Numerical Methods for Eigenvalue Problems

This textbook presents a number of the most important numerical methods for finding eigenvalues and eigenvectors of matrices. The authors discuss the central ideas underlying the different algorithms and introduce the theoretical concepts required to analyze their behaviour. Several programming examples allow the reader to experience the behaviour of the different algorithms first-hand. The book addresses students and lecturers of mathematics and engineering who are interested in the fundamental ideas of modern numerical methods and want to learn how to apply and extend these ideas to solve ne. Read more... Preface; 1 Introduction; 1.1 Example: Structural mechanics; 1.2 Example: Stochastic processes; 1.3 Example: Systems of linear differential equations; 2 Existence and properties of eigenvalues and eigenvectors; 2.1 Eigenvalues and eigenvectors; 2.2 Characteristic polynomials; 2.3 Similarity transformations; 2.4 Some properties of Hilbert spaces; 2.5 Invariant subspaces; 2.6 Schur decomposition; 2.7 Non-unitary transformations; 3 Jacobi iteration; 3.1 Iterated similarity transformations; 3.2 Two-dimensional Schur decomposition; 3.3 One step of the iteration; 3.4 Error estimates 3.5 Quadratic convergence4 Power methods; 4.1 Power iteration; 4.2 Rayleigh quotient; 4.3 Residual-based error control; 4.4 Inverse iteration; 4.5 Rayleigh iteration; 4.6 Convergence to invariant subspace; 4.7 Simultaneous iteration; 4.8 Convergence for general matrices; 5 QR iteration; 5.1 Basic QR step; 5.2 Hessenberg form; 5.3 Shifting; 5.4 Deflation; 5.5 Implicit iteration; 5.6 Multiple-shift strategies; 6 Bisection methods; 6.1 Sturm chains; 6.2 Gershgorin discs; 7 Krylov subspace methods for large sparse eigenvalue problems; 7.1 Sparse matrices and projection methods 7.2 Krylov subspaces7.3 Gram-Schmidt process; 7.4 Arnoldi iteration; 7.5 Symmetric Lanczos algorithm; 7.6 Chebyshev polynomials; 7.7 Convergence of Krylov subspace methods; 8 Generalized and polynomial eigenvalue problems; 8.1 Polynomial eigenvalue problems and linearization; 8.2 Matrix pencils; 8.3 Deflating subspaces and the generalized Schur decomposition; 8.4 Hessenberg-triangular form; 8.5 Deflation; 8.6 The QZ step; Bibliography; Index