orthogonal similarity transformation of a symmetric matrix into a diagonal plus semiseparable one with free choice of the diagonal in this paper we describe an orthogonal similarity transformation for transforming arbitrary symmetric matrices into a diagonal plus semiseparable matrix where we can freely choose the diagonal very recently an algorithm was proposed for transforming arbitrary symmetric matrices into similar semiseparable ones this reduction is strongly connected to the reduction to tridiagonal form the class of semiseparable matrices can be considered as a subclass of the diagonalplus semiseparable matrices therefore we can interpret the proposed algorithm here as an extension of the reduction to semiseparable form a numerical experiment is performed comparing thereby the accuracy of this reduction algorithm with respect to the accuracy of the traditional reduction to tridiagonal form and the reduction to semiseparable form the experiment indicates that all three reduction algorithms are equally accurate moreover it is shown in the experiments that asymptotically all the three approaches have the same complexity i e that they have the same factor preceding the n 3 term in the computational complexity finally we illustrate that special choices of the diagonal create a specific convergence behavior

random convex hulls in a product of balls the convex hull of a set of points sampled independently and uniformly from the cartesian product of balls of various dimensions is investigated bounds on the asymptotic behavior of the expected combinatorial complexity volume and mean width are derived when the distribution is held fixed and the sample size approaches infinity the expected combinational complexity and volume are found to depend up to constant factors only on the greatest dimension of any factor ball and the number of balls of that dimension on the other hand the expected mean width depends only on the number of balls and the dimensions of the product

structure ranks of matrices abstract the structure rank of a matrix i e the maximum order of a nonsingular submatrix all of whose entries are located in a given structure a subset of m n where m n are the row and column index sets respectively were previously studied for the off diagonal part here results on more general ranks e g off block diagonal strictly upper triangular and strictly upper block triangular are presented in particular it is shown that for a nonsingular matrix a in each of the last mentioned three cases the ranks of a and a 1 coincide

random polytopes and the efron stein jackknife inequality let k be a smooth convex body the convex hull of independent random points in k is a random polytope estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained the essential step is the use of the efron stein jackknife inequality for the variance of symmetric statistics consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope a conjecture of barany concerning random and best approximation of convex bodies is confirmed analogous results for random polytopes with vertices on the boundary of the convex body are given

central limit theorems for gaussian polytopes choose n random independent points in r d according to the standard normal distribution their convex hull k n is the gaussian random polytope we prove that the volume and the number of faces of k n satisfy the central limit theorem settling a well known conjecture in the field

a branch and bound algorithm to minimize the number of late jobs in a permutation flow shop abstract the problem of scheduling n jobs in an m machine permutation flow shop is considered each job requires processing on machines 1 m in that order the processing order of jobs is to be the same for each machine a due date for each job is specified that represents the time by which it should ideally be completed the objective is to schedule the jobs so that the number completed after their due dates is minimized a lower bound which is obtained by solving a single machine subproblem is derived also two improvement procedures which consider all subproblems together are developed a branch and bound algorithm that uses these lower bounds is described computational experience with the algorithm is reported

more on structure ranks of matrices abstract the structure rank of a matrix is the largest rank of a submatrix which lies within a specified structure we characterize those structures avoiding the main diagonal for which a matrix and its inverse have the same structure rank thereby generalizing some results of m fiedler

central limit theorems for random polygons in an arbitrary convex set we study the probability distribution of the area and the number of vertices of random polygons in a convex set kr 2 the novel aspect of our approach is that it yields uniform estimates for all convex sets kr 2 without imposing any regularity conditions on the boundary k our main result is a central limit theorem for both the area and the number of vertices settling a well known conjecture in the field we also obtain asymptotic results relating the growth of the expectation and variance of these two functionals

stochastische approximation konvexer polygone let k be a convex polygon with perimeter l and area f denote the perimeter of the convex hull of n uniformly and independently distributed points in k by ln and by fn its area as n tends to infinity l e l n c 1 k sqrt n o 1 n 1 varepsilon for any fixed 0 and f e f n c 2 k log n n o 1 n where the constants c1 k and c2 k are given explicitly this generalizes a result of renyi and sulanke

