Solving interval linear least squares problems by PPS-methods
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Solving interval linear least squares problems by PPS-methods Sergey P. Shary1
· Behnam Moradi2
Received: 7 February 2020 / Revised: 23 May 2020 / Accepted: 26 May 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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 A and an interval m-vector b, respectively. We have to compute, with a prescribed accuracy, outer coordinate-wise estimates of the set of all least squares solutions to Ax = b for A ∈ A and b ∈ b. 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 interval least squares problem under solution. 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. Keywords Interval systems of linear equations · Least squares problems · Outer estimation of solution set · PPS-methods Mathematics Subject Classification (2010) 65F20 · 65G40 · 65H10 · 93E24 Sergey P. Shary
[email protected] Behnam Moradi [email protected] 1
Federal Research Center for Information and Computational Technologies and Novosibirsk State University, Novosibirsk, Russia
2
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
Numerical Algorithms
1 Introduction The subject of our paper is the traditional linear least squares problem in which the input data are not precise and have interval uncertainty. We need to evaluate the variation in the solution of the linear least squares problem when its data changes within prescribed intervals. Let us be given an m × n-system of linear algebraic equations of the form ⎧ a11 x1 + a12 x2 + . . . + a1n xn = b1 , ⎪ ⎪ ⎪ ⎪ ⎨ a21 x1 + a22 x2 + . . . + a2n xn = b2 , (1) .. .. .. .. .. ⎪ . ⎪ . . . . ⎪ ⎪ ⎩ am1 x1 + am2 x2 + . . . + amn xn = bm , or, briefly, Ax = b
(2)
with an m × n-matrix A = (aij ), m ≥ n, and a right-hand side m-vector b = (bi ). This system of equations may or may not have the usual solution, but in our paper we will look for its least squares pseudo-solutions that minimizes the Euclidean norm of its residual, that is, m 1/2 Ax − b2 = ((Ax)i − bi )2 i=1
(see, e.g., [8]). In practice, the matrix A and vector b are often imprecise, and we only know interval bounds a ij and bi for the respective coefficients and right-hand side components, such that aij ∈ a ij and bi ∈ bi . Therefore, instead of the above systems of linear equations, we get an interval linear syst
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