1,721,057 research outputs found
On the computation of few eigenvalues of positive definite Hamiltonian matrices
No full textGiven a Hamiltonian matrix H = JS with S symmetric and positive definite, we analyze a symplectic Lanczos algorithm to transform −H^2 in a symmetric and positive definite tridiagonal matrix of half size. By means of two effective restarted procedures, this algorithm is then used to compute few extreme eigenvalues of H. Numerical examples are also reported to compare the presented techniques
Optimized cyclic reduction for the solution of linear tridiagonal systems on parallel computers
No full textAbstractA parallel version of the cyclic reduction algorithm for the solution of tridiagonal linear systems is presented. The original problem is divided into subproblems which may be solved almost independently. Synchronizations among the processors involved is only needed to solve a reduced tridiagonal system whose dimension depends on the number of processors.Numerical tests have been performed on a linear array of processors. The obtained speedups show that this is the best possible parallel implementation of the cyclic reduction and one of the fastest algorithms for the solution of tridiagonal systems on a parallel computer with medium grain parallelism
High order finite difference schemes for the solution of second order initial value problems
No full textThe numerical solution of second order ordinary differential equations with initial conditions is here approached by approximating each derivative by means of a set of finite difference schemes of high order. The stability properties of the obtained methods are discussed. Some numerical tests, reported to emphasize pros and cons of the approach, motivate possible choices on the use of these formulae
Parallel iterative solvers for boundary value methods
No full textA parallel variant of the block Gauss-Seidel iteration for the solution of block-banded linear systems is presented. The coefficient matrix is partitioned among the processors as in the domain decomposition methods and then it is split so that the resulting iterative method has the same spectral properties of the block Gauss-Seidel iteration. The parallel algorithm is applied to the solution of block-banded linear systems arising from the numerical discretization of initial value problems by means of Boundary Value Methods (BVMs). BVMs define a new approach for the solution of ordinary differential equations and seem to be attractive for their interesting stability properties and a possible parallel implementation. In this paper, we refer to BVMs based on the extended trapezoidal rules
Boundary value methods based on Adams-type methods
No full textThe aim of this paper is to derive Boundary Value Methods (BVMs) based on k-step Adams-type methods for the solution of initial value problems. BVMs lead to a discrete boundary value problem which needs one initial and k - 1 final conditions. We prove that the choice of boundary conditions, instead of the usual initial conditions, improves the stability properties of the classical Adams methods. For example, methods of order up to 6 are almost BV-A-stable, and those of order up to 9 are BV-A_0-stable
Variable step/order generalized upwind methods for the numerical solution of second order singular perturbation problems
No full textWe propose a simple and quite efficient code to solve singular perturbation problems when the perturbation parameter is very small. The code is based on generalized upwind methods of order ranging from 4 to 10 and uses highly variable stepsize to fit the boundary regions with relatively few points. An extensive numerical test section shows the effectiveness of the proposed technique on linear problems
Parallel numerical solution of ABD and BABD linear systems arising from BVPs
No full textWe consider linear systems with coefficient matrices having the ABD or the Bordered ABD (BABD) structures.
These systems arise in the discretization of BVPs for ordinary and partial differential equations with separated and non-separated boundary conditions, respectively. We describe the cyclic reduction algorithm for the solution of BABD linear systems which allowed us to write the codes BABDCR and GBABDCR (the latter code is suitable for matrices with a more generic BABD structure).
A comparison of the GBABDCR code with respect to the well-known sequential code COLROW on ABD linear systems is then analysed. We report some tests on an OpenMP Fortran 90 parallel version of the GBABDCR code and finally we discuss about the use of GBABDCR inside the BVP code BVP SOLVER
A parallel Gauss-Seidel method for block tridiagonal linear systems
No full textA parallel variant of the block Gauss-Seidel iteration is presented for the solution of Mock tridiagonal linear systems. In this method parallel computations derive from a block reordering of the coefficient matrix similar to that of the domain decomposition methods. It has been proved that the parallel Gauss-Seidel iteration has the same spectral properties of the sequential method and may be used for any sparsity pattern of the blocks of the linear system. The parallel algorithm is applied to the solution of linear systems arising from initial value problems when solved by means of boundary value methods and from elliptic partial differential equations
High order generalized upwind schemes and numerical solution of singular perturbation problems
No full textHigh even order generalizations of the traditional upwind method are introduced to solve second order ODE-BVPs without recasting the problem as a first order system. Both theoretical analysis and numerical comparison with central difference schemes of the same order show that these new methods may avoid typical oscillations and achieve high accuracy. Singular perturbation problems are taken into account to emphasize the main features of the proposed methods
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