PeriodicSchurDecompositions.jl

This Julia package provides implementations of the periodic Schur decomposition of matrix products of real element types and of the generalized periodic Schur decomposition for real and complex element types. Functions for reordering invariant subspaces are also provided. A Krylov-Schur scheme for computing extreme eigenvalues and corresponding subspaces of products of linear operators in large-dimensional spaces is also included.

Definitions

Periodic Schur decomposition

Given a series of $N\times N$ matrices $A_j,\ j=1,\ldots,p$, a periodic Schur decomposition (PSD) is a factorization of the form:

\[\begin{aligned} Q_1^\prime A_1 Q_2 &= T_1 \\ Q_2^\prime A_2 Q_3 &= T_2 \\ \vdots& \\ Q_p^\prime A_p Q_1 &= T_p \end{aligned}\]

where the $Q_j$ are unitary (orthogonal) and the $T_j$ are upper triangular, except that one of the $T_j$ is quasi-triangular for real element types. It furnishes the eigenvalues and invariant subspaces of the matrix product $\Pi_{j=1}^p A_j$.

The principal reason for using the PSD is that accuracy may be lost if one forms the product of the $A_j$ before eigen-analysis. For some applications the intermediate Schur vectors are also useful.

Operator ordering

For many applications it is more natural to pose the matrix product in the form $A_p A_{p-1}\ldots A_2 A_1$. In this case the more useful factorization is

\[\begin{aligned} Q_2^\prime A_1 Q_1 &= T_1 \\ Q_3^\prime A_2 Q_2 &= T_2 \\ \vdots& \\ Q_1^\prime A_p Q_p &= T_p. \end{aligned}\]

This ordering is accommodated with the ':L' (left) orientation argument to pschur!.

Generalized periodic Schur decomposition

Given a series of $N\times N$ matrices $A_j,\ j=1,\ldots,p$, and a signature vector $S$ where $s_j\in \{1,-1\}$, a generalized periodic Schur decomposition (GPSD) is a factorization of the formal product $\Pi_{j=1}^p A_j^{s_j}$ so that $Q_j^\prime A_j Q_{j+1} = T_j$ if $s_j = 1$ and $Q_{j+1}^\prime A_j Q_j = T_j$ if $s_j = -1$.

The GPSD is an extension of the QZ decomposition used for generalized eigenvalue problems. Thus formally infinite eigenvalues are not problematic.

References

A. Bojanczyk, G. Golub, and P. Van Dooren, "The periodic Schur decomposition. Algorithms and applications," Proc. SPIE 1996.

D. Kressner, thesis and assorted articles.

Acknowledgements

The meat of this package is largely a translation of implementations in the SLICOT library.