Showing papers in "SIAM Journal on Matrix Analysis and Applications in 2004"
TL;DR: General bounds for convergence for this convergence that include the effects of polynomial restarting are developed and impose no restrictions concerning the diagonalizability of the matrix or its degree of nonnormality are developed.
Abstract: The performance of Krylov subspace eigenvalue algorithms for large matrices can be measured by the angle between a desired invariant subspace and the Krylov subspace. We develop general bounds for this convergence that include the effects of polynomial restarting and impose no restrictions concerning the diagonalizability of the matrix or its degree of nonnormality. Associated with a desired set of eigenvalues is a maximum "reachable invariant subspace" that can be developed from the given starting vector. Convergence for this distinguished subspace is bounded in terms involving a polynomial approximation problem. Elementary results from potential theory lead to convergence rate estimates and suggest restarting strategies based on optimal approximation points (e.g., Leja or Chebyshev points); exact shifts are evaluated within this framework. Computational examples illustrate the utility of these results. Origins of superlinear effects are also described.
56 citations
TL;DR: The main result of this paper is that every matrix convex polynomial has degree two or less, more generally, if convexity holds only for all matrices X and Y in an "open set."
Abstract: A polynomial p (with real coefficients) in noncommutative variables is matrix convex provided \[ p(tX+(1-t)Y) \le tp(X)+(1-t)p(Y) \] for all $0 \le t \le 1$ and for all tuples X=(X1,. . .,Xg and Y=(Y1,. . .,Yg) of symmetric matrices on a common finite dimensional vector space of a sufficiently large dimension (depending upon p). The main result of this paper is that every matrix convex polynomial has degree two or less. More generally, the polynomial p has degree at most two if convexity holds only for all matrices X and Y in an "open set." An analogous result for nonsymmetric variables is also obtained.
Matrix convexity is an important consideration in engineering system theory. This motivated our work, and our results suggest that matrix convexity in conjunction with a type of "system scalability" produces surprisingly heavy constraints.
45 citations
TL;DR: It is shown via construction that the inverse problem is solvable for any k, given complex conjugately closed pairs of distinct eigenvalues and linearly independent eigenvectors, provided $k \leq n$.
Abstract: The inverse eigenvalue problem of constructing real and symmetric square matrices M, C, and K of size $n \times n$ for the quadratic pencil $Q(\lambda) = \lambda^2 M + \lambda C + K$ so that $Q(\lambda)$ has a prescribed subset of eigenvalues and eigenvectors is considered This paper consists of two parts addressing two related but different problemsThe first part deals with the inverse problem where M and K are required to be positive definite and semidefinite, respectively It is shown via construction that the inverse problem is solvable for any k, given complex conjugately closed pairs of distinct eigenvalues and linearly independent eigenvectors, provided $k \leq n$ The construction also allows additional optimization conditions to be built into the solution so as to better refine the approximate pencil The eigenstructure of the resulting $Q(\lambda)$ is completely analyzedThe second part deals with the inverse problem where M is a fixed positive definite matrix (and hence may be assumed to be t
42 citations
TL;DR: The overall conclusion is that the structure-preserving iterations and the scaled Newton iteration are all of practical interest, and which iteration is to be preferred is problem-dependent.
Abstract: For any matrix automorphism group G associated with a bilinear or sesquilinear form, Mackey, Mackey, and Tisseur have recently shown that the matrix sign decomposition factors of A ∈ G also lie in G; moreover, the polar factors of A lie in G if the matrix of the underlying form is unitary. Groups satisfying the latter condition include the complex orthogonal, real and complex symplectic, and pseudo-orthogonal groups. This work is concerned with exploiting the structure of G when computing the polar and matrix sign decompositions of matrices in G. We give sufficient conditions for a matrix iteration to preserve the group structure and show that a family of globally convergent rational Pade-based iterations of Kenney and Laub satisfy these conditions. The well-known scaled Newton iteration for computing the unitary polar factor does not preserve group structure, but we show that the approach of the iterates to the group is precisely tethered to the approach to unitarity, and that this forces a different and exploitable structure in the iterates. A similar relation holds for the Newton iteration for the matrix sign function. We also prove that the number of iterations needed for convergence of the structure-preserving methods can be precisely predicted by running an associated scalar iteration. Numerical experiments are given to compare the cubically and quintically converging iterations with Newton's method and to test stopping criteria. The overall conclusion is that the structure-preserving iterations and the scaled Newton iteration are all of practical interest, and which iteration is to be preferred is problem-dependent.