variance asymptotics for random polytopes in smooth convex bodies let k subset mathbb r d be a smooth convex set and let mathcal p lambda be a poisson point process on mathbb r d of intensity lambda the convex hull of mathcal p lambda cap k is a random convex polytope k lambda as lambda rightarrow infty we show that the variance of the number of k dimensional faces of k lambda when properly scaled converges to a scalar multiple of the affine surface area of k similar asymptotics hold for the variance of the number of k dimensional faces for the convex hull of a binomial process in k

a qz method based on semiseparable matrices this manuscript focusses on an alternative method for computing the eigenvalues of a pencil of two matrices based on semiseparable matrices an effective reduction of a matrix pair to lower semiseparable upper triangular form will be presented as well as a qz iteration for this matrix pair important to remark is that this reduction procedure also inherits a kind of nested subspace iteration as was the case when solving the standard eigenvalue problem with semiseparable matrices it will also be shown that the qz iteration for a semiseparable triangular matrix pair is closely related to the qz iteration for a hessenberg triangular matrix pair

computing all or some eigenvalues of symmetric mathcal h _ ell matrices we use a bisection method b parlett the symmetric eigenvalue problem prentice hall englewood cliffs nj 1980 p 51 to compute the eigenvalues of a symmetric mathcal h ell matrix m the number of negative eigenvalues of m mu i is computed via the ldl t factorization of m mu i for dense matrices the ldl t factorization is too expensive to yield an efficient eigenvalue algorithm in general but not so for mathcal h ell matrices in the special structure of mathcal h ell matrices there is an ldl t factorization with linear polylogarithmic complexity the bisection method requires only matrix size independent many iterations to find an eigenvalue up to the desired accuracy so that an eigenvalue can be found in linear polylogarithmic time for all n eigenvalues mathcal o n 2 log n 4 log vert m vert 2 epsilon text ev flops are needed to compute all eigenvalues with an accuracy epsilon text ev it is also possible to compute only eigenvalues in a specific interval or the j th smallest one numerical experiments demonstrate the efficiency of the algorithm in particular for the case when some interior eigenvalues are required a corrected pdf is attached to this article

variance asymptotics and scaling limits for random polytopes abstract let k be a convex set in r d and let k be the convex hull of a homogeneous poisson point process p of intensity on k when k is a simple polytope we establish scaling limits as for the boundary of k in a vicinity of a vertex of k and we give variance asymptotics for the volume and k face functional of k k 0 1 d 1 resolving an open question posed in 17 the scaling limit of the boundary of k and the variance asymptotics are described in terms of a germ grain model consisting of cone like grains pinned to the extreme points of a poisson point process on r d 1 r having intensity d e d h d h d v

a fast algorithm for simulating multiphase flows through periodic geometries of arbitrary shape this paper presents a new boundary integral equation bie method for simulating particulate and multiphase flows through periodic channels of arbitrary smooth shape in two dimensions the authors consider a particular system multiple vesicles suspended in a periodic channel of arbitrary shape to describe the numerical method and test its performance rather than relying on the periodic green s function as classical bie methods do the method combines the free space green s function with a small auxiliary basis and imposes periodicity as an extra linear condition as a result we can exploit existing free space solver libraries quadratures and fast algorithms and handle a large number of vesicles in a geometrically complex channel spectral accuracy in space is achieved using the periodic trapezoid rule and product quadratures while a first order semi implicit scheme evolves particles by treating the vesicle channel interactions explicitly new constraint correction formulas are introduced that pre

two connections between the sr and hr eigenvalue algorithms abstract the sr and hr algorithms are members of the family of gr algorithms for calculating eigenvalues and invariant subspaces of matrices this paper makes two connections between the sr and hr algorithms 1 an iteration of the sr algorithm on a 2 n 2 n symplectic butterfly matrix using shifts i 1 i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts i 1 i i 1 k 2 an iteration of the sr algorithm on a 2 n 2 n j tridagonal hamiltonian matrix using shifts i i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts 2 i i 1 k

equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points a general formula is proved which relates the equiaffine inner parallel curves of a plane convex body and the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points this formula is applied to improve some well known results in geometric probability for example an estimate which was established for a special case by l c g rogers is obtained with the best possible bound and an asymptotic formula due to a renyi and r sulanke is extended to an asymptotic expansion

qr like algorithms for eigenvalue problems in the year 2000 the dominant method for solving matrix eigenvalue problems is still the qr algorithm this paper discusses the family of gr algorithms with emphasis on the qr algorithm included are historical remarks an outline of what gr algorithms are and why they work and descriptions of the latest highly parallelizable versions of the qr algorithm now that we know how to parallelize it the qr algorithm seems likely to retain its dominance for many years to come

a unitary hessenberg qr based algorithm via semiseparable matrices in this paper we present a novel method for solving the unitary hessenberg eigenvalue problem in the first phase an algorithm is designed to transform the unitary matrix into a diagonal plus semiseparable form then we rely on our earlier adaptation of the qr algorithm to solve the dpss eigenvalue problem in a fast and robust way exploiting the structure of the problem enables us to yield a quadratic time using a linear memory space nonetheless the algorithm remains robust and converges as fast as the customary qr algorithm numerical experiments confirm the effectiveness and the robustness of our approach

a fast direct solver for boundary value problems on locally perturbed geometries abstract many applications including optimal design and adaptive discretization techniques involve solving several boundary value problems on geometries that are local perturbations of an original geometry this manuscript presents a fast direct solver for boundary value problems that are recast as boundary integral equations the idea is to write the discretized boundary integral equation on a new geometry as a low rank update to the discretized problem on the original geometry using the sherman morrison formula the inverse can be expressed in terms of the inverse of the original system applied to the low rank factors and the right hand side numerical results illustrate for problems where perturbation is localized the fast direct solver is three times faster than building a new solver from scratch

on the variable common due date minimal tardy jobs bicriteria two machine flow shop problem with ordered machines we consider a special case of the ordinary np hard two machine flow shop problem with the objective of determining simultaneously a minimal common due date and the minimal number of tardy jobs in panwalkar and koulamas 2012 5 the authors presented quadratic algorithm for the problem when each job has its smaller processing time on the first machine in this note we improve the running time of the algorithm to o n log n by efficient implementation using recently introduced modified binary tree data structure

random polytopes in a convex polytope independence of shape and concentration of vertices write d for the set of all convex bodies convex compact sets with nonempty interior in d define o g l d as the set of those k e 5 b d with vol k 1 fix k e g i d and choose points x l x e k randomly independently and according to the uniform distribution on k then k c o n v x l xn is a random polytope in k write e k n for the expectation of the random variable v o l k k n e k n shows how well k approximates k in volume on the average groemer grl proved that among all convex bodies k e o g l d the ellipsoids are approximated worst i e

minimizing tardy jobs in a flowshop with common due date abstract this paper considers the two machine flowshop scheduling problem where the total time for which a shop is available to complete a given number of jobs is constrained to a known number this problem is equivalent to the flowshop scheduling problem to minimize the total number of tardy jobs against a common due date it has been recently proven to be np hard in the ordinary sense a compact multidimensional knapsack problem formulation of the problem is introduced together with several structural properties a branch and bound procedure is proposed to find an optimal solution to the problem computational results show that the proposed algorithm is capable of finding optimal solutions to problems with up to 900 jobs

