Characterizing Matrices That Are Consistent with Given Solutions

Abstract

Abstract. For given vectors b ∈ C m and y ∈ C n we describe a unitary transformation approach to deriving the set F of all matrices F ∈ C m×n such that y is an exact solution to the compatible system F y = b. This is used for deriving minimal backward errors E and f such that (A+E)y = b+f when possibly noisy data A ∈ C m×n and b ∈ C m are given, and the aim is to decide if y is a satisfactory approximate solution to Ax = b. The approach might be different, but the above results are not new. However we also prove the apparently new result that two well known approaches to making this decision are theoretically equivalent, and discuss how such knowledge can be used in designing effective stopping criteria for iterative solution techniques. All these ideas generalize to the following formulations. We extend our constructive approach to derive a superset F ST LS+ of the set F ST LS of all matrices F ∈ C m×n such that y is a scaled total least squares solution to F y ≈ b. This is a new general result that specializes in two important ways. The ordinary least squares problem is an extreme case of the scaled total least squares problem, and we use our result to obtain the set F LS of all matrices F ∈ C m×n such that y is an exact least squares solution to F y ≈ b. This complements the original less-constructive derivation of Waldén, Karlson and Sun [Numerical Linear Algebra with Applications, 2:271-286 (1995)]. We do the equivalent for the data least squares problem-the other extreme case of the scaled total least squares problem. Not only can the results be used as indicated above for the compatible case, but the constructive technique we use could also be applicable to other backward problems-such as those for under-determined systems, the singular value decomposition, and the eigenproblem.Key words. matrix characterization, approximate solutions, iterative methods, linear algebraic equations, least squares, data least squares, total least squares, scaled total least squares, backward errors, stopping criteria.AMS subject classifications. 15A06, 15A29, 65F05, 65F10, 65F20, 65F25, 65G99. DOI. (Digital Object Identifier)1. Introduction. We will study a class of 'backward' problems for linear systems F y ≈ b. Specifically, given two vectors y and b we want to find the sets of all matrices F such that y is the exact solution (i.e., F y = b), the least squares (LS) solution, the data least squares (DLS) solution, and the scaled total least square (STLS) solution. We will propose a unified unitary transformation approach to handling these problems.Some of these problems have been investigated before. The result for the compatible case is well-known and the result for the least squares case was obtained elegantly by Waldén, Karlson and Sun in [31]. But while [31] presents, then proves, the least squares result, our approach shows how to derive a more general result in a fairly simple way, and we suspect that this constructive approach is not only easier to comprehend for non-mathematicians, but perhaps easier to apply to ot...

Backward perturbation analysis for scaled total least‐squares problems

“…Unfortunately, the inequality in (7) makes it difficult to derive a general expression for [ A, b] ∈C A,b ( ), hence we initially ignore it and consider a larger set The following result from Theorem 3.2 of [12] gives an explicit relation between A and b for any…”