Convergence of the shifted QR algorithm on 3×3 normal matrices

Summary This paper concerns the convergence properties of the shifted QR algorithm on 3×3 normal, Hessenberg matrices. The algorithm is viewed as an iteration on one dimensional subspaces. A class of matrices is characterized for which HQR2 is unable to approximate a solution.The author was partially supported by the NSF     

The QRPS algorithm: A generalization of the QR algorithm for the singular values decomposition of rectangular matrices

A generalization of the QR algorithm proposed by Francis [2] for square matrices is introduced for the singular values decomposition of arbitrary rectangular matrices. Geometrically the algorithm means the subsequent orthogonalization of the image of orthonormal bases produced in the course of the iteration. Our purpose is to show how to get a series of lower triangular matrices by alternate orthogonal-upper triangular decompositions in different dimensions and to prove the convergence of this series.     

Roundoff error analysis of algorithms based on Krylov subspace methods

We study the roundoff error propagation in an algorithm which computes the orthonormal basis of a Krylov subspace with Householder orthonormal matrices. Moreover, we analyze special implementations of the classical GMRES algorithm, and of the Full Orthogonalization Method. These techniques approximate the solution of a large sparse linear system of equations on a sequence of Krylov subspaces of small dimension. The roundoff error analyses show upper bounds for the error affecting the computed approximated solutions.This work was carried out with the financial contribution of the Human Capital and Mobility Programme of the European Union grant ERB4050PL921378.     

A numerical approach for solving weakly singular partial integro‐differential equations via two‐dimensional‐orthonormal Bernstein polynomials with the convergence analysis

In this paper, we develop an efficient matrix method based on two‐dimensional orthonormal Bernstein polynomials (2D‐OBPs) to provide approximate solution of linear and nonlinear weakly singular partial integro‐differential equations (PIDEs). First, we approximate all functions involved in the considerable problem via 2D‐OBPs. Then, by using the operational matrices of integration, differentiation, and product, the solution of Volterra singular PIDEs is transformed to the solution of a linear or nonlinear system of algebraic equations which can be solved via some suitable numerical methods. With a small number of bases, we can find a reasonable approximate solution. Moreover, we establish some useful theorems for discussing convergence analysis and obtaining an error estimate associated with the proposed method. Finally, we solve some illustrative examples by employing the presented method to show the validity, efficiency, high accuracy, and applicability of the proposed technique.     

Accurate downdating of a modified Gram-Schmidt QR decomposition

A new algorithm for downdating a QR decomposition is presented. We show that, when the columns in the Q factor from the Modified Gram-Schmidt QR decomposition of a matrixX are exactly orthonormal, the Gram-Schmidt downdating algorithm for the QR decomposition ofX is equivalent to downdating the full Householder QR decomposition of the matrixX augmented by ann ×n zero matrix on top. Using this relation, we derive an algorithm that improves the Gram-Schmidt downdating algorithm when the columns in the Q factor are not orthonormal. Numerical test results show that the new algorithm produces far more accurate results than the Gram-Schmidt downdating algorithm for certain ill-conditioned problems.This work was partially supported in part by the National Science Foundation grants CCR-9209726 and CCR-9509085.     

Accuracy and Stability of the Null Space Method for Solving the Equality Constrained Least Squares Problem

The null space method is a standard method for solving the linear least squares problem subject to equality constraints (the LSE problem). We show that three variants of the method, including one used in LAPACK that is based on the generalized QR factorization, are numerically stable. We derive two perturbation bounds for the LSE problem: one of standard form that is not attainable, and a bound that yields the condition number of the LSE problem to within a small constant factor. By combining the backward error analysis and perturbation bounds we derive an approximate forward error bound suitable for practical computation. Numerical experiments are given to illustrate the sharpness of this bound.     

Multivariate orthonormal interpolating scaling vectors

In this paper we introduce an algorithm for the construction of interpolating scaling vectors on ${\mathbb{R}}^d$ with compact support and orthonormal integer translates. Our method is substantiated by constructing several examples of bivariate scaling vectors for quincunx and box–spline dilation matrices. As the main ingredients of our recipe we derive some implementable conditions for accuracy and orthonormality of an interpolating scaling vector in terms of its mask.     

Efficient solution approaches for a discrete multi-facility competitive interaction model

In this paper, we present efficient solution approaches for discrete multi-facility competitive interaction model. Applying the concept of “Tangent Line Approximation” presented by the authors in their previous work, we develop efficient computational approaches—both exact and approximate (with controllable error bound α). Computational experiments show that the approximate approach (with small α) performs extremely well solving large scale problems while the exact approach performs very well for small to medium-sized problems.     

