This paper presents the theoretical background relevant to any method for producing a tridiagonal matrix similar to an arbitrary square matrix. Gragg's work on factoring Hankel matrices and the Kalman-Gilbert stru...
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This paper presents the theoretical background relevant to any method for producing a tridiagonal matrix similar to an arbitrary square matrix. Gragg's work on factoring Hankel matrices and the Kalman-Gilbert structure theorem from systems theory both find a place in the development. Tridiagonalization is equivalent to the application of the generalized Gram-Schmidt process to a pair of Krylov sequences. In Euclidean space proper normalization allows one to monitor a tight lower bound on the condition number of the transformation. The various possibilities for breakdown find a natural classification by the ranks of certain matrices. The theory is illustrated by some small examples and some suggestions for restarting are evaluated.
The Kalman filter is a sequential estimation procedure that combines a stochastic dynamical model with observations in order to update the model state and the associated uncertainty. In the situation where no measurem...
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The Kalman filter is a sequential estimation procedure that combines a stochastic dynamical model with observations in order to update the model state and the associated uncertainty. In the situation where no measurements are available the filter works as an uncertainty propagator. The most computationally demanding part of the Kalman filter is to propagate the covariance through the dynamical system, which may be completely infeasible in high-dimensional models. The reduced rank square-root (RRSQRT) filter is a special formulation of the Kalman filter for large-scale applications. In this formulation, the covariance matrix of the model state is expressed in a limited number of modes M. In the classical implementation of the RRSQRT filter the computational costs of the truncation step grow very fast with the number of modes (> M-3). In this work, a new approach based on the Lanzcos algorithm is formulated. It provides a more cost-efficient scheme and includes a precision coefficient that can be tuned for specific applications depending on the trade-off between precision and computational load. (c) 2004 Elsevier B.V. All rights reserved.
In this paper we present an algorithm for recursively generating orthogonal bivariate polynomials on a discrete set S subset of R-2. For this purpose we employ commuting pairs of real symmetric matrices H, K is an ele...
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In this paper we present an algorithm for recursively generating orthogonal bivariate polynomials on a discrete set S subset of R-2. For this purpose we employ commuting pairs of real symmetric matrices H, K is an element of R-n x n to obtain, in a certain sense, a two dimensional Hermitian lanczos method. The resulting algorithm relies on a recurrence having a slowly growing length. Practical implementation issues an applications are considered. The method can be generalized to compute orthogonal polynomials depending on an arbitrary number of variables.
This paper is a continuation of Part I [M. H. Gutknecht, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 594–639], where the theory of the “unsymmetric” lanczos biorthogonalization (BO) algorithm and the corresponding i...
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This paper is a continuation of Part I [M. H. Gutknecht, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 594–639], where the theory of the “unsymmetric” lanczos biorthogonalization (BO) algorithm and the corresponding iterative method BIORES for non-Hermitian linear systems was extended to the nongeneric case. The analogous extension is obtained here for the biconjugate gradient (or BIOMIN) method and for the related BIODIR method. Here, too, the breakdowns of these methods can be cured. As a preparation, mixed recurrence formulas are derived for a pair of sequences of formal orthogonal polynomials belonging to two adjacent diagonals in a nonnormal Padé table, and a matrix interpretation of these recurrences is developed. This matrix interpretation leads directly to a completed formulation of the progressive qd algorithm, valid also in the case of a nonnormal Padé table. Finally, it is shown how the cure for exact breakdown can be extended to near-breakdown in such a way that (in exact arithmetic) the well-conditioned formal orthogonal polynomials and the corresponding Krylov space vectors do not depend on the threshold specifying the near-breakdown.
We review in this article a recently proposed energy-global method that is capable of calculating the entire transition amplitude matrix with a single lanczos propagation. This method requires neither explicit computa...
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We review in this article a recently proposed energy-global method that is capable of calculating the entire transition amplitude matrix with a single lanczos propagation. This method requires neither explicit computation nor storage of the eigenfunctions, rendering it extremely memory efficient. Procedures are proposed to handle situations where "spurious" eigenvalues aggregate around true eigenvalues due to round-off errors. This method is amenable to both real-symmetric and complex-symmetric Hamiltonians. Applications to molecular spectra and reactive scattering are presented. Its relationships with other methods are also discussed.
In recent papers, numerous authors studied the solutions of symmetric positive definite Toeplitz systems Tx = b by the conjugate gradient method for different families of circulant preconditioners C. In this paper new...
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In recent papers, numerous authors studied the solutions of symmetric positive definite Toeplitz systems Tx = b by the conjugate gradient method for different families of circulant preconditioners C. In this paper new circulant/skewcirculant approximations are introduced to T and their properties are studied. The main interest is directed to the skewcirculant case. Furthermore, algorithms for computing the eigenvalues of T are formulated, based on the lanczos algorithm and Rayleigh quotient iteration. For some numerical examples the spectra of C-1 T are compared and the behaviour of the introduced eigenvalue algorithms is displayed.
In this paper, we consider ill-posed problems which discretize to linear least squares problems with matrices K of high dimensions. The algorithm proposed uses K only as an operator and does not need to explicitly sto...
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In this paper, we consider ill-posed problems which discretize to linear least squares problems with matrices K of high dimensions. The algorithm proposed uses K only as an operator and does not need to explicitly store or modify it. A method related to one of lanczos is used to project the problem onto a subspace for which K is bidiagonal. It is then an easy matter to solve the projected problem by standard regularization techniques. These ideas are illustrated with some integral equations of the first kind with convolution kernels, and sample numerical results are given.
This technical note proposes a novel profile likelihood method for estimating the covariance parameters in exploratory factor analysis of high-dimensional Gaussian datasets with fewer observations than number of varia...
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This technical note proposes a novel profile likelihood method for estimating the covariance parameters in exploratory factor analysis of high-dimensional Gaussian datasets with fewer observations than number of variables. An implicitly restarted lanczos algorithm and a limited-memory quasi-Newton method are implemented to develop a matrix-free framework for likelihood maximization. Simulation results show that our method is substantially faster than the expectation-maximization solution without sacrificing accuracy. Our method is applied to fit factor models on data from suicide attempters, suicide ideators, and a control group. for this article are available online.
algorithms are presented for computing some of the eigenvalues and their associated eigenvectors of the quadratic $\lambda $-matrix$M\lambda ^2 + C\lambda + K$. M, C and K are assumed to have special symmetrytype pr...
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algorithms are presented for computing some of the eigenvalues and their associated eigenvectors of the quadratic $\lambda $-matrix$M\lambda ^2 + C\lambda + K$. M, C and K are assumed to have special symmetrytype properties which insure that theory analogous to the standard symmetric eigenproblem exists. The algorithms are based on a generalization of the Rayleigh quotient and the, lanczos method for computing eigenpairs of standard symmetric eigenproblems. Monotone quadratic convergence of the basic method is proved. Test examples are presented.
An algorithm for computing the eigenvectors corresponding to the m algebraically smallest or largest eigenvalues of an $n \times n$ symmetric matrix ${\bf A}$ is described. The algorithm consists of repeated applicati...
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An algorithm for computing the eigenvectors corresponding to the m algebraically smallest or largest eigenvalues of an $n \times n$ symmetric matrix ${\bf A}$ is described. The algorithm consists of repeated applications of the Rayleigh-Ritz procedure to a sequence of subspaces of dimension $m + 1$ which converges to the desired subspace. The method is closely related to the lanczos method, but requires a constant amount of computation at each iteration. Applications of the algorithm include the adaptive covariance eigenstructure computation, in which the matrix ${\bf A}$ can change while the algorithm is in progress.
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