42 citations
TL;DR: In this article, the authors cast the sparse low-rank approximation problem in the framework of penalized optimization problems, and provided a detailed error analysis of the proposed algorithms together with some numerical experiments.
Abstract: In [SIAM J. Matrix Anal. Appl., 23 (2002), pp. 706--727], we developed numerical algorithms for computing sparse low-rank approximations of matrices, and we also provided a detailed error analysis of the proposed algorithms together with some numerical experiments. The low-rank approximations are constructed in a certain factored form with the degree of sparsity of the factors controlled by some user-specified parameters. In this paper, we cast the sparse low-rank approximation problem in the framework of penalized optimization problems. We discuss various approximation schemes for the penalized optimization problem which are more amenable to numerical computations. We also include some analysis to show the relations between the original optimization problem and the reduced one. We then develop a globally convergent discrete Newton-like iterative method for solving the approximate penalized optimization problems. We also compare the reconstruction errors of the sparse low-rank approximations computed by our new methods with those obtained using the methods in the earlier paper and several other existing methods for computing sparse low-rank approximations. Numerical examples show that the penalized methods are more robust and produce approximations with factors which have fewer columns and are sparser.
21 citations
TL;DR: The Neville elimination process is studied in relation to the existence of a totally positive factorization LS of a rectangular matrix.
Abstract: An n × m real matrix A is a totally positive matrix if all its minors are nonnegative. The Neville elimination process is studied in relation to the existence of a totally positive factorization LS of a rectangular matrix. An LS factorization is obtained for a totally positive matrix, where L is a lower echelon form matrix, of size n × k, and S is an upper echelon form matrix, of size k × m, and both L and S are totally positive matrices.
20 citations
TL;DR: Sufficient conditions are given under which derivative and proportional state feedback controls can be constructed so that the periodic closed-loop systems are regular and of index at most one.
Abstract: In this paper, we consider the regularization problem for the linear time-varying discrete-time periodic descriptor systems by derivative and proportional state feedback controls. Sufficient conditions are given under which derivative and proportional state feedback controls can be constructed so that the periodic closed-loop systems are regular and of index at most one. The construction procedures used to establish the theory are based on orthogonal and elementary matrix transformations and can, therefore, be developed to a numerically efficient algorithm. The problem of finite pole assignment of periodic descriptor systems is also studied.
19 citations
TL;DR: A method based on the solution of Hermitian 4 × 4 subproblems which generalizes the Jacobi-like method of Bunse-Gerstner and Fa{\ss}bender for Hamiltonian matrices is discussed.
Abstract: We discuss structure-preserving Jacobi-like algorithms for the solution of the indefinite generalized Hermitian eigenvalue problem. We discuss a method based on the solution of Hermitian 4 × 4 subproblems which generalizes the Jacobi-like method of Bunse-Gerstner and Fa{\ss}bender for Hamiltonian matrices. Furthermore, we discuss structure-preserving Jacobi-like methods based on the solution of non-Hermitian 2 × 2 subproblems. For these methods a local convergence proof is given. Numerical test results for the comparison of the proposed methods are presented.
18 citations
TL;DR: The focus is on the analogues of singular value and CS (cos -- sin) decompositions for general H-unitary and Lorentz matrices, and on the Analogues of Jordan form, in a suitable basis with certain orthonormality properties, for diagonalizable H- unitary andlorentzMatrices.
Abstract: Many properties of H-unitary and Lorentz matrices are derived using elementary methods. Complex matrices that are unitary with respect to the indefinite inner product induced by an invertible Hermitian matrix H are called H-unitary, and real matrices that are orthogonal with respect to the indefinite inner product induced by an invertible real symmetric matrix are called Lorentz. The focus is on the analogues of singular value and CS (cos -- sin) decompositions for general H-unitary and Lorentz matrices, and on the analogues of Jordan form, in a suitable basis with certain orthonormality properties, for diagonalizable H-unitary and Lorentz matrices. Several applications are given, including connected components of Lorentz similarity orbits, products of matrices that are simultaneously positive definite and H-unitary, products of reflections, and stability and robust stability.
16 citations
TL;DR: An a posteriori reorthogonalization technique based on a rank-k update of the computed vectors to recover a set of vectors orthogonal to machine precision level is proposed.
