Using Perturbed QR Factorizations To Solve Linear Least-Squares Problems

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We propose and analyze a new tool to help solve sparse linear least-squares problems min{sub x} {parallel}Ax-b{parallel}{sub 2}. Our method is based on a sparse QR factorization of a low-rank perturbation {cflx A} of A. More precisely, we show that the R factor of {cflx A} is an effective preconditioner for the least-squares problem min{sub x} {parallel}Ax-b{parallel}{sub 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 ... continued below

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Avron, Haim; Ng, Esmond G. & Toledo, Sivan March 21, 2008.

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We propose and analyze a new tool to help solve sparse linear least-squares problems min{sub x} {parallel}Ax-b{parallel}{sub 2}. Our method is based on a sparse QR factorization of a low-rank perturbation {cflx A} of A. More precisely, we show that the R factor of {cflx A} is an effective preconditioner for the least-squares problem min{sub x} {parallel}Ax-b{parallel}{sub 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 {cflx A}. 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 {cflx A}. 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.

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  • Journal Name: SIAM Journal on Matrix Analysis and Applications

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  • Report No.: LBNL-912E
  • Grant Number: DE-AC02-05CH11231
  • Office of Scientific & Technical Information Report Number: 938513
  • Archival Resource Key: ark:/67531/metadc899428

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  • March 21, 2008

Added to The UNT Digital Library

  • Sept. 27, 2016, 1:39 a.m.

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  • Nov. 8, 2016, 1:10 p.m.

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Avron, Haim; Ng, Esmond G. & Toledo, Sivan. Using Perturbed QR Factorizations To Solve Linear Least-Squares Problems, article, March 21, 2008; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc899428/: accessed October 22, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.