a fast direct solver for quasi periodic scattering problems abstract we consider the numerical solution of the scattering of time harmonic plane waves from an infinite periodic array of reflection or transmission obstacles in a homogeneous background medium in two dimensions boundary integral formulations are ideal since they reduce the problem to n unknowns on the obstacle boundary however for complex geometries and or higher frequencies the resulting dense linear system becomes large ruling out dense direct methods and often ill conditioned despite being 2nd kind rendering fast multipole based iterative schemes also inefficient we present an integral equation based solver with o n complexity which handles such ill conditioning using recent advances in fast direct linear algebra to invert hierarchically the isolated obstacle matrix this is combined with a recent periodizing scheme that is robust for all incident angles including wood s anomalies based upon the free space green s function kernel the resulting solver is extremely efficient when multiple incident angles are needed as occurs in many applications our numerical tests include a complicated obstacle several wavelengths in size with n 10 5 and solution error of 10 10 where the solver is 66 times faster per incident angle than a fast multipole based iterative solution and 600 times faster when incident angles are chosen to share bloch phases

two stage production scheduling with an outsourcing option abstract this paper considers a two stage production scheduling problem in which each activity requires two operations to be processed in stages 1 and 2 respectively there are two options for processing each operation the first is to produce it by utilizing in house resources while the second is to outsource it to a subcontractor for in house operations a schedule is constructed and its performance is measured by the makespan that is the latest completion time of operations processed in house operations by subcontractors are instantaneous but require outsourcing cost the objective is to minimize the weighted sum of the makespan and the total outsourcing cost this paper analyzes how the model s computational complexity changes according to unit outsourcing costs in both stages and describes the boundary between np hard and polynomially solvable cases finally this paper presents an approximation algorithm for one np hard case

on estimation of monotone and concave frontier functions abstract when analyzing the productivity of firms one may want to compare how the firms transform a set of inputs x typically labor energy or capital into an output y typically a quantity of goods produced the economic efficiency of a firm is then defined in terms of its ability to operate close to or on the production frontier the boundary of the production set the frontier function gives the maximal level of output attainable by a firm for a given combination of its inputs the efficiency of a firm may then be estimated via the distance between the attained production level and the optimal level given by the frontier function from a statistical viewpoint the frontier function may be viewed as the upper boundary of the support of the population of firms density in the input and output space it is often reasonable to assume that the production frontier is a concave monotone function then a famous estimator in the univariate input and output case is the data envelopment analysis dea estimato

a divide and conquer algorithm for the eigendecomposition of symmetric block diagonal plus semiseparable matrices we present a fast and numerically stable algorithm for computing the eigendecomposition of a symmetric block diagonal plus semiseparable matrix we report numerical experiments that indicate that our algorithm is significantly faster than the standard method which treats the given matrix as a general symmetric dense matrix

the lr cholesky algorithm for symmetric hierarchical matrices abstract we investigate the application of the lr cholesky algorithm to symmetric hierarchical matrices symmetric simple structured hierarchical matrices and symmetric hierarchically semiseparable hss matrices the data sparsity of these matrices make the otherwise expensive lr cholesky algorithm applicable as long as the data sparsity is preserved we will see in an example that the ranks of the low rank blocks grow and the data sparsity gets lost we will explain this behavior by applying a theorem on the structure preservation of diagonal plus semiseparable matrices under lr cholesky transformations therefore we have to give a new more constructive proof for the theorem we will show that the structure of h l matrices is almost preserved and so the lr cholesky algorithm is of almost quadratic complexity for h l matrices

random points and lattice points in convex bodies assume k subset of r d is a convex body and x is a large finite subset of k how many convex polytopes are there whose vertices belong to x is there a typical shape of such polytopes how well does the maximal such polytope which is actually the convex hull of x approximate k we are interested in these questions mainly in two cases the first is when x is a random sample of n uniform independent points from k in this case motivation comes from sylvester s famous four point problem and from the theory of random polytopes the second case is when x k boolean and z d where z d is the lattice of integer points in r d and the questions come from integer programming and geometry of numbers surprisingly or not so surprisingly the answers in the two cases are rather similar

central limit theorems for random polytopes let k be a smooth convex set the convex hull of independent random points in k is a random polytope central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity one essential step is to determine the precise asymptotic order of the occurring variances

