with a large matrix A of ill-determined rank. Thus, A has many “tiny” singular values of different orders of magnitude. In particular, A is severely ill-conditioned. Some of the singular values of A may be vanishing. We allow m ≥ n or m < n. The right-hand side vector b̃ is not required to be in the range of A. Linear systems of equations of the form (1) with a matrix of ill-determined rank are ...

This paper discusses the solution of large-scale linear discrete ill-posed problems with a noise-contaminated right-hand side. Tikhonov regularization is used to reduce the influence of the noise on the computed approximate solution. We consider problems in which the coefficient matrix is the sum of Kronecker products of matrices and present a generalized global Arnoldi method, that respects th...

The BiCG and QMR methods are well-known Krylov subspace iterative methods for the solution of linear systems of equations with a large nonsymmetric, nonsingular matrix. However, little is known of the performance of these methods when they are applied to the computation of approximate solutions of linear systems of equations with a matrix of ill-determined rank. Such linear systems are known as...

‎the global fom and gmres algorithms are among the effective‎ ‎methods to solve sylvester matrix equations‎. ‎in this paper‎, ‎we‎ ‎study these algorithms in the case that the coefficient matrices‎ ‎are real symmetric (real symmetric positive definite) and extract‎ ‎two cg-type algorithms for solving generalized sylvester matrix‎ ‎equations‎. ‎the proposed methods are iterative projection metho...

A necessary and sufficient condition is given for the quaternion matrix equations AiX + Y Bi = Ci (i = 1, 2) to have a pair of common solutions X and Y . As a consequence, the results partially answer a question posed by Y.H. Liu (Y.H. Liu, Ranks of solutions of the linear matrix equation AX + Y B = C, Comput. Math. Appl., 52 (2006), pp. 861-872).

A necessary and sufficient condition is given for the quaternion matrix equations AiX + Y Bi = Ci (i = 1, 2) to have a pair of common solutions X and Y . As a consequence, the results partially answer a question posed by Y.H. Liu (Y.H. Liu, Ranks of solutions of the linear matrix equation AX + Y B = C, Comput. Math. Appl., 52 (2006), pp. 861-872).

We address the problem of prior matrix estimation for the solution of `1-regularized ill-posed inverse problems. From a Bayesian viewpoint, we show that such a matrix can be regarded as an influence matrix in a multivariate `1-Laplace density function. Assuming a training set is given, the prior matrix design problem is cast as a maximum likelihood term with an additional sparsity-inducing term...