scholarly journals Parallel solution of symmetric positive definite systems with hyperbolic rotations

1986 ◽  
Vol 77 ◽  
pp. 75-111 ◽  
Author(s):  
Jean-Marc Delosme ◽  
Ilse C.F. Ipsen
Keyword(s):  
Positive Definite ◽  
Parallel Solution ◽  
Symmetric Positive Definite

scholarly journals Parallel Solution of Hierarchical Symmetric Positive Definite Linear Systems

2017 ◽  
Vol 2 (1) ◽  
pp. 201-212 ◽  
Author(s):  
José I. Aliaga ◽  
Rocío Carratalá-Sáez ◽  
Enrique S. Quintana-Ortí
Keyword(s):  
Linear Systems ◽  
Linear Algebra ◽  
High Performance ◽  
Parallel Implementation ◽  
Positive Definite ◽  
Cholesky Factorization ◽  
Task Parallelism ◽  
Parallel Solution ◽  
Symmetric Positive Definite ◽  
Performance Computing

AbstractWe present a prototype task-parallel algorithm for the solution of hierarchical symmetric positive definite linear systems via the ℋ-Cholesky factorization that builds upon the parallel programming standards and associated runtimes for OpenMP and OmpSs. In contrast with previous efforts, our proposal decouples the numerical aspects of the linear algebra operation from the complexities associated with high performance computing. Our experiments make an exhaustive analysis of the efficiency attained by different parallelization approaches that exploit either task-parallelism or loop-parallelism via a runtime. Alternatively, we also evaluate a solution that leverages multi-threaded parallelism via the parallel implementation of the Basic Linear Algebra Subroutines (BLAS) in Intel MKL.

A Symmetric Positive Definite Inverse Vibration Problem With Underdamped Modes

1995 ◽  
Author(s):  
Ladislav Starek ◽  
Daniel J. Inman ◽  
Deborah F. Pilkev
Keyword(s):  
Positive Definite ◽  
Inverse Eigenvalue Problem ◽  
Eigenvalues And Eigenvectors ◽  
Symmetric Positive Definite ◽  
Vibration Problem ◽  
Damped System ◽  
Stiffness Matrices ◽  
Inverse Eigenvalue ◽  
Inverse Vibration ◽  
Vector Differential Equation

Abstract This manuscript considers a symmetric positive definite inverse eigenvalue problem for linear vibrating systems described by a vector differential equation with constant coefficient matrices. The inverse problem of interest here is that of determining real symmetric, positive definite coefficient matrices assumed to represent the mass normalized velocity and position coefficient matrices, given a set of specified eigenvalues and eigenvectors. The approach presented here gives an alternative solution to a symmetric inverse vibration problem presented by Starek and Inman (1992) and extends these results to include the definiteness of the coefficient matrices. The new results give conditions which allow the construction of mass normalized damping and stiffness matrices based on given eigenvalues and eigenvectors for the case that each mode of the system is underdamped. The result provides an algorithm for determining a non-proportional damped system which will have symmetric positive definite coefficient matrices.

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