generalized oscillatory matrices we define a new class of generalized oscillatory matrices shortly go matrices over a noncommutative ring with identity and a positive part similarly to the classical case this class consists of square generalized totally nonnegative matrices gtn matrices of which some power is generalized totally positive using the previously defined ordering of invertible gtn matrices we study in particular so called basic go matrices which form in a sense atoms of this ordering

eigenstructure of order one quasiseparable matrices three term and two term recurrence relations abstract this paper presents explicit formulas and algorithms to compute the eigenvalues and eigenvectors of order one quasiseparable matrices various recursive relations for characteristic polynomials of their principal submatrices are derived the cost of evaluating the characteristic polynomial of an n n matrix and its derivative is only o n this leads immediately to several versions of a fast quasiseparable newton iteration algorithm in the hermitian case we extend the sturm property to the characteristic polynomials of order one quasiseparable matrices which yields to several versions of a fast quasiseparable bisection algorithm conditions guaranteeing that an eigenvalue of a order one quasiseparable matrix is simple are obtained and an explicit formula for the corresponding eigenvector is derived the method is further extended to the case when these conditions are not fulfilled several particular examples with tridiagonal almost unitary hessenberg and toeplitz matrices are considered the algorithms are based on new three term and two term recurrence relations for the characteristic polynomials of principal submatrices of order one quasiseparable matrices r it turns out that the latter new class of polynomials generalizes and includes two classical families i polynomials orthogonal on the real line that play a crucial role in a number of classical algorithms in numerical linear algebra and ii the szego polynomials that play a significant role in signal processing moreover new formulas can be seen as generalizations of the classical three term recurrence relations for the real orthogonal polynomials and of the two term recurrence relations for the szego polynomials

the combinatorial structure of random polytopes abstract choose n random points in r d let p n be their convex hull and denote by f i p n the number of i dimensional faces of p n a general method for computing the expectation of f i p n i 0 d 1 is presented this generalizes classical results of efron in the case i 0 and renyi and sulanke in the case i d 1 to arbitrary i for random points chosen in a smooth convex body a limit law for f i p n is proved as n for random points chosen in a polytope the expectation of f i p n is determined as n this implies an extremal property for random points chosen in a simplex

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 random approximation of set valued cadlag functions abstract in this paper we study the problem of estimating a cadlag function f whose values are compact convex sets for this purpose a random selection of points in the interval 0 1 is considered and for each selected point x a random sample in f x on the basis of this a sequence of approximants f n m is constructed where n and m are the respective sample sizes under general conditions rates of convergence are obtained for skorokhod s j 1 topology and in case of continuity of the estimated function also for the uniform one

generalized hessenberg matrices we define and study generalized hessenberg matrices i e square matrices which have subdiagonal rank one here subdiagonal rank means the maximum order of a nonsingular submatrix all of whose entries are in the subdiagonal part we prove that the property of being generalized hessenberg matrix is preserved by post and premultiplication by a nonsingular upper triangular matrix by inversion for invertible matrices etc we also study a special kind of generalized hessenberg matrices

a cholesky lr algorithm for the positive definite symmetric diagonal plus semiseparable eigenproblem abstract we present a cholesky lr algorithm with laguerre s shift for computing the eigenvalues of a positive definite symmetric diagonal plus semiseparable matrix by exploiting the semiseparable structure each step of the method can be performed in linear time

variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points we show that the random point measures induced by vertices in the convex hull of a poisson sample on the unit ball when properly scaled and centered converge to those of a mean zero gaussian field we establish limiting variance and covariance asymptotics in terms of the density of the poisson sample similar results hold for the point measures induced by the maximal points in a poisson sample the approach involves introducing a generalized spatial birth growth process allowing for cell overlap

the convex hull of the integer points in a large ball where fk p denotes the number of k dimensional faces of the polytope p the limit as r of the average of r 2 3f0 pr on an interval r r h is determined by balog and deshoullier bd and turns out to be 3 453 h must be large our main result extends 1 1 to any d 2 and to any fk pr with k 0 d 1 theorem 1 for every d 2 there are constants c1 d and c2 d such that for all k 0 d 1

