Abstract. For large-scale symmetric discrete ill-posed problems, MINRES and MR-II are commonly used iterative solvers. In this paper, we analyze their regularizing effects. We first prove that the regularized solutions by MINRES have filtered SVD forms. Then we show that (i) a hybrid MINRES that uses explicit regularization within projected problems is generally needed to compute a best possibl...

While there is no lack of efficient Krylov subspace solvers for Hermitian systems, few exist for complex symmetric, skew symmetric, or skew Hermitian systems, which are increasingly important in modern applications including quantum dynamics, electromagnetics, and power systems. For a large, consistent, complex symmetric system, one may apply a non-Hermitian Krylov subspace method disregarding ...

The minimum residual method, MINRES, is an iterative method for solving a n x n linear system, Ax = b, where is A is a symmetric matrix. It searches for a vector, xk, in the k th Krylov subspace that minimizes the residual, rk = b − Axk. Due to the symmetric nature, the vectors are computed to be the Lanczos vectors, which simplifies to a three-term recurrence, where vk+1 is computed as a linea...

A variant of the MINRES method, often referred to as the MR-II method, has in the last few years become a popular iterative scheme for computing approximate solutions of large linear discrete ill-posed problems with a symmetric matrix. It is important to terminate the iterations sufficiently early in order to avoid severe amplification of measurement and round-off errors. We present a new L-cur...

The solution of least squares support vector machines LS-SVMs is characterized by a specific linear system, that is, a saddle point system. Approaches for its numerical solutions such as conjugate methods Sykens and Vandewalle 1999 and null space methods Chu et al. 2005 have been proposed. To speed up the solution of LS-SVM, this paper employs the minimal residual MINRES method to solve the abo...

A deflated and restarted Lanczos algorithm to solve hermitian linear systems, and at the same time compute eigenvalues and eigenvectors for application to multiple right-hand sides, is described. For the first right-hand side, eigenvectors with small eigenvalues are computed while simultaneously solving the linear system. Two versions of this algorithm are given. The first is called Lan-DR and ...

For MinRes and SymmLQ it is essential to compute a QR decomposition of a tridiagonal coefficient matrix gained in the Lanczos process. This QR decomposition is constructed by an update scheme applying in every step a single Givens rotation. Using complex Householder reflections we generalize this idea to block tridiagonal matrices that occur in generalizations of MinRes and SymmLQ to block meth...

The conjugate gradient method (CG) has long been the workhorse for inner-iterations of second-order algorithms large-scale nonconvex optimization. Prominent examples include line-search based algorithms, e.g., Newton-CG, and those on a trust-region framework, CG-Steihaug. This is mainly thanks to CG's several favorable properties, including certain monotonicity properties its inherent ability d...

In this paper we present three theorems which give insight into the regularizing properties of MINRES. While our theory does not completely characterize the regularizing behavior of the algorithm, it provides a partial explanation of the observed behavior of the method. Unlike traditional attempts to explain the regularizing properties of Krylov subspace methods, our approach focuses on converg...