A Class of Incomplete Orthogonal Factorization Methods. II: Implementation and Results

We present, implement and test several incomplete QR factorization methods based on Givens rotations for sparse square and rectangular matrices. For square systems, the approximate QR factors are used as right-preconditioners for GMRES, and their performance is compared to standard ILU techniques. For rectangular matrices corresponding to linear least-squares problems, the approximate R factor is used as a right-preconditioner for CGLS. A comprehensive discussion is given about the uses, advantages and shortcomings of the preconditioners. AMS subject classification (2000) 65F10, 65F25, 65F50.Received May 2002. Revised October 2004. Communicated by Åke Björck.     

An approximate solution for linear boundary-value problems with slowly varying coefficients

In this paper, an approximate closed-form solution for linear boundary-value problems with slowly varying coefficient matrices is obtained. The derivation of the approximate solution is based on the freezing technique, which is commonly used in analyzing the stability of slowly varying initial-value problems as well as solving them. The error between the approximate and the exact solutions is given, and an upper bound on the norm of the error is obtained. This upper bound is proportional to the rate of change of the coefficient matrix of the boundary-value problem. The proposed approximate solution is obtained for a two-point boundary-value problem and is compared to its solution obtained numerically. Good agreement is observed between the approximate and the numerical solutions, when the rate of change of the coefficient matrix is small.     

小波分析中一个逆问题

本文证明:如果来自多尺度分析(伸缩因子为矩阵)的小波是标准正交的,那么相对应的尺度函数也是标准正交的,其中函数f_s(x)∈L~2(R~n)(s=1,2,…,r,r是正整数)的标准正交性是指f_s(x)的整平移所构成的函数族为L~2(R~n)的标准正交系。结果表明,如果我们想从多尺度分析出发构造正交小波,那么该多尺度分析必须有正交尺度函数。     

An inexact spectral bundle method for convex quadratic semidefinite programming

We present an inexact spectral bundle method for solving convex quadratic semidefinite optimization problems. This method is a first-order method, hence requires much less computational cost in each iteration than second-order approaches such as interior-point methods. In each iteration of our method, we solve an eigenvalue minimization problem inexactly, and solve a small convex quadratic semidefinite program as a subproblem. We give a proof of the global convergence of this method using techniques from the analysis of the standard bundle method, and provide a global error bound under a Slater type condition for the problem in question. Numerical experiments with matrices of order up to 3000 are performed, and the computational results establish the effectiveness of this method.     

An efficient and accurate parallel algorithm for the singular value problem of bidiagonal matrices

Summary. In this paper we propose an algorithm based on Laguerre's iteration, rank two divide-and-conquer technique and a hybrid strategy for computing singular values of bidiagonal matrices. The algorithm is fully parallel in nature and evaluates singular values to tiny relative error if necessary. It is competitive with QR algorithm in serial mode in speed and advantageous in computing partial singular values. Error analysis and numerical results are presented. Received March 15, 1993 / Revised version received June 7, 1994     

An algorithm based on the generalized D-gap function for equilibrium problems

The equilibrium problem (EP) can be reformulated as an unconstrained minimization problem through the generalized D-gap function. In this paper, we propose an algorithm for minimizing the problem and analyze some convergence properties of the proposed algorithm. Under some reasonable conditions, we show that the iteration sequence generated by the algorithm is globally convergent and converges to a solution to the EP and the generalized D-gap function provides a global error bound for the algorithm.     

Multiscale Haar Orthonormal Matrices with the Corresponding Riesz Products and a Characterization of Cantor-Type Languages

We introduce a class of multiscale orthonormal matrices H(m) of order m×m, m = 2, 3,... . For m = 2 ^N, N = 1, 2,..., we get the well known Haar wavelet system. The term "multiscale" indicates that the construction of H(m) is achieved in different scales by an iteration process, determined through the prime integer factorization of m and by repetitive dilation and translation operations on matrices. The new Haar transforms allow us to detect the underlying ergodic structures on a class of Cantor-type sets or languages. We give a sufficient condition on finite data of lengthm, or step functions determined on the intervals [k/m, (k + 1)/m) , k = 0,...,m − 1 of [0, 1), to be written as a Riesz-type product in terms of the rows of H(m). This allows us to approximate in the weak-* topology continuous measures by Riesz-type products.     

