Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization
Applied Numerical Mathematics - Numerical algorithms, parallelism and applications
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The solution of the total least squares (TLS) problems, $\min_{E,f}\|(E,f)\|_F$ subject to (A+E)x=b+f, can in the generic case be obtained from the right singular vector corresponding to the smallest singular value $\sigma_{n+1}$ of (A, b). When A is large and sparse (or structured) a method based on Rayleigh quotient iteration (RQI) has been suggested by Björck. In this method the problem is reduced to the solution of a sequence of symmetric, positive definite linear systems of the form $(A^TA-\bar\sigma^2I)z=g$, where $\bar\sigma$ is an approximation to $\sigma_{n+1}$. These linear systems are then solved by a {\em preconditioned} conjugate gradient method (PCGTLS). For TLS problems where A is large and sparse a (possibly incomplete) Cholesky factor of ATA can usually be computed, and this provides a very efficient preconditioner. The resulting method can be used to solve a much wider range of problems than it is possible to solve by using Lanczos-type algorithms directly for the singular value problem. In this paper the RQI-PCGTLS method is further developed, and the choice of initial approximation and termination criteria are discussed. Numerical results confirm that the given algorithm achieves rapid convergence and good accuracy.}