  This paper presents a new method for computing all eigenvalues and
eigenvectors of quadratic matrix pencil. It is an upgrade of the quadeig
algorithm by Hammarling, Munro and Tisseur, which attempts to reveal and remove
by deflation certain number of zero and infinite eigenvalues before QZ
iterations. Proposed modifications of the quadeig framework are designed to
enhance backward stability and to make the process of deflating infinite and
zero eigenvalues more numerically robust. In particular, careful preprocessing
allows scaling invariant/component-wise backward error and thus better
condition number. Further, using an upper triangular version of the Kronecker
canonical form enables deflating additional infinite eigenvalues, in addition
to those inferred from the rank of leading coefficient matrix. Theoretical
analysis and empirical evidence from thorough testing of the software
implementation confirm superior numerical performances of the proposed method.
