrix analog of the ordinary QR algorithm with double shift. As all other block-matrix algorithms, this algorithm requires adjustment to achieve optimal performance.

You can adjust a value of NS (internal parameter of the subroutine InternalSchurDecomposition) by defining a number of shifts in one iteration. By increasing the number of shifts, the algorithm performance rises, reaching its maximum between NS=4 and NS=16. If NS is more than 16, algorithm slows down considerably. Interval ranges can differ according to different systems, but the situation is mostly same. The default value of NS provides a good performance on most systems, but if the performance is critical, it is worth calibrating this parameter. It should be noted that the optimal parameter value depends both on the system characteristics and on the properties of the matrices processed.

The subroutine UpperHessenbergSchurDecomposition performs the Schur decomposition. It returns matrices T and S. Commonly, we need to know the block structure of matrix T. It could be easily recognized: since all the elements below the blocks are zeros, elements a[i+1,i] which are equal to 0 show the block border.

Algorithm implemented on the basis of the subroutine DHSEQR (LAPACK 3.0 library).