LDU Factorization Research Articles

A New Modeling Method for Multiple-Relay Wireless Power Transfer System Considering Cross-Coupling

The cross-coupling among coils makes it complex to analyze and understand the circuit characters of multiple-relay wireless power transfer (MR-WPT) systems. In order to describe and explain the interaction among cross-coupled coils, this paper proposes a novel method named the transfer coefficient model (TCM), which is derived by the LDU factorization of the system's impedance matrix. This model can describe the interaction among cross-coupled coil loops and the composition of each coil current, with clear physical meanings and recursion forms for programming. As a result, the input and output characters of MR-WPT systems can be obtained and explained explicitly. This paper presents the comprehensive and recursion form of TCM for an arbitrary number of coils, followed by the analysis of MR-WPT systems under various operation modes. A five-coil prototype is built to demonstrate the system characters obtained from the TCM. The experimental results coincide with the theoretical analysis, indicating that the TCM is an effective approach for analyzing MR-WPT systems.

Structured condition numbers and statistical condition estimation for the LDU factorization

In this article, we consider the structured condition numbers for LDU, factorization by using the modified matrix-vector approach and the differential calculus, which can be represented by sets of parameters. By setting the specific norms and weight parameters, we present the expressions of the structured normwise, mixed, componentwise condition numbers and the corresponding results for unstructured ones. In addition, we investigate the statistical estimation of condition numbers of LDU factorization using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, we compare the structured condition numbers with the corresponding unstructured ones in numerical experiments.

Numerical methods for accurate computation of the eigenvalues of Hermitian matrices and the singular values of general matrices

This paper offers a review of numerical methods for computation of the eigenvalues of Hermitian matrices and the singular values of general and some classes of structured matrices. The focus is on the main principles behind the methods that guarantee high accuracy even in the cases that are ill-conditioned for the conventional methods. First, it is shown that a particular structure of the errors in a finite precision implementation of an algorithm allows for a much better measure of sensitivity and that computation with high accuracy is possible despite a large classical condition number. Such structured errors incurred by finite precision computation are in some algorithms e.g. entry-wise or column-wise small, which is much better than the usually considered errors that are in general small only when measured in the Frobenius matrix norm. Specially tailored perturbation theory for such structured perturbations of Hermitian matrices guarantees much better bounds for the relative errors in the computed eigenvalues. Secondly, we review an unconventional approach to accurate computation of the singular values and eigenvalues of some notoriously ill-conditioned structured matrices, such as e.g. Cauchy, Vandermonde and Hankel matrices. The distinctive feature of accurate algorithms is using the intrinsic parameters that define such matrices to obtain a non-orthogonal factorization, such as the LDU factorization, and then computing the singular values of the product of thus computed factors. The state of the art software is discussed as well.

Numerical Investigation of Preconditioning for Iterative Methods in Linear Systems Obtained by Extended Element-Free Galerkin Method

In an extended element-free Galerkin method (X-EFG), the essential and natural boundary conditions can be imposed by the collocation based method. However, a coefficient matrix of linear systems obtained by X-EFG become asymmetric, although a symmetric structure exists in a part of the coefficient matrix. In fact, the structure of the coefficient matrices almost becomes symmetric when the size of linear systems is large. Hence, efficient effects may be obtained by using preconditioning for symmetric matrices. The purpose of the present study is to investigate effects of preconditioning for symmetric matrices to linear systems obtained by X-EFG. To this end, the incomplete Cholesky factorization (IC) is applied to the linear systems by regarding the coefficient matrix as symmetric one. In numerical experiments, it is found that the linear systems obtained by X-EFG can be efficiently solved by using IC as preconditioning for GMRES(m) and Bi-CGSTAB. In some cases, the efficiency of IC is superior than that of the incomplete LDU factorization as preconditioning for these iterative methods.

QR decomposition of almost strictly sign regular matrices

In this paper, LDU factorization of nonsingular sign regular and almost strictly sign regular matrices is analyzed. Besides, nonsingular almost strictly sign regular matrices are characterized through the QR factorization. Illustrative numerical examples are included.

Overlapping for preconditioners based on incomplete factorizations and nested arrow form

In this paper, we discuss the usage of overlapping techniques for improving the convergence of preconditioners based on incomplete factorizations. To enable parallelism, these preconditioners are usually applied after the input matrix is permuted into a nested arrow form using k-way nested dissection. This graph partitioning technique uses k-way partitionning by vertex separator to recursively partition the graph of the input matrix into k subgraphs using a subset of its vertices called a separator. The overlapping technique is then based on algebraically extending the associated subdomains of these subgraphs and their corresponding separators obtained from k-way nested dissection by their direct neighbours. A similar approach is known to accelerate the convergence of domain decomposition methods, where the input matrix is partitioned into a number of independent subdomains using k-way vertex partitioning of a graph by edge separators, a different graph decomposition technique. We discuss the effect of the overlapping technique on the convergence of two classes of preconditioners, on the basis of nested factorization and block incomplete LDU factorization.