cones generated by random points on half spheres and convex hulls of poisson point processes let u 1 u 2 ldots be random points sampled uniformly and independently from the d dimensional upper half sphere we show that as n rightarrow infty the f vector of the d 1 dimensional convex cone c n generated by u 1 ldots u n weakly converges to a certain limiting random vector without any normalization we also show convergence of all moments of the f vector of c n and identify the limiting constants for the expectations we prove that the expected grassmann angles of c n can be expressed through the expected f vector this yields convergence of expected grassmann angles and conic intrinsic volumes and answers thereby a question of barany et al random struct algorithms 50 1 3 22 2017 https doi org 10 1002 rsa 20644 our approach is based on the observation that the random cone c n weakly converges after a suitable rescaling to a random cone whose intersection with the tangent hyperplane of the half sphere at its north pole is the convex hull of the poisson point process with power law intensity function proportional to vert x vert d gamma where gamma 1 we compute the expected number of facets the expected intrinsic volumes and the expected t functional of this random convex hull for arbitrary gamma 0

a direct solver with o n complexity for variable coefficient elliptic pdes discretized via a high order composite spectral collocation method a numerical method for solving elliptic pdes with variable coefficients on two dimensional domains is presented the method is based on high order composite spectral approximations and is designed for problems with smooth solutions the resulting system of linear equations is solved using a direct as opposed to iterative solver that has optimal o n complexity for all stages of the computation when applied to problems with nonoscillatory solutions such as the laplace and the stokes equations numerical examples demonstrate that the scheme is capable of computing solutions with a relative accuracy of 10 10 or better for challenging problems such as highly oscillatory helmholtz problems and convection dominated convection diffusion equations in terms of speed it is demonstrated that a problem with a nonoscillatory solution that was discretized using 10 8 nodes can be solved in 115 minutes on a personal workstation with two quad core 3 3 ghz cpus since the solver is direct and the solution

two machine flow shop scheduling problem with an outsourcing option abstract we consider a two machine flow shop problem in which each job is processed through an in house system or outsourced to a subcontractor a schedule is established for the in house jobs and performance is measured by the makespan jobs processed by subcontractors require paying an outsourcing cost the objective is to minimize the sum of the makespan and total outsourcing costs we show that the problem is np hard in the ordinary sense we consider a special case in which each job has a processing requirement and each machine a characteristic value in this case the time a job occupies a machine is equal to the job s processing requirement plus a setup time equal to the characteristic value of that machine we introduce some optimality conditions and present a polynomial time algorithm to solve the special case

convergence of algorithms of decomposition type for the eigenvalue problem we develop the theory of convergence of a generic gr algorithm for the matrix eigenvalue problem that includes the qr lr sr and other algorithms as special cases our formulation allows for shifts of origin and multiple gr steps the convergence theory is based on the idea that the gr algorithm performs nested subspace iteration with a change of coordinate system at each step thus the convergence of the gr algorithm depends on the convergence of certain sequences of subspaces it also depends on the quality of the coordinate transformation matrices as measured by their condition numbers we show that with a certain obvious shifting strategy the gr algorithm typically has a quadratic asymptotic convergence rate for matrices possessing certain special types of structure cubic convergence can be achieved

a constraint generation approach for two machine shop problems with jobs selection abstract we consider job selection problems in two stage flow shops and job shops the aim is to select the best job subset with a given cardinality to minimize the makespan these problems are known to be ordinary np hard and the current state of the art algorithms can solve flow shop problems with up to 3000 jobs we introduce a constraint generation approach to the integer linear programming ilp formulation of these problems according to which the constraints associated with nearly all potential critical paths are relaxed and then only the ones violated by the relaxed solution are sequentially reinstated the proposed approach is capable of solving problems with up to 100 000 jobs

