A total least squares method for Toeplitz systems of equations

A Newton method to solve total least squares problems for Toeplitz systems of equations is considered. When coupled with a bisection scheme, which is based on an efficient algorithm for factoring Toeplitz matrices, global convergence can be guaranteed. Circulant and approximate factorization preconditioners are proposed to speed convergence when a conjugate gradient method is used to solve linear systems arising during the Newton iterations. The work of the second author was partially supported by a National Science Foundation Postdoctoral Research Fellowship.     

On generalized successive overrelaxation methods for augmented linear systems

For the augmented system of linear equations, Golub, Wu and Yuan recently studied an SOR-like method (BIT 41(2001)71–85). By further accelerating it with another parameter, in this paper we present a generalized SOR (GSOR) method for the augmented linear system. We prove its convergence under suitable restrictions on the iteration parameters, and determine its optimal iteration parameters and the corresponding optimal convergence factor. Theoretical analyses show that the GSOR method has faster asymptotic convergence rate than the SOR-like method. Also numerical results show that the GSOR method is more effective than the SOR-like method when they are applied to solve the augmented linear system. This GSOR method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration. Besides, we also demonstrate a complete version about the convergence theory of the SOR-like method. Subsidized by The Special Funds For Major State Basic Research Projects (No. G1999032803) and The National Natural Science Foundation (No. 10471146), P.R. China     

BTTB preconditioners for BTTB least squares problems

In this paper, we consider solving the least squares problem min_x‖b-Tx‖₂ by using preconditioned conjugate gradient (PCG) methods, where T is a large rectangular matrix which consists of several square block-Toeplitz-Toeplitz-block (BTTB) matrices and b is a column vector. We propose a BTTB preconditioner to speed up the PCG method and prove that the BTTB preconditioner is a good preconditioner. We then discuss the construction of the BTTB preconditioner. Numerical examples, including image restoration problems, are given to illustrate the efficiency of our BTTB preconditioner. Numerical results show that our BTTB preconditioner is more efficient than the well-known Level-1 and Level-2 circulant preconditioners.     

On the weighting method for least squares problems with linear equality constraints

The weighting method for solving a least squares problem with linear equality constraints multiplies the constraints by a large number and appends them to the top of the least squares problem, which is then solved by standard techniques. In this paper we give a new analysis of the method, based on the QR decomposition, that exhibits many features of the algorithm. In particular it suggests a natural criterion for chosing the weighting factor. This work was supported in part by the National Science Foundation under grant CCR 95503126.     

Symmetric block-SOR methods for rank-deficient least squares problems

In this article, we develop symmetric block successive overrelaxation (S-block-SOR) methods for finding the solution of the rank-deficient least squares problems. We propose an S2-block-SOR and an S3-block-SOR method for solving such problems and the convergence of these two methods is studied. The comparisons between the S2-block and the S3-block methods are presented with some numerical examples.     

Quadratically constrained least squares and quadratic problems

Summary We consider the following problem: Compute a vectorx such that Ax–b₂=min, subject to the constraint x₂=. A new approach to this problem based on Gauss quadrature is given. The method is especially well suited when the dimensions ofA are large and the matrix is sparse.It is also possible to extend this technique to a constrained quadratic form: For a symmetric matrixA we consider the minimization ofx ^T A x–2b ^T x subject to the constraint x₂=.Some numerical examples are given.This work was in part supported by the National Science Foundation under Grant DCR-8412314 and by the National Institute of Standards and Technology under Grant 60NANB9D0908.     

Preconditioned GAOR methods for solving weighted linear least squares problems

In this paper, we present the preconditioned generalized accelerated overrelaxation (GAOR) method for solving linear systems based on a class of weighted linear least square problems. Two kinds of preconditioning are proposed, and each one contains three preconditioners. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the convergence rate of the preconditioned GAOR methods is indeed better than the rate of the original method, whenever the original method is convergent. Finally, a numerical example is presented in order to confirm these theoretical results.     

