Efficient pre-processing in the parallel block-Jacobi SVD algorithm
Parallel Computing - Parallel matrix algorithms and applications (PMAA'04)
Robust treatment of simultaneous collisions
ACM SIGGRAPH 2008 papers
On iterative QR pre-processing in the parallel block-Jacobi SVD algorithm
Parallel Computing
A Matrix Computation View of FastMap and RobustMap Dimension Reduction Algorithms
SIAM Journal on Matrix Analysis and Applications
MINRES-QLP: A Krylov Subspace Method for Indefinite or Singular Symmetric Systems
SIAM Journal on Scientific Computing
LSMR: An Iterative Algorithm for Sparse Least-Squares Problems
SIAM Journal on Scientific Computing
Algorithm 937: MINRES-QLP for symmetric and Hermitian linear equations and least-squares problems
ACM Transactions on Mathematical Software (TOMS)
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In this paper we introduce a new decomposition called the pivoted QLP decomposition. It is computed by applying pivoted orthogonal triangularization to the columns of the matrix X in question to get an upper triangular factor R and then applying the same procedure to the rows of R to get a lower triangular matrix L. The diagonal elements of R are called the R-values of X; those of L are called the L-values. Numerical examples show that the L-values track the singular values of X with considerable fidelity---far better than the R-values. At a gap in the L-values the decomposition provides orthonormal bases of analogues of row, column, and null spaces provided of X. The decomposition requires no more than twice the work required for a pivoted QR decomposition. The computation of R and L can be interleaved, so that the computation can be terminated at any suitable point, which makes the decomposition especially suitable for low-rank determination problems. The interleaved algorithm also suggests a new, efficient 2-norm estimator.