an o n2 algorithm for the variable common due date minimal tardy jobs bicriteria two machine flow shop problem with ordered machines abstract we consider the ordinary np hard two machine flow shop problem with the objective of determining simultaneously a minimal common due date and the minimal number of tardy jobs we present an o n 2 algorithm for the problem when the machines are ordered that is when each job has its smaller processing time on the first second machine we also discuss the applicability of the proposed algorithm to the corresponding single objective problem in which the common due date is given

an extended qr solver for large profiled matrices a new method for the solution of the standard eigenvalue problem with large symmetric profile matrices is presented the method is based on the well known qr method for dense matrices a new flexible and reliable extension of the method is developed that is highly suited for the independent computation of any set of eigenvalues in order to analyze the weak convergence of the method in the presence of clustered eigenvalues the qr method is studied two effective stable and numerically cheap extensions are introduced to overcome the troublesome stagnation of the convergence a repeated preconditioning process in combination with jacobi rotations in the parts of the matrix with the strongest convergence is developed to significantly improve both local and global convergence the extensions preserve the profile structure of the matrix the efficiency of the new method is demonstrated with several examples

a multiple shift qr step for structured rank matrices eigenvalue computations for structured rank matrices are the subject of many investigations nowadays there exist methods for transforming matrices into structured rank form qr algorithms for semiseparable and semiseparable plus diagonal form methods for reducing structured rank matrices efficiently to hessenberg form and so forth eigenvalue computations for the symmetric case involving semiseparable and semiseparable plus diagonal matrices have been thoroughly explored a first attempt for computing the eigenvalues of nonsymmetric matrices via intermediate hessenberg like matrices i e a matrix having all subblocks in the lower triangular part of rank at most one was restricted to the single shift strategy unfortunately this leads in general to the use of complex shifts switching thereby from real to complex operations this paper will explain a general multishift implementation for hessenberg like matrices semiseparable matrices are a special case and hence also admit this approach besides a general multishift qr step this will also admit restriction to real computations when computing the eigenvalues of arbitrary real matrices details on the implementation are provided as well as numerical experiments proving the viability of the presented approach

heuristics for a flowshop scheduling problem with stepwise job objective function abstract in this work we introduce the flowshop scheduling problem with delivery dates and cumulative payoffs this problem is a variation of the flowshop scheduling problem with job release dates that maximizes the total payoff with a stepwise job objective function this paper contributes towards proposing a mathematical formulation for this new problem and an original constructive heuristic additionally we develop a new benchmark of 300 hard instances which are available in a public repository besides we provide primal and dual bounds for them extensive computational experiments were carried out taking into account classical constructive heuristics and hybrid local searches results show the merit of the ff heuristic when compared to other classical heuristics for flowshop scheduling problems additionally we compare an iterated local search ils an iterated greedy search igs and a biased random key genetic algorithm brkga both customized for the studied problem and a commercial mixed integer programming solver the comparison between these methods showed that the brkga starting with a solution from ff heuristic is able to find the best solutions in a very short period of time iterated local search and the commercial solver presented significantly worse results than the aforementioned methods

on the orthogonality of eigenvectors computed by divide and conquer techniques a detailed analysis on the accuracy issues in calculating the eigensystems of rank 1 perturbed diagonal systems is presented such calculations are the core of the divide and conquer technique prop

inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices abstract matrices represented as a sum of diagonal and semiseparable ones are considered here these matrices belong to the class of structured matrices which arises in numerous applications fast o n algorithms for their inversion were developed before under additional restrictions which were a source of instability our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms in this paper the case of semiseparable matrices of order one is considered

eigen solving via reduction to dpr1 matrices highly effective polynomial root finders have been recently designed based on eigen solving for dpr1 that is diagonal rank one matrices we extend these algorithms to eigen solving for the general matrix by reducing the problem to the case of the dpr1 input via intermediate transition to a tpr1 that is triangular rank one matrix our transforms use substantially fewer arithmetic operations than the qr classical algorithms but employ non unitary similarity transforms of a tpr1 matrix whose representation tends to be numerically unstable we however operate with tpr1 matrices implicitly as with the inverses of hessenberg matrices in this way our transform of an input matrix into a similar dpr1 matrix partly avoids numerical stability problems and still substantially decreases arithmetic cost versus the qr algorithm