Bounds for the least squares distance using scaled total least squares

Summary. The standard approaches to solving overdetermined linear systems construct minimal corrections to the data to make the corrected system compatible. In ordinary least squares (LS) the correction is restricted to the right hand side c, while in scaled total least squares (STLS) [14,12] corrections to both c and B are allowed, and their relative sizes are determined by a real positive parameter . As , the STLS solution approaches the LS solution. Our paper [12] analyzed fundamentals of the STLS problem. This paper presents a theoretical analysis of the relationship between the sizes of the LS and STLS corrections (called the LS and STLS distances) in terms of . We give new upper and lower bounds on the LS distance in terms of the STLS distance, compare these to existing bounds, and examine the tightness of the new bounds. This work can be applied to the analysis of iterative methods which minimize the residual norm, and the generalized minimum residual method (GMRES) [15] is used here to illustrate our theory. Received July 20, 2000 / Revised version received February 28, 2001 / Published online July 25, 2001     

Algebraic relations between the total least squares and least squares problems with more than one solution

Summary This paper completes our previous discussion on the total least squares (TLS) and the least squares (LS) problems for the linear systemAX=B which may contain more than one solution [12, 13], generalizes the work of Golub and Van Loan [1,2], Van Huffel [8], Van Huffel and Vandewalle [11]. The TLS problem is extended to the more general case. The sets of the solutions and the squared residuals for the TLS and LS problems are compared. The concept of the weighted squares residuals is extended and the difference between the TLS and the LS approaches is derived. The connection between the approximate subspaces and the perturbation theories are studied.It is proved that under moderate conditions, all the corresponding quantities for the solution sets of the TLS and the modified LS problems are close to each other, while the quantities for the solution set of the LS problem are close to the corresponding ones of a subset of that of the TLS problem.This work was financially supported by the Education Committee, People's Republic of China     

Accelerated Gauss-Newton algorithms for nonlinear least squares problems

Recent theoretical and practical investigations have shown that the Gauss-Newton algorithm is the method of choice for the numerical solution of nonlinear least squares parameter estimation problems. It is shown that when line searches are included, the Gauss-Newton algorithm behaves asymptotically like steepest descent, for a special choice of parameterization. Based on this a conjugate gradient acceleration is developed. It converges fast also for those large residual problems, where the original Gauss-Newton algorithm has a slow rate of convergence. Several numerical test examples are reported, verifying the applicability of the theory.     

On the analysis of the unsymmetric successive overrelaxation method when applied top-cyclic matrices

Summary A Determinantal Invariance, associated with consistently ordered weakly cyclic matrices, is given. The DI is then used to obtain a new functional equation which relates the eigenvalues of a particular block Jacobi iteration matrix to the eigenvalues of its associated Unsymmetric Successive Overrelaxation (USSOR) iteration matrix. This functional equation as well as the theory of nonnegative matrices and regular splittings are used to obtain convergence and divergence regions of the USSOR method.     

Algebraic connections between the least squares and total least squares problems

Summary In this paper the closeness of the total least squares (TLS) and the classical least squares (LS) problem is studied algebraically. Interesting algebraic connections between their solutions, their residuals, their corrections applied to data fitting and their approximate subspaces are proven.All these relationships point out the parameters which mainly determine the equivalences and differences between the two techniques. These parameters also lead to a better understanding of the differences in sensitivity between both approaches with respect to perturbations of the data.In particular, it is shown how the differences between both approaches increase when the equationsAXB become less compatible, when the length ofB orX is growing or whenA tends to be rank-deficient. They are maximal whenB is parallel with the singular vector ofA associated with its smallest singular value. Furthermore, it is shown how TLS leads to a weighted LS problem, and assumptions about the underlying perturbation model of both techniques are deduced. It is shown that many perturbation models correspond with the same TLS solution.Senior Research Assistant of the Belgian N.F.W.O. (National Fund of Scientific Research)     

Constrained least squares methods for linear timevarying DAE systems

Summary This paper presents a family of methods for accurate solution of higher index linear variable DAE systems, . These methods use the DAE system and some of its first derivatives as constraints to a least squares problem that corresponds to a Taylor series ofy, or an approximative equality derived from a Pade' approximation of the exponential function. Accuracy results for systems transformable to standard canonical form are given. Advantages, disadvantages, stability properties and implementation of these methods are discussed and two numerical examples are given, where we compare our results with results from more traditional methods.     

