An Algorithm for Generalized Matrix Eigenvalue Problems.

@article{Moler1973AnAF,
  title={An Algorithm for Generalized Matrix Eigenvalue Problems.},
  author={Cleve B. Moler and G. W. Stewart},
  journal={SIAM Journal on Numerical Analysis},
  year={1973},
  volume={10},
  pages={241-256}
}
  • C. Moler, G. Stewart
  • Published 1 April 1973
  • Computer Science
  • SIAM Journal on Numerical Analysis
A new method, called the $QZ$ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = \lambda Bx$ with general square matrices A and B. Particular attention is paid to the degeneracies which result when B is singular. No inversions of B or its submatrices are used. The algorithm is a generalization of the $QR$ algorithm, and reduces to it when $B = I$. Problems involving higher powers of $\lambda $ are also mentioned. 
A General Matrix Eigenvalue Algorithm
TLDR
The main emphasis is upon the algorithm’s generality as well as its bearing upon the generalized singular value problem $A^T Ax = \mu ^2 B^T Bx$.
Balancing the Generalized Eigenvalue Problem
  • R. Ward
  • Computer Science, Mathematics
  • 1981
TLDR
The three-step algorithm is specifically designed to precede the $QZ$-type algorithms, but improved performance is expected from most eigensystem solvers.
An extension of the QZ algorithm for solving the generalized matrix eigenvalue problem
TLDR
This algorithm is an extension of Moler and Stewart's QZ algorithm with some added features for saving time and operations and should be preferred over existing algorithms which attempt to solve the class of generalized eigenproblems where both matrices are singular or nearly singular.
A Generalized Eigenvalue Approach for Solving Riccati Equations
A numerically stable algorithm is derived to compute orthonormal bases for any deflating subspace of a regular pencil $\lambda B - A$. The method is based on an update of the $QZ$-algorithm, in order
The Combination Shift $QZ$ Algorithm
An extension of the $QZ$ algorithm, called the combination shift $QZ$ algorithm, is presented for solving the generalized matrix eigenvalue problem $Ax = \lambda Bx$ with real square matrices A and
Numerical Solution of a Quadratic Matrix Equation
This paper is concerned with the efficient numerical solution of the matrix equation $AX^2 + BX + C = 0$, where A, B, C and X are all square matrices. Such a matrix X is called a solvent. This
On the Generalized Schur Decomposition of a Matrix Pencil for Parallel Computation
TLDR
An algorithm to solve the generalized eigenvalue problem for arbitrary complex matrices A, B is developed, a generalization of an unsymmetric Jacobi method of Eberlein, thus admitting efficient parallel implementations on certain parallel architectures.
...
...

References

SHOWING 1-3 OF 3 REFERENCES
An Algorithm for the Ill-Conditioned Generalized Eigenvalue Problem
The spectrum of $Ax - \lambda Bx = 0$ consists of stable and unstable eigenvalues, which undergo, respectively, small and large changes in response to small changes in A and B. The algorithm isolates
Unitary Triangularization of a Nonsymmetric Matrix
TLDR
This note points out that the same result can be obtained with fewer arithmetic operations, and, in particular, for inverting a square matrix of order N, at most 2(N-1) square roots are required.
The algebraic eigenvalue problem
Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of