Hessenberg matrix for sums of Hermitian positive definite matrices and weighted shifts

On Adaptive Weighted Polynomial Preconditioning for Hermitian Positive Definite Matrices

The conjugate gradient algorithm for solving Hermitian positive definite linear systems is usually combined with preconditioning in order to speed up convergence. In recent years, there has been a revival of polynomial preconditioning, motivated by the attractive features of the method on modern architectures. Standard techniques for choosing the preconditioning polynomial are based only on bou...

On SSOR-like preconditioners for non-Hermitian positive definite matrices

We construct, analyze and implement SSOR-like preconditioners for non-Hermitian positive definite system of linear equations when its coefficient matrix possesses either a dominant Hermitian part or a dominant skew-Hermitian part. We derive tight bounds for eigenvalues of the preconditioned matrices and obtain convergence rates of the corresponding SSOR-like iteration methods as well as the cor...

Matrices with Positive Definite Hermitian Part : Inequalities and Linear

The Hermitian and skew-Hermitian parts of a square matrix A are deened by H(A) (A + A)=2 and S(A) (A ? A)=2: We show that the function f(A) = (H(A ?1)) ?1 is convex with respect to the Loewner partial order on the cone of matrices with positive deenite Hermitian part. That is, for any matrices A and B with positive deenite Hermitian part ff(A) + f(B)g=2 ? f(fA + Bg=2) is positive semideenite: U...

Joint Approximate Diagonalization of Positive Definite Hermitian Matrices

Pure States, Positive Matrix Polynomials and Sums of Hermitian Squares

Let M be an archimedean quadratic module of real t× t matrix polynomials in n variables, and let S ⊆ R be the set of all points where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and S × P(R). This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol a...