QR algorithm with two-sided Rayleigh quotient shifts

We introduce the two-sided Rayleigh quotient shift to the QR algorithm for non-Hermitian matrices to achieve a cubic local convergence rate. For the singly shifted case, the two-sided Rayleigh quotient iteration is incorporated into the QR iteration. A modified version of the method and its truncated version are developed to improve the efficiency. Based on the observation that the Francis double-shift QR iteration is related to a 2D Grassmann–Rayleigh quotient iteration, A doubly shifted QR algorithm with the two-sided 2D Grassmann–Rayleigh quotient double-shift is proposed. A modified version of the method and its truncated version are also developed. Numerical examples are presented to show the convergence behavior of the proposed algorithms. Numerical examples also show that the truncated versions of the modified methods outperform their counterparts including the standard Rayleigh quotient single-shift and the Francis double-shift.     

On the weighting method for mixed least squares–total least squares problems

It is well known that the standard algorithm for the mixed least squares–total least squares (MTLS) problem uses the QR factorization to reduce the original problem into a standard total least squares problem with smaller size, which can be solved based on the singular value decomposition (SVD). In this paper, the MTLS problem is proven to be closely related to a weighted total least squares problem with its error‐free columns multiplied by a large weighting factor. A criterion for choosing the weighting factor is given; and for the sake of stability in solving the MTLS problem, the Cholesky factorization‐based inverse (Cho‐INV) iteration and Rayleigh quotient iteration are also considered. For large‐scale MTLS problems, numerical tests show that Cho‐INV is superior to the standard QR‐SVD method, especially for the case with big gap between the desired and undesired singular values and the case when the coefficient matrix has much more error‐contaminated columns. Rayleigh quotient iteration behaves more efficient than QR‐SVD for most cases and fails occasionally, and in some cases, it converges much faster than Cho‐INV but still less efficient due to its higher computation cost.     

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

A novel collocation method based on Genocchi wavelet is presented for the numerical solution of fractional differential equations and time‐fractional partial differential equations with delay. In this work, to achieve the approximate solution with height accuracy, we employed the operational matrix of integer derivative and the pseudo‐operational matrix of fractional derivative in Caputo sense. Also, based on Genocchi function properties, we presented delay and pantograph operational matrices of Genocchi wavelet functions (GWFs). Due to operational and pseudo‐operational matrices, the equations under this study can be turned into nonlinear algebraic equations with the unknown GWF coefficients. For illustrating the upper bound of error for the proposed method, we estimate the error in the sense of Sobolev space. In addition, to demonstrate the efficacy of the pseudo‐operational matrix of fractional derivative, we investigate the upper bound of error for the mentioned matrix. Finally, the algorithm based on the proposed approach is implemented for some numerical experiments to confirm accuracy and applicability.     

A new a posteriori error estimate for convection–reaction–diffusion problems

A new a posteriori error estimate is derived for the stationary convection–reaction–diffusion equation. In order to estimate the approximation error in the usual energy norm, the underlying bilinear form is decomposed into a computable integral and two other terms which can be estimated from above using elementary tools of functional analysis. Two auxiliary parameter-functions are introduced to construct such a splitting and tune the resulting bound. If these functions are chosen in an optimal way, the exact energy norm of the error is recovered, which proves that the estimate is sharp. The presented methodology is completely independent of the numerical technique used to compute the approximate solution. In particular, it is applicable to approximations which fail to satisfy the Galerkin orthogonality, e.g. due to an inconsistent stabilization, flux limiting, low-order quadrature rules, round-off and iteration errors, etc. Moreover, the only constant that appears in the proposed error estimate is global and stems from the Friedrichs–Poincaré inequality. Numerical experiments illustrate the potential of the proposed error estimation technique.     

The Worst-Case GMRES for Normal Matrices

We study the convergence of GMRES for linear algebraic systems with normal matrices. In particular, we explore the standard bound based on a min-max approximation problem on the discrete set of the matrix eigenvalues. This bound is sharp, i.e. it is attainable by the GMRES residual norm. The question is how to evaluate or estimate the standard bound, and if it is possible to characterize the GMRES-related quantities for which this bound is attained (worst-case GMRES). In this paper we completely characterize the worst-case GMRES-related quantities in the next-to-last iteration step and evaluate the standard bound in terms of explicit polynomials involving the matrix eigenvalues. For a general iteration step, we develop a computable lower and upper bound on the standard bound. Our bounds allow us to study the worst-case GMRES residual norm as a function of the eigenvalue distribution. For hermitian matrices the lower bound is equal to the worst-case residual norm. In addition, numerical experiments show that the lower bound is generally very tight, and support our conjecture that it is to within a factor of 4/π of the actual worst-case residual norm. Since the worst-case residual norm in each step is to within a factor of the square root of the matrix size to what is considered an “average” residual norm, our results are of relevance beyond the worst case. This revised version was published online in July 2006 with corrections to the Cover Date.