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

limit laws for sums of functions of subtrees of random binary search trees we consider sums of functions of subtrees of a random binary search tree and obtain general laws of large numbers and central limit theorems these sums correspond to random recurrences of the quicksort type x n stackrel cal l x i n x n 1 i n y n n ge 1 where in is uniformly distributed on 0 1 n 1 yn is a given random variable x k stackrel cal l x k for all k and given in xin and x n 1 in are independent conditions are derived such that x n mu n sigma sqrt n stackrel cal l rightarrow cal n 0 1 the normal distribution for some finite constants mu and sigma

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

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

a note on the height of binary search trees let h n be the height of a binary search tree with n nodes constructed by standard insertions from a random permutation of 1 n it is shown that h n log n c 4 31107 in probability as n where c is the unique solution of c log 2 e c 1 c 2 also for all p 0 lim n e h p n log p n c p finally it is proved that s n log n c 0 3733 in probability where c is defined by c log 2 e c 1 c 1 and s n is the saturation level of the same tree that is the number of full levels in the tree

fragmentation of random trees we study fragmentation of a random recursive tree into a forest by repeated removal of nodes the initial tree consists of n nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly selected existing node as nodes are removed from the tree one at a time the tree dissolves into an ensemble of separate trees namely a forest we study statistical properties of trees and nodes in this heterogeneous forest and find that the fraction of remaining nodes m characterizes the system in the limit we obtain analytically the size density of trees of size s the size density has power law tail with exponent therefore the tail becomes steeper as further nodes are removed and the fragmentation process is unusual in that exponent increases continuously with time we also extend our analysis to the case where nodes are added as well as removed and obtain the asymptotic size density for growing trees

on the variance of the height of random binary search trees let h n be the height of a random binary search tree on n nodes we show that there exists a constant alpha 4 31107 ldots such that rm p h n alpha log n beta log log n to 0 where beta 15 alpha ln 2 93 2933 ldots the proof uses the second moment method and does not rely on properties of branching processes we also show that var h n o log log n 2

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

on hypersingular surface integrals in the symmetric galerkin boundary element method application to heat conduction in exponentially graded materials a symmetric galerkin formulation and implementation for heat conduction in a three dimensional functionally graded material is presented the green s function of the graded problem in which the thermal conductivity varies exponentially in one co ordinate is used to develop a boundary only formulation without any domain discretization the main task is the evaluation of hypersingular and singular integrals which is carried out using a direct limit to the boundary approach however due to complexity of the green s function for graded materials the usual direct limit procedures have to be modified incorporating taylor expansions to obtain expressions that can be integrated analytically several test examples are provided to verify the numerical implementation the results of test calculations are in good agreement with exact solutions and corresponding finite element method simulations copyright 2004 john wiley sons ltd

smoothed analysis of binary search trees binary search trees are one of the most fundamental data structures while the height of such a tree may be linear in the worst case the average height with respect to the uniform distribution is only logarithmic the exact value is one of the best studied problems in average case complexity we investigate what happens in between by analysing the smoothed height of binary search trees randomly perturb a given adversarial sequence and then take the expected height of the binary search tree generated by the resulting sequence as perturbation models we consider partial permutations partial alterations and partial deletions on the one hand we prove tight lower and upper bounds of roughly q 1 p n p for the expected height of binary search trees under partial permutations and partial alterations where n is the number of elements and p is the smoothing parameter this means that worst case instances are rare and disappear under slight perturbations on the other hand we examine how much a perturbation can increase the height of a binary search tree i e how much worse well balanced instances can become

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

the variance of the height of binary search trees by using analytic tools it is shown that the variance e hnehn 2 of the height hn of binary search trees of size n is bounded

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

stratification in the preferential attachment network we study structural properties of trees grown by preferential attachment in this mechanism nodes are added sequentially and attached to existing nodes at a rate that is strictly proportional to the degree we classify nodes by their depth n defined as the distance from the root of the tree and find that the network is strongly stratified most notably the distribution f n k of nodes with degree k at depth n has a power law tail f n k k n the exponent grows linearly with depth where the brackets denote an average over all nodes therefore nodes that are closer to the root are better connected and moreover the degree distribution strongly varies with depth similarly the in component size distribution has a power law tail and the characteristic exponent grows linearly with depth qualitatively these behaviors extend to a class of networks that grow by redirection

universal limit laws for depths in random trees random binary search trees b ary search trees median of 2k 1 trees quadtrees simplex trees tries and digital search trees are special cases of random split trees for these trees we offer a universal law of large numbers and a limit law for the depth of the last inserted point as well as a law of large numbers for the height

