TY - JOUR

T1 - Using perturbed qr factorizations to solve linear least-squares problems

AU - Avron, Haim

AU - Ng, Esmond

AU - Toledo, Sivan

PY - 2009

Y1 - 2009

N2 - We propose and analyze a new tool to help solve sparse linear least-squares problems minx \\Ax - b\\2. Our method is based on a sparse QR factorization of a low-rank perturbation Â of A. More precisely, we show that the R factor of Â is an effective preconditioner for the least-squares problem minx \\Ax - b\\2, when solved using LSQR. We propose applications for the new technique. When A is rank deficient, we can add rows to ensure that the preconditioner is well conditioned without column pivoting. When A is sparse except for a few dense rows, we can drop these dense rows from A to obtain Â. Another application is solving an updated or downdated problem. If R is a good preconditioner for the original problem A, it is a good preconditioner for the updated/downdated problem Â. We can also solve what-if scenarios, where we want to find the solution if a column of the original matrix is changed/removed. We present a spectral theory that analyzes the generalized spectrum of the pencil (A≠ A, R≠ R) and analyze the applications.

AB - We propose and analyze a new tool to help solve sparse linear least-squares problems minx \\Ax - b\\2. Our method is based on a sparse QR factorization of a low-rank perturbation Â of A. More precisely, we show that the R factor of Â is an effective preconditioner for the least-squares problem minx \\Ax - b\\2, when solved using LSQR. We propose applications for the new technique. When A is rank deficient, we can add rows to ensure that the preconditioner is well conditioned without column pivoting. When A is sparse except for a few dense rows, we can drop these dense rows from A to obtain Â. Another application is solving an updated or downdated problem. If R is a good preconditioner for the original problem A, it is a good preconditioner for the updated/downdated problem Â. We can also solve what-if scenarios, where we want to find the solution if a column of the original matrix is changed/removed. We present a spectral theory that analyzes the generalized spectrum of the pencil (A≠ A, R≠ R) and analyze the applications.

KW - Iterative linear least-squares solvers

KW - Preconditioning sparse qr

UR - http://www.scopus.com/inward/record.url?scp=72449138061&partnerID=8YFLogxK

U2 - 10.1137/070698725

DO - 10.1137/070698725

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AN - SCOPUS:72449138061

SN - 0895-4798

VL - 31

SP - 674

EP - 693

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 2

ER -