Estimation of Backward Perturbation Bounds for Linear Least Squares Problem
نویسنده
چکیده
Waldén, Karlson, and Sun found an elegant explicit expression of backward error for the linear least squares problem. However, it is difficult to compute this quantity as it involves the minimal singular value of certain matrix. In this paper we present a simple estimation to this bound which can be easily computed especially for large problems. Numerical results demonstrate the validity of the estimation.
منابع مشابه
Backward Perturbation Bounds for Linear Least Squares Problems
Recently, Higham and Wald en, Karlson, and Sun have provided formulas for computing the best backward perturbation bounds for the linear least squares problem. In this paper we provide several backward perturbation bounds that are easier to compute and optimal up to a factor less than 2. We also show that any least squares algorithm that is stable in the sense of Stewart is necessarily a backwa...
متن کاملOptimal backward perturbation bounds for the linear least squares problem
Dedicated to William Kahan and Beresford Parlett on the occasion of their 60th birthdays Let A be an m n matrix, b be an m-vector, and x̃ be a purported solution to the problem of minimizing kb Axk2. We consider the following open problem: find the smallest perturbation E of A such that the vector x̃ exactly minimizes kb (A+E)xk2. This problem is completely solved whenE is measured in the Frobeni...
متن کاملAccuracy 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...
متن کاملBackward Error Bounds for Constrained Least Squares Problems ∗
We derive an upper bound on the normwise backward error of an approximate solution to the equality constrained least squares problem minBx=d ‖b − Ax‖2. Instead of minimizing over the four perturbations to A, b, B and d, we fix those to B and d and minimize over the remaining two; we obtain an explicit solution of this simplified minimization problem. Our experiments show that backward error bou...
متن کاملAnalysis of a linearly constrained least squares algorithm for adaptive beamforming
DTrj QUXIAJAT flSCFTED 3 The problem of linearly constrained least squares has many applications in signal processing. In this paper, we present a perturbation analysis of a linearly constrrined least squares algorithm for adaptive beaniforming. The perturbation bounds for the solution as well as for the alaest residual element are derived. We also propose an error estimation scheme for the res...
متن کامل