We develop a hierarchical matrix construction algorithm using matrix–vector multiplications, based on the randomized singular value decomposition of low-rank matrices. The algorithm uses OðlognÞ applications of the matrix on structured random test vectors and Oðn lognÞ extra computational cost, where n is the dimension of the unknown matrix. Numerical examples on constructing Green’s functions ...

We study the distributions θ,p of the random sums 3 " ε n θn, where ε " , ε # ,... are i.i.d. Bernoulli-p and θ is the inverse of a Pisot number (an algebraic integer β whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p ̄ .5, θ,p is a singular measure with exact Hausdorff dimension less than 1. We show that in all cases the Hausdorff dimension can be expresse...

Sparsified Randomization Monte Carlo (SRMC) algorithms for solving systems of linear algebraic equations introduced in our previous paper [34] are discussed here in a broader context. In particular, I present new randomized solvers for large systems of linear equations, randomized singular value (SVD) decomposition for large matrices and their use for solving inverse problems, and stochastic si...

We extend the Golub-Kahan algorithm for computing the singular value decomposition of bidiagonal matrices to triangular matrices R. Our algorithm avoids the explicit formation of R T R or RRT. We derive a relation between left and right singular vectors of triangular matrices and use it to prove monotonic convergence of singular values and singular vectors. The convergence rate for singular val...

In this paper, we first give a lower and upper bounds for singular values of a 2×2 positive semidefinite block matrices. Then, we give some weakly majorization inequalities of singular values positive semidefinite block matrices. Also, we present inequalities involving the direct sum and sum of positive semidefinite matrices.

We study n × n symmetric random matrices H, possibly discrete, with iid abovediagonal entries. We show that H is singular with probability at most exp(−nc), and ‖H−1‖ = O(√n). Furthermore, the spectrum of H is delocalized on the optimal scale o(n−1/2). These results improve upon a polynomial singularity bound due to Costello, Tao and Vu, and they generalize, up to constant factors, results of T...