Modification and Maintenance of ULV Decompositions

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). It is useful in the many applications of the SVD where we are only interested in separating the right singular subspace associated with “large” singular values from that associated with “small” ones.

[1]  Franklin T. Luk,et al.  Advanced Signal Processing Algorithms, Architectures, and Implementations XVIII , 1991 .

[2]  Hasan Erbay,et al.  Two-step Gram-Schmidt downdating methods , 2001, SPIE Optics + Photonics.

[3]  Per Christian Hansen,et al.  UTV Tools: Matlab templates for rank-revealing UTV decompositions , 1999, Numerical Algorithms.

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

[5]  Sabine Van Huffel,et al.  Total least squares problem - computational aspects and analysis , 1991, Frontiers in applied mathematics.

[6]  L. Foster.  Rank and null space calculations using matrix decomposition without column interchanges , 1986 .

[7]  G. W. Stewart,et al.  An updating algorithm for subspace tracking , 1992, IEEE Trans. Signal Process..

[8]  G. Golub,et al.  Linear least squares solutions by householder transformations , 1965 .

[9]  Beresford N. Parlett,et al.  An implementation of the dqds algorithm (positive case) , 2000 .

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

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

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

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

[14]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[15]  James Demmel,et al.  Accurate Singular Values of Bidiagonal Matrices , 1990, SIAM J. Sci. Comput..

[16]  J. H. Wilkinson.  Global convergene of tridiagonal QR algorithm with origin shifts , 1968 .

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

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

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

[20]  Å. Björck.  Numerics of Gram-Schmidt orthogonalization , 1994 .

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

[22]  Axel Ruhe.  Numerical aspects of gram-schmidt orthogonalization of vectors , 1983 .

[23]  Hasan Erbay,et al.  Modified Gram-Schmidt-based downdating technique for ULV decompositions with applications to recursive TLS problems , 1999, Optics & Photonics.

[24]  James R. Bunch,et al.  Bounding the Subspaces from Rank Revealing Two-Sided Orthogonal Decompositions , 1995, SIAM J. Matrix Anal. Appl..

[25]  G. Stewart,et al.  A block QR algorithm and the singular value decomposition , 1993 .

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

[27]  K. V. Fernando,et al.  Accurately Counting Singular Values of Bidiagonal Matrices and Eigenvalues of Skew-Symmetric Tridiagonal Matrices , 1999, SIAM J. Matrix Anal. Appl..

[28]  B. Parlett,et al.  Accurate singular values and differential qd algorithms , 1994 .

[29]  A. K. Cline,et al.  Generalizing the LINPACK Condition Estimator , 1981 .

[30]  T. Chan.  Rank revealing QR factorizations , 1987 .

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

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

[33]  Sabine Van Huffel,et al.  Recent advances in total least squares techniques and errors-in-variables modeling , 1997 .