Singular Value Decomposition Solution Research Articles

Computing Truncated Singular Value Decomposition Least Squares Solutions by Rank Revealing QR-Factorizations

Solutions to rank deficient least squares problems are conveniently expressed in terms of the singular value decomposition (SVD) of the coefficient matrix. When the matrix is nearly rank deficient, a common procedure is to neglect its smallest singular values, which leads to the truncated SVD (TSVD) solution. In this paper, an efficient method is presented for computing the TSVD solution via a QR-factorization, without the need for computing a complete SVD. The numerical rank of the matrix is determined by means of a rank revealing QR-factorization, which provides upper and lower bounds on the small singular values and approximations to the corresponding singular vectors, which are then refined by inverse subspace iteration and used in conjunction with the QR factors to compute the TSVD solution.

Truncated Singular Value Decomposition Solutions to Discrete Ill-Posed Problems with Ill-Determined Numerical Rank

Tikhonov regularization is a standard method for obtaining smooth solutions to discrete ill-posed problems. A more recent method, based on the singular value decomposition (SVD), is the truncated SVD method. The purpose of this paper is to show, under mild conditions, that the success of both truncated SVD and Tikhonov regularization depends on satisfaction of a discrete Picard condition, involving both the matrix and the right-hand side. When this condition is satisfied, then both methods are guaranteed to produce smooth solutions that are very similar.