Abstract
In this article, we present a QR updating procedure as a solution approach for linear least squares problem with equality constraints. We reduce the constrained problem to unconstrained linear least squares and partition it into a small subproblem. The QR factorization of the subproblem is calculated and then we apply updating techniques to its upper triangular factor R to obtain its solution. We carry out the error analysis of the proposed algorithm to show that it is backward stable. We also illustrate the implementation and accuracy of the proposed algorithm by providing some numerical experiments with particular emphasis on dense problems.
Highlights
We consider the linear least squares problem with equality constraints (LSE) min x Ax – b, subject to Bx = d, ( )where A ∈ Rm×n, b ∈ Rm, B ∈ Rp×n, d ∈ Rp, x ∈ Rn with m + p ≥ n ≥ p and · denotes the Euclidean norm
In direct elimination and nullspace methods, the LSE problem is first transformed into unconstrained linear least squares (LLS) problem and it is solved via normal equations or QR factorization methods
6 Conclusion The solution of linear least squares problems with equality constraints is studied by updated techniques based on QR factorization
Summary
Where A ∈ Rm×n, b ∈ Rm, B ∈ Rp×n, d ∈ Rp, x ∈ Rn with m + p ≥ n ≥ p and · denotes the Euclidean norm. In direct elimination and nullspace methods, the LSE problem is first transformed into unconstrained linear least squares (LLS) problem and it is solved via normal equations or QR factorization methods. The solution of the LSE problem is approximated by solving the weighted LLS problem. Which is an unconstrained weighted LLS problem where γ ≥ A / B M given in [ ] and approximated its solution by updating Householder QR factorization. 2.1 The method of weighting This method is based on the observations that while solving LSE problem ( ) we are interested that some equations are to be exactly satisfied This can be achieved by multiplying large weighted factor γ to those equations. The method of weighting is useful as it allows for the use of subroutines for LLS problems to approximate the solution of LSE problem.
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