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

the height of a random binary search tree let h n be the height of a random binary search tree on n nodes we show that there exist constants 4 311 and 1 953 such that e h n ln n ln ln n o 1 we also show that var h n o 1

squarish k d trees we modify the k d tree on 0 1 d by always cutting the longest edge instead of rotating through the coordinates this modification makes the expected time behavior of lower dimensional partial match queries behave as perfectly balanced complete k d trees on n nodes this is in contrast to a result of flajolet and puech j assoc comput mach 33 1986 pp 371 407 who proved that for standard random k d trees with cuts that rotate among the coordinate axes the expected time behavior is much worse than for balanced complete k d trees we also provide results for range searching and nearest neighbor search for our trees

on robson s convergence and boundedness conjectures concerning the height of binary search trees let cn denote the number of nodes in a random binary search tree of n nodes at the maximal level in this paper we present a direct proof of robson s boundedness conjecture saying that the expected values e cn remain bounded as n we also prove that e cn is asymptotically multiplicatively periodic which shows that robson s convergence conjecture that is e cn is convergent is only true if the limiting periodic function c x is constant interestingly it can be shown that c x is almost constant in the sense that possible oscillations are very small however it seems to be a difficult problem to decide whether c x is really constant or not we present similar properties for the variance of the height var hn too

on growing random binary trees abstract a sequence tn n 1 of nested binary trees generated by an infinite sequence of i i d random variables is studied two absolute constants 1 2 are shown to exist 0 37 lim h n ln n 1 lim h n ln n 2 with probability one here hn and hn are respectively the lengths of the shortest and the longest branches of the tree tn

diffusion limited aggregation on a tree we study the following growth model on a regular d ary tree points at distance n adjacent to the existing subtree are added with probabilities proportional to n where 1 is a positive real parameter the heights of these clusters are shown to increase linearly with their total size this complements known results that show the height increases only logarithmically when 1 results are obtained using stochastic monotonicity and regeneration results which may be of independent interest our motivation comes from two other ways in which the model may be viewed as a problem in first passage percolation and as a version of diffusion limited aggregation dla adjusted so that fingering occurs

the height of a binary search tree the limiting distribution perspective we study the height of the binary search tree the most fundamental data structure used for searching we assume that the binary search tree is built from a random permutation of n elements under this assumption we study the limiting distribution of the height as n we show that the distribution has six asymptotic regions scales these correspond to different ranges of k and n where pr hn k is the height distribution in the critical region the so called central region where most of the probability mass is concentrated the limiting distribution satisfies a non linear integral equation while we cannot solve this equation exactly we show that both tails of the distribution are roughly of a double exponential form from our analysis we conclude that the average height e hn a log n 3 2 a a 1 log log n where a 4 311 is the unique solution of x logx x xlog2 1 0 x 1 while the variance var hn o 1 the second term in the expansion of e kn and the rate of growth of the variance were also recently obtained by b reed who used probabilistic arguments while m drmota established the growth of the variance by analytic methods our analysis makes certain assumptions about the forms of some asymptotic expansions as well as their asymptotic matching

stochastic analysis of tree like data structures the purpose of this article is to present two types of data structures binary search trees and usual combinatorial binary trees although they constitute the same set of rooted trees they are constructed via completely different rules and thus the underlying probabilistic models are different too both kinds of data structure can be analysed by probabilistic and stochastic tools binary search trees more or less with martingales and binary trees which can be considered as a special case of galton watson trees with stochastic processes it is also an aim of this article to demonstrate the strength of analytical methods in specific parts of probability theory related to combinatorial problems and we especially make use of the concept of generating functions one reason for the use of generating functions is that recursive combinatorial descriptions can be translated to relations for generating functions and another is that analytical properties of these generating functions can be used to derive asymptotic probabilistic relations

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

a diffusion limit for a class of randomly growing binary trees binary trees are grown by adding one node at a time an available node at height i being added with probability proportional to c i c 1 we establish both a strong law of large numbers and a central limit theorem for the vector x t x i t where x i t is the proportion of nodes of height i that are available at time t we show in fact that there is a deterministic process x i t such that sum x i t x i t text converges to 0 a text s text and such that if c2 tfrac 1 2 z i n t 2 n 2 x n 1 tc n x n 1 tc n and z n t z i n t then z n t converges weakly to a gaussian diffusion z t the results are applied to establish asymptotic normality in the unbiased coin tossing case for an entropy estimation procedure due to j ziv and to obtain results on the growth of the maximum height of the tree