1,721,099 research outputs found
A new version of the arithmetic mean method for solving block tridiagonal linear systems
No full textIn this report we consider a new version of the arithmetic mean method for solving large block tridiagonal linear systems. The iterative method converges for systems with symmetric positive definite or positive real matrices or irreducible L-matrices with a strong diagonal dominance. When the coefficient matrix is symmetric positive definite, an additive preconditioner for the conjugate gradient method is derived.The Fortran 77 code carried out on multivector computer Cray Y-MP implementing the algorithm above, are reported in appendix
On a class of iterative methods for large-scale convex quadratic programs
No full textWe analyse the efficiency of a class of iterative methods for solving large scale convex quadratic programs. These methods, known as splitting methods and projection methods, require to solve a sequence of easy strictly convex quadratic programming subproblems obtained by splitting the matrix of the objective function. We describe in details the techniques used for generating the large and sparse test problems on which the computational behaviour of the methods is studied
A polynomial preconditioner for block tridiagonal matrices
No full textThis paper is concerned with the solution of block tridiagonal linear systems by the preconditioned conjugate gradient (PCG) method. If we consider a block AGE splitting of the coefficient matrix, it is possible to derive an additive polynomial preconditioner and to give conditions for such preconditioner to be symmetric positive definite.Numerical experiments on diffusion problem are carried out on CRAY Y-MP in order to evaluate the effectiveness of the parallel polynomial preconditioner
Variable projection methods for large convex quadratic programming
No full textWe propose two projection-type methods for solving large quadratic programs. The main feature of these iterative schemes consists in using, at each iteration, a variable projection parameter instead of a fixed one as in the classical projection methods. The convergence may be obtained without restrictive conditions on the projection parameters by using appropriate correction rules that imply, at each iteration, a sufficient decrease in the objective function. The first method uses a correction rule on the descent direction produced by the projection step, while in the second method, the correction formula works adaptively on the value of the variable projection parameter. We give convergence results for the general case of inexact solution of the inner subproblems. The numerical behaviour of the methods is strictly dependent on the sequence of the projection parameters. We introduce a practical nonexpensive updating rule for these parameters and evaluate its effectiveness on large scale test problems
The arithmetic mean preconditioner for multivector computers
No full textIn this paper we consider the arithmetic mean preconditioner for the conjugate gradient method. This preconditioner is designed to be implemented on a multiprocessor system that can execute concurrently different tasks on vector processors.Some spectral bounds for the preconditioner have been obtained. A comparison between this preconditioner and the SSOR preconditioner has been carried out on different test-matrices. Also the m-step preconditioners generated by the arithmetic mean preconditioner and by the SSOR preconditioner have been compared
An overview on projection-type methods for convex large-scale quadratic programs
No full textA well-known approach for solving large and sparse linearly constrained quadratic programming (QP) problems is given by the splitting and projection methods. After a survey on these classical methods, we show that they can be unified in a general iterative scheme consisting in to solve a sequence of QP subproblems with the constraints of the original problem and an easily solvable Hessian matrix. A convergence theorem is given for this general scheme. In order to improve the numerical performance of these methods, we introduce two variants of a projection- type scheme that use a variable projection parameter at each step. The two variable projection methods differ in the strategy used to assure a sufficient decrease of the objective function at each iteration. We prove, under very general hypotheses, the convergence of these schemes and we propose two practical, nonexpensive and efficient updating rules for the projection parameter. An extensive numerical experimentation shows the effectiveness of the variable projection-type methods
The two-stage arithmetic mean method on Cray T3D
No full textIl metodo della Media Aritmetica è particolarmente adatto per risolvere su sistemi multiprocessori sistemi lineari sparsi di dimensioni elevate con struttura a banda. Infatti ogni iterazione consiste nel risolvere due sistemi indipendenti diagonali a blocchi. Quando per le soluzioni di tali sistemi si calcolano solo delle approssimazioni mediante un numero prefissato di passi di uno schema iterativo, si genera una procedura detta metodo della Media Aritmetica a Due Stadi. Una serie di esperimenti numerici effettuati su Cray T3D permette di valutare l’efficienza di questo schema, evidenziandone le caratteristiche parallele
On the efficiency of splitting and projection methods for large strictly convex quadratic programs
No full textIn this paper we analyse the behaviour of the classical splitting and projection methods for solving large-scale strictly convex quadratic programming problems with linear constraints. The drawbacks of these classical methods are overcome by the recent modified projection-type and variable projection methods. These new approaches have the same complexity and a similar structure: each iteration consists of a projection step followed by a correction formula. Neverthless, on the contrary of the modified projection-type methods, the variable projection method does not require to prefix any scalar parameters and is weakly dependent on a priori scaling of the objective function. The results of a numerical experimentation permit to compare the new approaches with the classical splitting and projection methods and to evaluate the effectivenes of the variable projection method as solver of large quadratic programs
Computation of minimal eigenpair in the large sparse generalized eigen-problem using vector computer
No full textThis report is concerned with the computation of the minimal eigenpair of the generalized eigen-problem. An iterative method of first degree with the arithmetic mean preconditioner and the SSOR preconditioner has been analyzed. These preconditioners have been evaluated also for the method of Rayleigh quotient minimization.The Fortran 77 codes carried out on multivector computer Cray Y-MP implementing the algorithms described above, are reported in appendix
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