Solving interval linear least squares problems by PPS-methods
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In our work, we consider the linear least squares problem for m× n-systems of linear equations Ax = b, m≥ n, such that the matrix A and right-hand side vector b can vary within an interval m× n-matrix and an interval m-vector respectively. We have to compute, as sharp as possible, an interval enclosure of the set of all least squares solutions to Ax = b when A and b independently vary within their interval bounds. Our article is devoted to the development of the so-called PPS-methods (based on Partitioning of the Parameter Set) to solve the above problem. We reduce the normal equation system, associated with the linear lest squares problem, to a special extended matrix form and produce a symmetric interval system of linear equations that is equivalent to the original interval least squares problem. To solve such symmetric system, we propose a new construction of PPS-methods, called ILSQ-PPS, which estimates the enclosure of the solution set with practical efficiency. To demonstrate the capabilities of the ILSQ-PPS method, we present a number of numerical tests and compare their results with those obtained by other methods.
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