Efficient preconditioner updates for unsymmetric shifted linear systems

In this paper, we introduce a preconditioning strategy for unsymmetric shifted linear systems (A+αI)x=b, which is a generalization of the scheme proposed by Bellavia et al. (2011). By modifying the nonzero entries in the LDU factorization of A, we give a series of preconditioners with the same sparsity pattern as the seed preconditioner. Theoretical analyses show that when α→0 or ∞, the eigenvalues of the preconditioned systems will cluster about 1. Some practical examples, including non-Hermitian eigenvalue problems arising from the convection diffusion equation, are given to illustrate the efficiency of the preconditioners.

A New Perturbation Bound for the LDU Factorization of Diagonally Dominant Matrices

This work introduces a new perturbation bound for the $L$ factor of the LDU factorization of (row) diagonally dominant matrices computed via the column diagonal dominance pivoting strategy. This strategy yields $L$ and $U$ factors which are always well-conditioned and, so, the LDU factorization is guaranteed to be a rank-revealing decomposition. The new bound together with those for the $D$ and $U$ factors in [F. M. Dopico and P. Koev, Numer. Math., 119 (2011), pp. 337--371] establish that if diagonally dominant matrices are parameterized via their diagonally dominant parts and off-diagonal entries, then tiny relative componentwise perturbations of these parameters produce tiny relative normwise variations of $L$ and $U$ and tiny relative entrywise variations of $D$ when column diagonal dominance pivoting is used. These results will allow us to prove in a follow-up work that such perturbations also lead to strong perturbation bounds for many other problems involving diagonally dominant matrices.

MRILDU: An Improvement to ILUT Based on Incomplete LDU Factorization and Dropping in Multiple Rows

We provide an improvement MRILDU to ILUT for general sparse linear systems in the paper. The improvement is based on the consideration that relatively large elements should be kept down as much as possible. To do so, two schemes are used. Firstly, incomplete LDU factorization is used instead of incomplete LU. Besides, multiple rows are computed at a time, and then dropping is applied to these rows to extract the relatively large elements in magnitude. Incomplete LDU is not only fairer when there are large differences between the elements of factorsLandU, but also more natural for the latter dropping in multiple rows. And the dropping in multiple rows is more profitable, for there may be large differences between elements in different rows in each factor. The provided MRILDU is comparable to ILUT in storage requirement and computational complexity. And the experiments for spare linear systems from UF Sparse Matrix Collection, inertial constrained fusion simulation, numerical weather prediction, and concrete sample simulation show that it is more effective than ILUT in most cases and is not as sensitive as ILUT to the parameterp, the maximum number of nonzeros allowed in each row of a factor.

Relative Perturbation Theory for Diagonally Dominant Matrices

In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices.

Accurate computations of matrices with bidiagonal decomposition using methods for totally positive matrices

SUMMARYA class of sign‐symmetric P‐matrices including all nonsingular totally positive matrices and their inverses as well as tridiagonal nonsingular H‐matrices is presented and analyzed. These matrices present a bidiagonal decomposition that can be used to obtain algorithms to compute with high relative accuracy their singular values, eigenvalues, inverses, or their LDU factorization. Copyright © 2012 John Wiley & Sons, Ltd.

Accurate and efficient LDU decomposition of diagonally dominant M-matrices

NA † Abstract. An efficient method for the computation to high relative accuracy of the LDU decomposition of an n × n row diagonally dominant M-matrix is presented, assuming that the off-diagonal entries and row sums are given. This method costs an additional O(n2) elementary operations over the cost of Gaussian elimination, and leads to a lower triangular, column diagonally dominant matrix and an upper triangular, row diagonally dominant matrix. Comparisons with other methods in the literature are commented and illustrated. 1. Introduction. Recent advances in Numerical Linear Algebra have shown that certain classes of matrices allow computation of certain matrix functions to high relative accuracy, independently of the size of the classical condition number. Some of these classes of matrices are defined by special sign or other structure and require knowledge of some natural parameters to high relative accuracy. In most of those cases, accurate spectral computation (eigenvalues, singular values) is assured once we have an accurate matrix factorization with a suitable pivoting. For instance, the bidiagonal decomposition in the case of totally nonnegative matrices (see also (5), (10)) or an LDU factorization after a symmetric pivoting in the case of diagonally dominant matrices (cf. (4), (13), (14)). Let us focus now on the problem considered in this paper. An algorithm published in (2) computes with high relative accuracy the LDU factorization of an n × n row diagonally dominant M-matrix, if the off-diagonal entries and the row sums are given. The trick is to modify Gaussian elimination to compute the off-diagonal entries and the row sums of each Schur complement without performing subtractions. In addition, symmetric complete pivoting was used in (4) in order to obtain well conditioned L and U factors (U is even row diagonally dominant). This factorization is a special case of a

Perturbation theory for the LDU factorization and accurate computations for diagonally dominant matrices

We present a structured perturbation theory for the LDU factorization of (row) diagonally dominant matrices and we use this theory to prove that a recent algorithm of Ye (Math Comp 77(264):2195–2230, 2008) computes the L, D and U factors of these matrices with relative errors less than 14n 3 u, where u is the unit roundoff and n × n is the size of the matrix. The relative errors for D are componentwise and for L and U are normwise with respect the “max norm” $${\|A\|_M = \max_{ij} |a_{ij}|}$$. These error bounds guarantee that for any diagonally dominant matrix A we can compute accurately its singular value decomposition and the solution of the linear system Ax = b for most vectors b, independently of the magnitude of the traditional condition number of A and in O(n 3) flops.

