ACM Transactions on Mathematical Software (TOMS)
On the Compatibility of a Given Solution With the Data of a Linear System
Journal of the ACM (JACM)
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Design, implementation and testing of extended and mixed precision BLAS
ACM Transactions on Mathematical Software (TOMS)
Error bounds from extra-precise iterative refinement
ACM Transactions on Mathematical Software (TOMS)
A Partial Condition Number for Linear Least Squares Problems
SIAM Journal on Matrix Analysis and Applications
A fast and robust mixed-precision solver for the solution of sparse symmetric linear systems
ACM Transactions on Mathematical Software (TOMS)
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We present the algorithm, error bounds, and numerical results for extra-precise iterative refinement applied to overdetermined linear least squares (LLS) problems. We apply our linear system refinement algorithm to Björck’s augmented linear system formulation of an LLS problem. Our algorithm reduces the forward normwise and componentwise errors to O(ϵw), where ϵw is the working precision, unless the system is too ill conditioned. In contrast to linear systems, we provide two separate error bounds for the solution x and the residual r. The refinement algorithm requires only limited use of extra precision and adds only O(mn) work to the O(mn2) cost of QR factorization for problems of size m-by-n. The extra precision calculation is facilitated by the new extended-precision BLAS standard in a portable way, and the refinement algorithm will be included in a future release of LAPACK and can be extended to the other types of least squares problems.