A modified Gram-Schmidt-based downdating technique for ULV decompositions with applications to recursive TLS problems

The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition (SVD). The ULVD can be updated and downdated much faster than the SVD, hence its utility in the solution of recursive total least squares (TLS) problems. However, the robust implementation of ULVD after the addition and deletion of rows (called updating and downdating, respectively) is not altogether straightforward. When updating or downdating the ULVD, the accurate computation of the subspaces necessary to solve the TLS problem is of great importance. In this paper, algorithms are given to compute simple parameters that can often show when good subspaces have been computed.

[1]  W. Kahan,et al.  The Rotation of Eigenvectors by a Perturbation. III , 1970 .

[2]  J. H. Wilkinson The algebraic eigenvalue problem , 1966 .

[3]  Jesse L. Barlow,et al.  Solving recursive TLS problems using the rank-revealing ULV decomposition , 1997 .

[4]  C. Lawson,et al.  Extensions and applications of the Householder algorithm for solving linear least squares problems , 1969 .

[5]  G. Stewart,et al.  Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization , 1976 .

[6]  Jesse L. Barlow,et al.  Modifying two-sided orthogonal decompositions: algorithms, implementation, and applications , 1996 .

[7]  Hasan Erbay,et al.  Recursive ULV decomposition , 2000, SPIE Optics + Photonics.

[8]  G. Stewart Updating a Rank-Revealing ULV Decomposition , 1993, SIAM J. Matrix Anal. Appl..

[9]  John G. Lewis,et al.  Proceedings of the Fifth SIAM Conference on Applied Linear Algebra , 1994 .

[10]  J. Barlow,et al.  An efficient rank detection procedure for modifying the ULV decomposition , 1998 .

[11]  Ricardo D. Fierro,et al.  The Total Least Squares Problem: Computational Aspects and Analysis (S. Van Huffel and J. Vandewalle) , 1993, SIAM Rev..

[12]  Stanley C. Eisenstat,et al.  Downdating the Singular Value Decomposition , 1995, SIAM J. Matrix Anal. Appl..

[13]  H. Zha,et al.  An algorithm and a stability theory for downdating the ULV decomposition , 1996 .

[14]  Sabine Van Huffel,et al.  Total least squares algorithms based on rank-revealing complete orthogonal decompositions , 1997 .

[15]  Haesun Park,et al.  Accurate downdating of a modified Gram-Schmidt QR decomposition , 1996 .