preconditioned krylov solvers for bea the performance of a number of preconditioned krylov methods is analysed for a large variety of boundary element formulations low and high order element two dimensional 2 d and three dimensional 3 d regular singular and hypersingular collocation and symmetric galerkin single and multi zone thermal and elastic continuous and discontinuous boundary formulations with and without condensation are considered preconditioned conjugate gradient cg solvers in standard form and a form effectively operating on the normal equations cgn generalized minimal residual gmres conjugate gradient squared cgs and stabilized bi conjugate gradient bi cgstab krylov solvers are employed in this study both the primitive and preconditioned matrix operators are depicted graphically to illustrate the relative amenability of the alternative formulations to solution via kryiov methods and to contrast and explain their computational performances a notable difference between 2 d and 3 d bea operators is readily visualized in this manner numerical examples are presented and the relative conditioning of the various discrete bea operators is reflected in the performance of the krylov equation solvers a preconditioning scheme which was found to be uncompetitive in the collocation bea context is shown to make iterative solution of symmetric galerkin bea problems more economical than employing direct solution techniques we conclude that the preconditioned krylov techniques are competitive with or superior to direct methods in a wide range of boundary formulated problems and that their performance can be partially correlated with certain problem characteristics

trends in systolic and cellular computation abstract a profile is given of current research as it pertains to computational mathematics on very large scale integration vlsi array processors in this type of parallel computers the cells of the array operate in single instruction multiple data simd mode and algorithms are executed in systolic or cellular fashion the focus of the presentation is on linear algebraic techniques a systolization process is illustrated by matching an arbitrary gaxpy operation onto a fixed size square array processor two recent systolic methods for solving systems of linear equations one iterative and the other direct are described a cellular algorithm for a fast fourier transform based on a new implementation on rectangular array processors of the perfect shuffle permutation is then derived using annotated lists of recent references snapshots of active research areas are given on systolic linear solvers the singular value decomposition artificial neural networks and the simulated annealing algorithm

primal dual strategy for constrained optimal control problems an algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed it is based on an active set strategy involving primal as well as dual variables for discretized problems sufficient conditions for convergence in finitely many iterations are given numerical examples are given and the role of the strict complementarity condition is discussed

comparisons of gaussian elimination algorithms on a cray y mp abstract the results of various implementations of gaussian elimination on full matrices on a single processor cray y mp are presented and discussed it is shown that when the manufacturer supplied blas kernels are used the difference between thebest versions of level 2 blas and level 3 blas blocked implementations is almost negligible it is suggested that to improve the performance of blocked gaussian elimination it is possible to utilize strassen s matrix multiplication algorithm

numerical solution of differential algebraic equations in mechanical systems simulation abstract the numerical solution of the differential algebraic equations of motion of mechanical systems offers many computational challenges in this paper we describe progress which has been made in understanding the formulation of the equations of motion from the viewpoint of numerical stability and outline some of the difficulties which must be resolved for efficient and reliable numerical methods in real time simulation of mechanical systems

projective block lanczos algorithm for dense hermitian eigensystems projection operators are used to effect deflation by restriction and it is argued that this is an optimal lanczos algorithm for memory minimization algorithmic optimization is constrained to dense hermitian eigensystems where a significant number of the extreme eigenvectors must be obtained reliably and completely the defining constraints are operator algebra without a matrix representation and semi orthogonalization without storage of krylov vectors other semi orthogonalization strategies for lanczos algorithms and conjugate gradient techniques are evaluated within these constraints large scale sparse complex numerical experiments are performed on clusters of magnetic dipoles a quantum many body system that is not block diagonalizable plane wave density functional theory of beryllium clusters provides examples of dense complex eigensystems use of preconditioners and spectral transformations is evaluated in a preprocessor prior to a high accuracy self consistent field calculation

