Generalized matrix functions (GMFs) extend the concept of a matrix function to rectangular matrices via the singular value decomposition. Several applications involving directed graphs, Hamiltonian dynamical systems, and optimization problems with low-rank constraints require the action of a GMF of a large, sparse matrix on a vector. We present a new method for applying GMFs to vectors based on...

In theory, the Lanczos algorithm generates an orthogonal basis of corresponding Krylov subspace. However, in finite precision arithmetic orthogonality and linear independence computed vectors is usually lost quickly. this paper we study a class matrices starting having special nonzero structure that guarantees exact computations whenever floating point satisfying IEEE 754 standard used. Analogo...

it is well known that if the coefficient matrix in a linear system is large and sparse or sometimes not readily available, then iterative solvers may become the only choice. the block solvers are an attractive class of iterative solvers for solving linear systems with multiple right-hand sides. in general, the block solvers are more suitable for dense systems with preconditioner. in this paper,...

The present paper is concerned with developing tensor iterative Krylov subspace methods to solve large multi-linear equations. We use the T-product for two tensors define tubal global Arnoldi and Golub-Kahan bidiagonalization algorithms. Furthermore, we illustrate how tensor-based approaches can be exploited ill-posed problems arising from recovering blurry multichannel (color) images videos, u...

This paper is concerned with estimating the solutions of numerically ill-posed least squares problems through Tikhonov regularization. Given a priori estimates on the covariance structure of errors in the measurement data b, and a suitable statistically-chosen σ, the Tikhonov regularized least squares functional J(σ) = ‖Ax − b‖2Wb + 1/σ 2‖D(x − x0)‖2, evaluated at its minimizer x(σ), approximat...

• First, the lower bidiagonal matrix A11 with nonzero bidiagonal elements has full column rank and its singular values are simple. Consequently, any zero singular values or repeats that A has must appear in A22. • Second, A11 has minimal dimensions, and A22 has maximal dimensions, over all orthogonal transformations giving the block structure in (2), without any additional assumptions on the st...

The reduction of a large-scale symmetric linear discrete ill-posed problem with multiple right-hand sides to smaller block tridiagonal matrix can easily be carried out by the application small number steps Lanczos method. We show that subdiagonal blocks reduced converge zero fairly rapidly increasing number. This quick convergence indicates there is little advantage in expressing solutions prob...

The L-curve criterion is often applied to determine a suitable value of the regularization parameter when solving ill-conditioned linear systems of equations with a right-hand side contaminated by errors of unknown norm. However, the computation of the L-curve is quite costly for large problems; the determination of a point on the L-curve requires that both the norm of the regularized approxima...