Estimation of optimal backward perturbation bounds for the linear least squares problem

In this paper a method of estimating the optimal backward perturbation bound for the linear least squares problem is presented. In contrast with the optimal bound, which requires a singular value decomposition, this method is better suited for practical use on large problems since it requiresO(mn) operations. The method presented involves the computation of a strict lower bound for the spectral norm and a strict upper bound for the Frobenius norm which gives a gap in which the optimal bounds for the spectral and the Frobenius norm must be. Numerical tests are performed showing that this method produces an efficient estimate of the optimal backward perturbation bound.     

Optimal successive overrelaxation iterative methods for P-cyclic matrices

Summary We consider linear systems whose associated block Jacobi matricesJ _p are weakly cyclic of indexp. In a recent paper, Pierce, Hadjidimos and Plemmons [13] proved that the block two-cyclic successive overrelaxation (SOR) iterative method is numerically more effective than the blockq-cyclic SOR-method, 2<qp, if the eigenvalues ofJ _p ^p are either all non-negative or all non-positive. Based on the theory of stationaryp-step methods, we give an alternative proof of their theorem. We further determine the optimal relaxation parameter of thep-cyclic SOR method under the assumption that the eigenvalues ofJ _p ^p are contained in a real interval, thereby extending results due to Young [19] (for the casep=2) and Varga [15] (forp>2). Finally, as a counterpart to the result of Pierce, Hadjidimos and Plemmons, we show that, under this more general assumption, the two-cyclic SOR method is not always superior to theq-cyclic SOR method, 2<qp.Dedicated to R. S. Varga on the occasion of his 60th birthdayResearch supported in part by the Deutsche Forschungsgemeinschaft     

Component-wise perturbation analysis and error bounds for linear least squares solutions

Perturbation bounds for the linear least squares problem min _x Ax –b₂ corresponding tocomponent-wise perturbations in the data are derived. These bounds can be computed using a method of Hager and are often much better than the bounds derived from the standard perturbation analysis. In particular this is true for problems where the rows ofA are of widely different magnitudes. Generalizing a result by Oettli and Prager, we can use the bounds to compute a posteriori error bounds for computed least squares solutions.     

A factorization method for the solution of constrained linear least squares problems allowing subsequent data changes

Summary In this paper we describe how to use Gram-Schmidt factorizations of Daniel et al. [1] to realize the method of [8] for solving linearly constrained linear least squares problems. The main advantage of using these factorizations is that it is relatively easy to take data changes into account, if necessary.The algorithm is compared numerically with two other codes, one of them published by Lawson and Hanson [3]. Further computational tests show the efficiency of the proposed methods, if a few data of the original problem are changed subsequently.This paper was sponsored by Deutsche Forschungsgemeinschaft, Bonn-Bad Godesberg     

About existence and uniqueness of the restricted total least squares estimation

Summary Total Least Squares (TLS) is an estimation method for the solutiona of the linear system when both data sets and are subject to error. The TLS-method minimizes the functional with weighting parameter . In this paper the TLS-functional is analyzed by the technique of Lagrangian multipliers. The main part of the work deals with the case when the estimatea is restricted by an inequality of the formD a–b0, D a diagonal matrix. It is shown that there exists a unique estimatea if the weighting parameter is chosen sufficiently large.     

Continuous-time accelerated block successive overrelaxation methods for time-dependent Stokes equations

In order to solve the time-dependent Stokes equation, we follow the “Method of Lines” to obtain structured linear constant-coefficient differential–algebraic equations (DAEs). By taking advantage of the structure, we propose a class of waveform relaxation methods, called continuous-time accelerated block SOR (CABSOR) methods, for solving the obtained DAEs. The new methods are theoretically analyzed. The theory is applied to a two-dimensional time-dependent Stokes equation and verified by numerical experiments.     

Some results on condition numbers of the scaled total least squares problem

Under the Golub-Van Loan condition for the existence and uniqueness of the scaled total least squares (STLS) solution, a first order perturbation estimate for the STLS solution and upper bounds for condition numbers of a STLS problem have been derived by Zhou et al. recently. In this paper, a different perturbation analysis approach for the STLS solution is presented. The analyticity of the solution to the perturbed STLS problem is explored and a new expression for the first order perturbation estimate is derived. Based on this perturbation estimate, for some STLS problems with linear structure we further study the structured condition numbers and derive estimates for them. Numerical experiments show that the structured condition numbers can be markedly less than their unstructured counterparts.