recursive blocked algorithms and hybrid data structures for dense matrix library software matrix computations are both fundamental and ubiquitous in computational science and its vast application areas along with the development of more advanced computer systems with complex memory hierarchies there is a continuing demand for new algorithms and library software that efficiently utilize and adapt to new architecture features this article reviews and details some of the recent advances made by applying the paradigm of recursion to dense matrix computations on today s memory tiered computer systems recursion allows for efficient utilization of a memory hierarchy and generalizes existing fixed blocking by introducing automatic variable blocking that has the potential of matching every level of a deep memory hierarchy novel recursive blocked algorithms offer new ways to compute factorizations such as cholesky and qr and to solve matrix equations in fact the whole gamut of existing dense linear algebra factorization is beginning to be reexamined in view of the recursive paradigm use of recursion has led to using new hybrid data structures and optimized superscalar kernels the results we survey include new algorithms and library software implementations for level 3 kernels matrix factorizations and the solution of general systems of linear equations and several common matrix equations the software implementations we survey are robust and show impressive performance on today s high performance computing systems

hybrid ikebe newton s iteration for inverting general nonsingular hessenberg matrices after a concise survey the expanded ikebe algorithm for inverting the lower half plus the superdiagonal of an n n unreduced upper hessenberg matrix h is extended to general nonsingular upper hessenberg matrices by computing in the reduced case a block diagonal form of the factor matrix hl in the inverse factorization h 1 h l u 1 this factorization enables us to propose hybrid and accurate nongaussian procedures for computing h 1 thus hl is computed directly in the aim to be used as a fine initial guess for newton s iteration which converges to h 1 in a suitable number of iterations

a parallel alternating direction implicit preconditioning method abstract the alternating direction implicit adi iterative method is an efficient iterative method to solve systems of linear equations due to its extremely fast convergence the adi method has also been used successfully as a preconditioner in some other iterative methods such as the preconditioned conjugate gradient in this paper a parallel algorithm for the adi preconditioning is proposed in this algorithm several steps of the adi iteration are computed simultaneously this means that several tridiagonal systems that are traditionally solved sequentially are now solved concurrently the high performance of this algorithm is achieved by increasing the degree of parallelism and reducing memory contention the algorithm can easily be implemented in a multiprocessor architecture experiments have been conducted on the myrias sps 2 computer with 64 processors and good performance of this algorithm is observed

blocking neville elimination algorithm for exploiting cache memories neville elimination is a method for solving a linear system of equations that introduces zeros in a matrix column by adding to each row an adequate multiple of the previous one in this paper we explore block algorithms for neville elimination which take into account the memory hierarchies of a computer these algorithms try to manage the memory movements to optimize them thus the matrix of the system is divided following three different strategies blocking by rows columns or submatrices in each case we study the performance of the algorithm according to the ratio of floating point operations to memory references q theoretical estimations show that q depends on data partitioning being submatrix blocks the best choice

algebraic and numerical techniques for the computation of matrix determinants we review modify and combine together several numerical and algebraic techniques in order to compute the determinant of a matrix or the sign of such a determinant the resulting algorithms enable us to obtain the solution by using a lower precision of computations and relatively few arithmetic operations the problem has important applications to computational geometry

newton s method for large bound constrained optimization problems we analyze a trust region version of newton s method for bound constrained problems our approach relies on the geometry of the feasible set not on the particular representation in terms of constraints the convergence theory holds for linearly constrained problems and yields global and superlinear convergence without assuming either strict complementarity or linear independence of the active constraints we also show that the convergence theory leads to an efficient implementation for large bound constrained problems

a new gaussian elimination based algorithm for parallel solution of linear equations abstract in this paper a variant of gaussian elimination ge called successive gaussian elimination sge algorithm for parallel solution of linear equations is presented unlike the conventional ge algorithm the sge algorithm does not have a separate back substitution phase which requires o n steps using o n processors or o log 2 2 n steps using o n 3 processors for solving a system of linear algebraic equations it replaces the back substitution phase by only one step division and possesses numerical stability through partial pivoting further in this paper the sge algorithm is shown to produce the diagonal form in the same amount of parallel time required for producing triangular form using the conventional parallel ge algorithm finally the effectiveness of the sge algorithm is demonstrated by studying its performance on a hypercube multiprocessor system

