Robustness of Computed GUPTRI Form and Error Bounds.

The GUPTRI algorithm and software is numerically stable in the sense that it computes the exact Kronecker structure (generalized Schur-staircase form) of a nearby pencil $A' - \lambda B'$. $\delta \equiv {\Vert(A - A', B - B')\Vert}_E$ is an upper bound on the distance to the closest $\{A + \delta A, B + \delta B\}$ with the KCF of $\{A', B'\}$. An accurate estimate of $\delta$ is the square root of the sum of the squares of all singular values interpreted as zeros during the reduction to GUPTRI form.

We refer to sections 5 and 6 in [122] for a more detailed discussion of the computed GUPTRI form and the associated error bounds for reducing subspaces and generalized eigenvalues. For example, section 6 presents a sample usage of all four routines for an application in control theory (computing the controllable subspace and uncontrollable modes).