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.     

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.     

Perturbation bounds for the linear least squares problem subject to linear inequality constraints

We study the effect on the solution to a linear least squares problem with linear inequality and equality constraints when the data defining the problem are perturbed. The existence and uniqueness of a solution are investigated. If the matrices involved have full rank, then a detailed bound is obtained by the duality theory for quadratic programming. Sufficient conditions are derived for an estimate of the perturbation in the solution to hold in terms of the largest perturbation in the data.     

Optimal backward perturbation bounds for the linear least-squares problem with multiple right-hand sides

Let A be an mn matrix, B be an mr matrix, and $$\stackrel{\¯}{X}$$be an approximate solution to the problem of minimizing ||AX–B||_F.In this note we consider the following open problem: find anexplicit expression of the optimal backward perturbation bound_F$${\eta }_{F}\left(\tilde{X}\right)$$ dehned by where is a positive number. This problem is solved when $$\tilde{X}$$is of full column rank.     

On perturbation bounds of the linear complementarity problem

In this paper, new perturbation bounds for linear complementarity problems are presented based on the sign patterns of the solution of the equivalent modulus equations. The new bounds are shown to be the generalization of the existing ones.     

Minimal distance upper bounds for the perturbation of least squares problems in Hilbert spaces

Let X and Y be Hilbert spaces, and let T : X → Y be a bounded linear operator with closed range. In this paper, we present an optimal perturbation result on the least squares solutions to the operator equation Tx = y under the most general condition.     

Solving the minimal least squares problem subject to bounds on the variables

A computational procedure is developed for determining the solution of minimal length to a linear least squares problem subject to bounds on the variables. In the first stage, a solution to the least squares problem is computed and then in the second stage, the solution of minimal length is determined. The objective function in each step is minimized by an active set method adapted to the special structure of the problem.The systems of linear equations satisfied by the descent direction and the Lagrange multipliers in the minimization algorithm are solved by direct methods based on QR decompositions or iterative preconditioned conjugate gradient methods. The direct and the iterative methods are compared in numerical experiments, where the solutions are sought to a sequence of related, minimal least squares problems subject to bounds on the variables. The application of the iterative methods to large, sparse problems is discussed briefly.This work was supported by The National Swedish Board for Technical Development under contract dnr 80-3341.     

Improved componentwise verified error bounds for least squares problems and underdetermined linear systems

Recently Miyajima presented algorithms to compute componentwise verified error bounds for the solution of full-rank least squares problems and underdetermined linear systems. In this paper we derive simpler and improved componentwise error bounds which are based on equalities for the error of a given approximate solution. Equalities are not improvable, and the expressions are formulated in a way that direct evaluation yields componentwise and rigorous estimates of good quality. The computed bounds are correct in a mathematical sense covering all sources of errors, in particular rounding errors. Numerical results show a gain in accuracy compared to previous results.     

On the perturbation of least squares problems with equality constraints

Some new perturbation results are presented for least squares problems with equality constraints, in which relative errors are obtained on perturbed solutions, least squares residuals, and vectors of Lagrange multipliers of the problem, based on the equivalence of the problem to a usual least squares problem and optimal perturbation results for orthogonal projections.     

Error bounds for least squares approximation by polynomials

Let f ε Cⁿ⁺¹[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials q_k associated with a distribution dα on [−1, 1]. It is shown that if q_n+1/q_n max(q_n+1(1)/q_n(1), −q_n+1(−1)/q_n(−1)), then f − H[f] f^{n + 1} · q_n+1/q_{n + 1}^{(n + 1)}, where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)^α (1 + t)^β dt, α, β > −1, the condition on q_n+1/q_n is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.     

Exact and stable least squares solution to the linear programming problem

A linear programming problem is transformed to the finding an element of polyhedron with the minimal norm. According to A. Cline [6], the problem is equivalent to the least squares problem on positive ortant. An orthogonal method for solving the problem is used. This method was presented earlier by the author and it is based on the highly developed least squares technique. First of all, the method is meant for solving unstable and degenerate problems. A new version of the artifical basis method (M-method) is presented. Also, the solving of linear inequality systems is considered.     

Towards backward perturbation bounds for approximate dual Krylov subspaces

Given a matrix A,n by n, and two subspaces K and L of dimension m, we consider how to determine a backward perturbation E whose norm is as small as possible, such that k and L are Krylov subspaces of A+E and its adjoint, respectively. We first focus on determining a perturbation matrix for a given pair of biorthonormal bases, and then take into account how to choose an appropriate biorthonormal pair and express the Krylov residuals as a perturbation of the matrix A. Specifically, the perturbation matrix is globally optimal when A is Hermitian and K=L. The results show that the norm of the perturbation matrix can be assessed by using the norms of the Krylov residuals and those of the biorthonormal bases. Numerical experiments illustrate the efficiency of our strategy.     

A criterion for truncation of theQR-decomposition algorithm for the singular linear least squares problem

A possible improvement of the Faddeev-Kublanovskaja-Faddeeva lower bound for the least singular value ofR by using additional information aboutR is discussed. A fast algorithm is given for calculating such a bound using the diagonal elements and the elements of largest modulus in each row ofR.     

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.     

On the condition number of the total least squares problem

This paper concerns singular value decomposition (SVD)-based computable formulas and bounds for the condition number of the total least squares (TLS) problem. For the TLS problem with the coefficient matrix $A$ and the right-hand side $b$ , a new closed formula is presented for the condition number. Unlike an important result in the literature that uses the SVDs of both $A$ and $[A,\ b]$ , our formula only requires the SVD of $[A,\ b]$ . Based on the closed formula, both lower and upper bounds for the condition number are derived. It is proved that they are always sharp and estimate the condition number accurately. A few lower and upper bounds are further established that involve at most the smallest two singular values of $A$ and of $[A,\ b]$ . Tightness of these bounds is discussed, and numerical experiments are presented to confirm our theory and to demonstrate the improvement of our upper bounds over the two upper bounds due to Golub and Van Loan as well as Baboulin and Gratton. Such lower and upper bounds are particularly useful for large scale TLS problems since they require the computation of only a few singular values of $A$ and $[A, \ b]$ other than all the singular values of them.     

The condition numbers for weighted Moore-Penrose inverse and weighted linear least squares problem

Condition numbers play an important role in numerical analysis. Classical normise condition numbers are used to measure the size of both input perturbations and output errors. In this paper, we study the weighted normwise relative condition numbers for the weighted Moore-Penrose inverse and the weighted linear least-squares (WLS) problems in the case of the full-column rank matrix. The bounds or formulas for the weighted condition numbers are presented. The obtained results can be viewed as extensions of the earlier works studied by others.     

On condition numbers for Moore–Penrose inverse and linear least squares problem involving Kronecker products

In this paper, we investigate the normwise, mixed, and componentwise condition numbers and their upper bounds for the Moore–Penrose inverse of the Kronecker product and more general matrix function compositions involving Kronecker products. We also present the condition numbers and their upper bounds for the associated Kronecker product linear least squares solution with full column rank. In practice, the derived upper bounds for the mixed and componentwise condition numbers for Kronecker product linear least squares solution can be efficiently estimated using the Hager–Higham Algorithm. Copyright © 2012 John Wiley & Sons, Ltd.     

A relative backward perturbation theorem for the eigenvalue problem

Summary A-posteriori bound is given for the computed eigenpair ( ), of the eigenvalue problemAx=x, which is shown to be more realistic than the available one. A simple expression is further presented for calculating the backward error.