Best linear unbiased prediction (BLUP) has been used to estimate the fixed effects and random effects of complex traits. Traditionally, genomic relationship matrix-based (GRM) and random marker-based BLUP analyses are prevalent to estimate the genetic values of complex traits. We used three methods: GRM-based prediction (G-BLUP), random marker-based prediction using an identity matrix (so-calle...

Langrangian Support Vector Machine (LSVM) and Least Squares Support Vector Machine (LSSVM) are two quick and effective classification methods. In this paper, we first introduce the mathematical models for LSVM and LSSVM and analyze their properties. In the nonlinear case, Sherman-Morrison-Woodbury identity is not used to compute the inversion of a matrix. According to block computation of a mat...

A long-standing issue in the Bayesian statistical approach to the phase problem in X-ray crystallography is to solve an entropy maxi-mization subproblem eeciently in every iteration of phase estimation. The entropy maximization problem is a semi-innnite convex program and can be solved in a nite dual space by using a standard Newton's method. However, the Newton's method is too expensive for th...

In this paper we deal with the solution, by means of preconditioned conjugate gradient (PCG) methods, of n × n symmetric Toeplitz systems An(f)x = b with nonnegative generating function f . Here the function f is assumed to be continuous and strictly positive, or is assumed to have isolated zeros of even order. In the first case we use as preconditioner the natural and the optimal τ approximati...

Two-level overlapping Schwarz methods for elliptic partial differential equations combine local solves on overlapping domains with a global solve of a coarse approximation of the original problem. To obtain robust methods for equations with highly varying coefficients, it is important to carefully choose the coarse approximation. Recent theoretical results by the authors have shown that bases f...

It has been known that the sparse approximate inverse preconditioning procedures SPAI and PSAI(tol) are costly to construct preconditioners for a large sparse nonsymmetric linear system with the coefficient matrix having at least one relatively dense column. This is also true for SPAI and the recently proposed sparse approximate inverse preconditioning procedure RSAI(tol) procedure when the mat...

The high computational cost involved in modeling of the progressive fracture simulations using large discrete lattice networks stems from the requirement to solve a new large set of linear equations every time a new lattice bond is broken. To address this problem, we propose an algorithm that combines the multiple-rank sparse Cholesky downdating algorithm with the rank-p inverse updating algori...

The linear support vector machine can be posed as a quadratic program in a variety of ways. In this paper, we look at a formulation using the two-norm for the misclassification error that leads to a positive definite quadratic program with a single equality constraint when the Wolfe dual is taken. The quadratic term is a small rank update to a positive definite matrix. We reformulate the optima...

Our randomized preprocessing enables pivoting-free and orthogonalization-free solution of homogeneous linear systems of equations, which leads to significant acceleration of the known algorithms in the cases of both general and structured input matrices. E.g., in the case of Toeplitz inputs, we decrease the estimated solution time from quadratic to nearly linear, and our tests show dramatic dec...

In this paper we deal with the solution, by means of preconditioned conjugate gradient (PCG) methods, of n × n symmetric Toeplitz systems An(f)x = b with nonnegative generating function f . Here the function f is assumed to be continuous and strictly positive, or is assumed to have isolated zeros of even order. In the first case we use as preconditioner the natural and the optimal τ approximati...