Abstract: The modified Gram--Schmidt algorithm is a well-known and widely used procedure to orthogonalize the column vectors of a given matrix. When applied to ill-conditioned matrices in floating point arithmetic, the orthogonality among the computed vectors may be lost. In this work, we propose an a posteriori reorthogonalization technique based on a rank-k update of the computed vectors. The level of orthogonality of the set of vectors built gets better when k increases and finally reaches the machine precision level for a large enough k. The rank of the update can be tuned in advance to monitor the orthogonality quality. We illustrate the efficiency of this approach in the framework of the seed-GMRES technique for the solution of an unsymmetric linear system with multiple right-hand sides. In particular, we report experiments on numerical simulations in electromagnetic applications where a rank-one update is sufficient to recover a set of vectors orthogonal to machine precision level.
12 citations
TL;DR: Multiplicative backward stability results are presented for two algorithms which compute the singular value decomposition of dense matrices, the classical one-sided Jacobi algorithm, with a stringent stopping criterion, and an algorithm which uses one- sided Jacobi to compute high accurate singularvalue decompositions of matrices given as rank-revealing factorizations.
Abstract: Multiplicative backward stability results are presented for two algorithms which compute the singular value decomposition of dense matrices. These algorithms are the classical one-sided Jacobi algorithm, with a stringent stopping criterion, and an algorithm which uses one-sided Jacobi to compute high accurate singular value decompositions of matrices given as rank-revealing factorizations. When multiplicative backward errors are small, the multiplicative perturbation theory for the singular value decomposition developed in the last decade can be applied to get high accuracy bounds on the errors of the computed singular values and vectors.
TL;DR: An algorithm to compute the smallest even and odd eigenvalues of a real symmetric positive-definite Toeplitz matrix is proposed, which is based on the factorization of the characteristic polynomial into an even and an oddPolynomial.
Abstract: We propose an algorithm to compute the smallest even and odd eigenvalues of a real symmetric positive-definite Toeplitz matrix, which is based on the factorization of the characteristic polynomial into an even and an odd polynomial. Newton's method is used to compute the smallest even and odd eigenvalues as the smallest roots of the even and odd characteristic polynomials, respectively.
TL;DR: A determinant that encompasses the classical Vandermonde determinant, the generalized Vandermondes, and the recently introduced exponential VandermondE determinant when the exponents are nonnegative integers is introduced.
Abstract: We introduce a determinant that encompasses the classical Vandermonde determinant, the generalized Vandermonde determinant, and the recently introduced exponential Vandermonde determinant when the exponents are nonnegative integers. An explicit factorization of such a determinant will be established. This factorization enables us to develop a computationally tractable necessary and sufficient condition for the existence of a unique solution of a Hermite ($\ell$-point) discrete boundary value problem.
TL;DR: It is proved that the closure of the union of all the spectra of preconditioned matrix sequences of the form T_n^{-1}(g)T_n(f) can have gaps if the essential range of f/g is not connected.
Abstract: Different than for the case of Toeplitz matrix sequences $\{T_n(f)\}$, $f\in L^1$, we can prove that the closure of the union of all the spectra of preconditioned matrix sequences of the form $\{T_n^{-1}(g)T_n(f)\}$, $f,g\in L^1$, $g\ge 0$, can have gaps if the essential range of f/g is not connected. The result has important consequences on the practical use of band Toeplitz preconditioners widely used in the literature both for (multilevel) ill-conditioned positive definite and (multilevel) indefinite Toeplitz linear systems.
TL;DR: The spectral structure of P and K is investigated and the eigenvalues of K are characterized as zeros of a rational function, and it is proved that, for normal A, $\sigma(K)$ resides in the convex hull of $\s Sigma(A)$.
Abstract: Let A=P+K be an n × n complex matrix with $P = \frac12(A-HAH)$ and $K=\frac12(A+HAH)$, H being a unitary involution. Having characterized all unitary involutions, we investigate the spectral structure of P and K and, in particular, characterize the eigenvalues of K as zeros of a rational function, and prove that, for normal A, $\sigma(K)$ resides in the convex hull of $\sigma(A)$. We also demonstrate that this need not be true when A is not normal.
TL;DR: It is one of the basic facts of linear time-invariant systems theory that any two minimal realizations are connected in the best possible way: by system similarity.
Abstract: It is one of the basic facts of linear time-invariant systems theory that any two minimal (canonical) realizations are connected in the best possible way: by system similarity. We study five different types of possible connections between two arbitrary realizations of a transfer function, and are interested in questions of existence (sufficient and/or necessary conditions), uniqueness, and description of all (or of a possibly large class of) connecting operators or pairs of operators. In the case of the existence of nonnegative realizations we seek nonnegative connecting pairs or operators.