Numerical Methods for General and Structured Eigenvalue ProblemsThe purpose of this book is to describe recent developments in solving eig- value problems, in particular with respect to the QR and QZ algorithms as well as structured matrices. Outline Mathematically speaking, the eigenvalues of a square matrix A are the roots of its characteristic polynomial det(A??I). An invariant subspace is a linear subspace that stays invariant under the action of A. In realistic applications, it usually takes a long process of simpli?cations, linearizations and discreti- tions before one comes up with the problem of computing the eigenvalues of a matrix. In some cases, the eigenvalues have an intrinsic meaning, e.g., for the expected long-time behavior of a dynamical system; in others they are just meaningless intermediate values of a computational method. The same applies to invariant subspaces, which for example can describe sets of initial states for which a dynamical system produces exponentially decaying states. Computing eigenvalues has a long history, dating back to at least 1846 when Jacobi [172] wrote his famous paper on solving symmetric eigenvalue problems. Detailed historical accounts of this subject can be found in two papers by Golub and van der Vorst [140, 327]. |
Contents
The QZ Algorithm | 67 |
The KrylovSchur Algorithm 113 | 112 |
Structured Eigenvalue Problems | 131 |
A Background in Control Theory | 215 |
B Software | 225 |
References 233 | 232 |
Index | 253 |
Common terms and phrases
â â â aggressive early deflation applied Arnoldi decomposition Arnoldi method ba ba backward error backward stable balancing block cyclic block matrix bulge pair columns computing eigenvalues convergence deflating subspaces denotes diagonal blocks flops Francis shifts ĥ ĥ Hamiltonian matrix Hessenberg form Hessenberg matrix Hessenberg-triangular form Householder matrix implementation implicit shifted implies invariant subspace Kågström Krylov decomposition Krylov subspace Krylov-Schur algorithm LAPACK LAPACK routine Lemma Linear Algebra Linear Algebra Appl Math Matrix Anal matrix pair matrix Q Mehrmann norm orthogonal matrix orthogonal symplectic matrix perturbation product eigenvalue problem proof QR algorithm QR decomposition real Schur form reducing reordering restarting Riccati equations Ritz values Rn×n Rnxn Schur decomposition Section shifted QR iteration SIAM singular value skew-Hamiltonian matrix Softcover structured condition number subdiagonal subdiagonal entry submatrix swapping Sylvester equation symmetric symplectic matrix Theorem triangular matrix unreduced Update upper triangular vector zero