Direct model reference adaptive control using K p = LDU factorization for multivariable discrete-time plants

For a large class of discrete-time multivariable plants with arbitrary relative degrees, the design and analysis of the direct model reference adaptive control scheme are investigated under less restrictive assumptions. The algorithm is based on a new parametrization derived from the high frequency gain matrix factorization Kp = LDU under the condition that the signs of the leading principal minors ofKp are known. By reproving the discrete-time \( \mathcal{L}_p \) and \( \mathcal{L}_{2\delta } \) norm relationship between inputs and outputs, establishing the properties of discrete-time adaptive law, defining the normalizing signal, and relating the signal with all signals in the closed-loop system, the stability and convergence of the discrete-time multivariable model reference adaptive control scheme are analyzed rigorously in a systematic fashion as in the continuous-time case.

Stochastic Preconditioning for Diagonally Dominant Matrices

This paper presents a new stochastic preconditioning approach for large sparse matrices. For the class of matrices that are rowwise and columnwise irreducibly diagonally dominant, we prove that an incomplete $\rm{LDL^T}$ factorization in a symmetric case or an incomplete LDU factorization in an asymmetric case can be obtained from random walks and used as a preconditioner. It is argued that our factor matrices have better quality, i.e., better accuracy-size trade-offs, than preconditioners produced by existing incomplete factorization methods. Therefore a resulting preconditioned Krylov-subspace iterative solver requires less computation than traditional methods to solve a set of linear equations with the same error tolerance. The advantage increases for larger and denser matrices. These claims are verified by numerical tests, and we provide techniques that can potentially extend the theory to non-diagonally-dominant matrices.

Perturbation bounds for triangular and full rank factorizations

We give componentwise bounds for the perturbations of the LU and LDU factorizations.These bounds are valid for all perturbations which keeps nonsingularity and LU-factorizability. It is shown that the new perturbation bounds are sharper than the earlier results. The perturbation bounds are then applied to full rank factorizations produced by the rank reduction procedure. The result indicates that these full rank factorizations are stable if the LDU factorization of a certain matrix is also stable.

Robust direct model reference adaptive control using KP = LDU factorization for multivariable plants

Motivated by the idea of high frequency gain matrix KP = LDU factorization, the purpose of this paper is to study the problem of robust direct model reference adaptive control for more general multivariable plants with unmodeled dynamics and disturbances. For multivariable plants, by reproving some properties similar to those in robust adaptive control theory for SISO plants and redefining the normalizing signal, the relationship between all the signals in the closed-loop plant and the normalizing signal can be established, and the stability and robustness of the closed-loop plant are analyzed rigorously from a qualitative point of view.

Computing the singular value decomposition with high relative accuracy

We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, which in general only guarantee correct digits in the singular values with large enough magnitudes. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as finite element problems and quantum mechanics, it is the smallest singular values that have physical meaning, and should be determined accurately by the data. Many recent papers have identified special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite different, motivating us to seek a common perturbation theory and common algorithm. We provide these in this paper, and show that high relative accuracy is possible in many new cases as well. The briefest way to describe our results is that we can compute the SVD of G to high relative accuracy provided we can accurately factor G=XDY T where D is diagonal and X and Y are any well-conditioned matrices; furthermore, the LDU factorization frequently does the job. We provide many examples of matrix classes permitting such an LDU decomposition.

LDU factorization results for bi-infinite and semi-infinite scalar and block Toeplitz matrices

In this article various existence results for theLDU-factorization of semi-infinite and bi-infinite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonormalization of splines is indicated. Both banded and non-banded Toeplitz matrices are considered. Extensive use is made of matrix polynomial theory. Results on the approximation by theLDU-factorizations of finite sections are discussed. The generalization of the results to theLDU-factorization of multi-index Toeplitz matrices is outlined.

Combining direct and inverse factors for solving sparse network equations in parallel

A new parallel algorithm for solving the forward and back substitution part of the solution of sparse network equations for power systems is proposed. The approach assumes that the network equations are solved by LDU factorization and sparse vector techniques. The parallelization approach is based on the factorization path tree and its main feature is the switching from direct factor to inverse factors to best exploit the parallelism during the solution. The algorithm can be easily mapped on a multiprocessor architecture that exhibits both global and local memories. An illustrative example and validation results showing the effectiveness of the algorithm are also presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>