TY - JOUR

T1 - Solving interval linear least squares problems by PPS-methods

AU - Shary, Sergey P.

AU - Moradi, Behnam

PY - 2020/9/4

Y1 - 2020/9/4

N2 - 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.

AB - 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.

KW - Interval systems of linear equations

KW - Least squares problems

KW - Outer estimation of solution set

KW - PPS-methods

KW - SYSTEMS

UR - http://www.scopus.com/inward/record.url?scp=85090316742&partnerID=8YFLogxK

U2 - 10.1007/s11075-020-00958-x

DO - 10.1007/s11075-020-00958-x

M3 - Article

AN - SCOPUS:85090316742

JO - Numerical Algorithms

JF - Numerical Algorithms

SN - 1017-1398

ER -