parallel algorithms for dense linear algebra computations scientific and engineering research is becoming increasingly dependent upon the development and implementation of efficient parallel algorithms on modern high performance computers numerical linear algebra is an indispensable tool in such research and this paper attempts to collect and describe a selection of some of its more important parallel algorithms the purpose is to review the current status and to provide an overall perspective of parallel algorithms for solving dense banded or block structured problems arising in the major areas of direct solution of linear systems least squares computations eigenvalue and singular value computations and rapid elliptic solvers a major emphasis is given here to certain computational primitives whose efficient execution on parallel and vector computers is essential in order to obtain high performance algorithms

multigrid algorithms for variational inequalities we present an iterative scheme for obstacle problems described by variational inequalities at each step of the iteration the method leads to a reduced linear resp nonlinear algebraic system whi

an infeasible active set method for quadratic problems with simple bounds a primal dual active set method for quadratic problems with bound constraints is presented based on a guess on the active set a primal dual pair x s is computed that satisfies the first order optimality condition and the complementarity condition if x s is not feasible a new active set is determined and the process is iterated sufficient conditions for the iterations to stop in a finite number of steps with an optimal solution are provided computational experience indicates that this approach often requires only a few less than 10 iterations to find the optimal solution

more integrable birational mappings we study birational mappings generated by matrix inversion and permutation of the entries of q q matrices for q 3 we have performed a systematic examination of all the permutations of 3 3 matrices in order to find integrable mappings of three different kinds and finite order mappings this exhaustive analysis gives among 30 462 classes of mappings 27 new integrable classes of birational mappings and 36 classes yielding finite order recursions associated with these mappings an exhaustive analysis with a constraint on the diagonal entries has also been performed for 4 4 matrices we have found 8306 new classes of integrable mappings all these new examples show that integrability can actually correspond to non involutive permutations the analysis of the integrable cases specific of a particular size of the matrix and a careful examination of the non involutive permutations could shed some light on integrability of such birational mappings it seems that one has the following result the non involutive examples are specific of a given matrix size 3 3 matrix and the permutations which yield integrable mappings for arbitrary matrix size are always involutions

neville elimination a study of the efficiency using checkerboard partitioning abstract it is well known that checkerboard partitioning can exploit more concurrency than striped partitioning because the matrix computation can be divided among more processors than in the case of striping in this work we analyze the performance of neville method when a checkerboard partitioning is used focusing on the special case of block cyclic checkerboard partitioning this method is an alternative to gaussian elimination and it has been proved to be very useful for some classes of matrices such as totally positive matrices the performance of this parallel system is measured in terms of the efficiency the fraction of time for which a processor is usefully employed which in our model is close to one when the optimum block size is used also we have executed our algorithms on a parallel pc cluster observing that both efficiencies theoretical and empirical are quite similar

on the spectrum of the sor operator for symmetric positive definite matrices one attractive feature of the classical successive overrelaxation sor iterative method is its algorithmic simplicity whereas for linear systems with symmetric positive definite matrices a the theorem of ostrowski guarantees convergence as long as the relaxation parameter 0 2 results on the optimal choice of can only be obtained under certain additional assumtions on a here for positive definite systems with coefficient matrices a i l lt in which l is not necessarily strictly lower triangular we study the behavior of the spectrum of the sor operator l as 0 and 2 and describe enclosure sets which may be used to estimate the spectral radius of l for 0 2 numerical experiments show why results on the optimal relaxation parameter are difficult to obtain in the general case of positive definite systems

an iterative two step algorithm for linear complementarity problems we propose an algorithm for the numerical solution of large scale symmetric positive definite linear complementarity problems each step of the algorithm combines an application of the successive overrelaxation method with projection to determine an approximation of the optimal active set with the preconditioned conjugate gradient method to solve the reduced residual systems of linear equations convergence of the iterates to the solution is proved in the experimental part we compare the efficiency of the algorithm with several other methods as test example we consider the obstacle problem with different obstacles for problems of dimension up to 24 000 variables the algorithm finds the solution in less then 7 iterations where each iteration requires about 10 matrix vector multiplications