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Purpose


LA_GGESX computes for a pair of $n\times n$ real or complex matrices $( A, B )$ the (generalized) real or complex Schur form, the generalized eigenvalues in the form of scalar pairs $(\alpha, \beta)$, and, optionally, the left and/or right Schur vectors.
If $A$ and $B$ are real then the real-Schur form is computed, otherwise the complex-Schur form is computed. The real-Schur form is a pair of real matrices $(S, T)$ such that 1) $S$ has block upper triangular form, with $1 \times 1$ and $2 \times 2$ blocks along the main diagonal, 2) $T$ has upper triangular form with nonnegative elements on the main diagonal, and 3) $S = Q^T A Z$ and $T = Q^T B Z$, where $Q$ and $Z$ are orthogonal matrices. The $2 \times 2$ blocks of $S$ are ``standardized'' by making the corresponding elements of $T$ have the form

\begin{displaymath}\left[ \begin{array}{cc}
a & 0 \\ 0 & b \end{array} \right] \end{displaymath}

The complex-Schur form is a pair of matrices $(S, T)$ such that 1) $S$ has upper triangular form, 2) $T$ has upper triangular form with nonnegative elements on the main diagonal, and 3) $S =
Q^H A Z$ and $T = Q^H B Z$, where $Q$ and $Z$ are unitary matrices.
In both cases the columns of $Q$ and $Z$ are called, respectively, the left and right Schur vectors.
A generalized eigenvalue of the pair $( A, B )$ is, roughly speaking, a scalar of the form $\lambda = \alpha/\beta$ such that the matrix $A - \lambda B$ is singular. It is usually represented as the pair $(\alpha, \beta)$, as there is a reasonable interpretation of the case $\beta = 0$ (even if $\alpha = 0$)
LA_GGESX also computes two reciprocal condition numbers for the average of the selected eigenvalues and reciprocal condition numbers for the right and left deflating subspaces corresponding to the selected eigenvalues.

next up previous contents index
Next: Arguments Up: Generalized Nonsymmetric Eigenvalue Problems Previous: LA_GGESX   Contents   Index
Susan Blackford 2001-08-19