Updating QR factorization procedure for solution of linear least squares problem with equality constraints

Area of Science:

• Numerical analysis
• Linear algebra
• Computational mathematics

Background:

• Linear least squares problems with equality constraints are common in data analysis and scientific computing.
• Existing methods may face challenges with large or dense datasets.

Purpose of the Study:

• To present a novel QR updating procedure for solving linear least squares problems with equality constraints.
• To demonstrate the backward stability and accuracy of the proposed algorithm.

Main Methods:

• The constrained problem is reduced to an unconstrained linear least squares subproblem.
• QR factorization is applied to the subproblem, and updating techniques are used on the upper triangular factor R.
• Error analysis is performed to assess backward stability.

Main Results:

• The proposed QR updating procedure effectively solves the constrained linear least squares problem.
• The algorithm is shown to be backward stable through error analysis.
• Numerical experiments confirm the implementation and accuracy, especially for dense problems.

Conclusions:

• The QR updating procedure offers an efficient and stable solution for linear least squares problems with equality constraints.
• The method is particularly well-suited for dense problems.
• This approach contributes to robust numerical methods in linear algebra.