Fast Algorithms for Structured Least Squares and Total Least Squares Problems
نویسندگان
چکیده
We consider the problem of solving least squares problems involving a matrix M of small displacement rank with respect to two matrices Z 1 and Z 2. We develop formulas for the generators of the matrix M (H) M in terms of the generators of M and show that the Cholesky factorization of the matrix M (H) M can be computed quickly if Z 1 is close to unitary and Z 2 is triangular and nilpotent. These conditions are satisfied for several classes of matrices, including Toeplitz, block Toeplitz, Hankel, and block Hankel, and for matrices whose blocks have such structure. Fast Cholesky factorization enables fast solution of least squares problems, total least squares problems, and regularized total least squares problems involving these classes of matrices.
منابع مشابه
Superlinearly convergent exact penalty projected structured Hessian updating schemes for constrained nonlinear least squares: asymptotic analysis
We present a structured algorithm for solving constrained nonlinear least squares problems, and establish its local two-step Q-superlinear convergence. The approach is based on an adaptive structured scheme due to Mahdavi-Amiri and Bartels of the exact penalty method of Coleman and Conn for nonlinearly constrained optimization problems. The structured adaptation also makes use of the ideas of N...
متن کاملExact and approximate solutions of fuzzy LR linear systems: New algorithms using a least squares model and the ABS approach
We present a methodology for characterization and an approach for computing the solutions of fuzzy linear systems with LR fuzzy variables. As solutions, notions of exact and approximate solutions are considered. We transform the fuzzy linear system into a corresponding linear crisp system and a constrained least squares problem. If the corresponding crisp system is incompatible, then the fuzzy ...
متن کاملNew Fast Algorithms for Structured Linear Least Squares Problems
We present new fast algorithms for solving the Toeplitz and the Toeplitz-plus-Hankel least squares problems. These algorithms are based on a new fast algorithm for solving the Cauchy-like least squares problem. We perform an error analysis and provide conditions under which these algorithms are numerically stable. We also develop implementation techniques that signiicantly reduce the execution ...
متن کاملFast Iteratively Reweighted Least Squares Algorithms for Analysis-Based Sparsity Reconstruction
In this paper, we propose a novel algorithm for analysis-based sparsity reconstruction. It can solve the generalized problem by structured sparsity regularization with an orthogonal basis and total variation regularization. The proposed algorithm is based on the iterative reweighted least squares (IRLS) model, which is further accelerated by the preconditioned conjugate gradient method. The con...
متن کاملSoftware for structured total least squares problems: User’s guide
The package contains ANSI C software with Matlab mex interface for structured total least squares estimation problems. The allowed structures in the data matrix are block-Toeplitz, block-Hankel, unstructured, and noise free. Combinations of blocks with this structures can be specified. The computational complexity of the algorithms is O(m), where m is the sample size.
متن کامل