rank structures preserved by the qr algorithm the singular case in an earlier paper we introduced the classes of polynomial and rank structures both of them preserved by applying a shifted qr step on a matrix a in the present paper we further investigate the case of rank structures we show that even if a is a singular matrix a new qr iterate can be constructed having the same rank structure as the matrix a itself to this end we introduce the concepts of effectively eliminating qr decompositions and sparse givens patterns both of them being of independent interest

structures preserved by the qr algorithm in this paper we investigate some classes of structures that are preserved by applying a shifted qr step on a matrix a we will handle two classes of such structures the first we call polynomial structures for example a matrix being hermitian or hermitian up to a rank one correction and the second we call rank structures which are encountered for example in all kinds of what we could call hessenberg like and lower semiseparable like matrices an advantage of our approach is that we define a structure by decomposing it as a collection of building stones which we call structure blocks this allows us to state the results in their natural most general context

eigenvalue computation for unitary rank structured matrices in this paper we describe how to compute the eigenvalues of a unitary rank structured matrix in two steps first we perform a reduction of the given matrix into hessenberg form next we compute the eigenvalues of this resulting hessenberg matrix via an implicit qr algorithm along the way we explain how the knowledge of a certain shift correction term to the structure can be used to speed up the qr algorithm for unitary hessenberg matrices and how this observation was implicitly used in a paper due to william b gragg we also treat an analogue of this observation in the hermitian tridiagonal case

on the area and perimeter of a random convex hull in a bounded convex set suppose k is a compact convex set in 2 and x i 1 i n is a random sample of points in the interior of k under general assumptions on k and the distribution of the x i we study the asymptotic properties of certain statistics of the convex hull of the sample

limit theorems for the typical poisson voronoi cell and the crofton cell with a large inradius in this paper we are interested in the behavior of the typical poisson voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity we prove a law of large numbers for its number of vertices and the area of the cell outside the disk moreover for the latter we establish a central limit theorem as well as moderate deviation type results the proofs deeply rely on precise connections between poisson voronoi tessellations convex hulls of poisson samples and germ grain models in the unit ball besides we derive analogous facts for the crofton cell of a stationary poisson line process in the plane

supporting points processes and some of their applications we introduce a stochastic point process of s supporting points and prove that upon rescaling it converges to a gaussian field the notion of s supporting points specializes for adequately chosen s to pareto or more generally cone extremal points or to vertices of convex hulls or to centers of generalized voronoi tessellations in the models of large scale structure of the universe based on burgers equation the central limit theorems proven here imply i a the asymptotic normality for the number of convex hull vertices in large poisson sample from a simple polyhedra or for the number of pareto vector extremal points in poisson samples with independent coordinates

the qr iteration method for hermitian quasiseparable matrices of an arbitrary order abstract the qr iteration method for tridiagonal matrices is in the heart of one classical method to solve the general eigenvalue problem in this paper we consider the more general class of quasiseparable matrices that includes not only tridiagonal but also companion comrade unitary hessenberg and semiseparble matrices a fast qr iteration method exploiting the hermitian quasiseparable structure and thus generalizing the classical tridiagonal scheme is presented the algorithm is based on an earlier work y eidelman and i gohberg a modification of the dewilde van der veen method for inversion of finite structured matrices linear algebra appl 343 344 2002 419 450 and it applies to the general case of hermitian quasiseparable matrices of an arbitrary order the algorithm operates on generators i e a linear set of parameters defining the quasiseparable matrix and the storage and the cost of one iteration are only linear the results of some numerical experiments are presented an application of this method to solve the general eigenvalue problem via quasiseparable matrices will be analyzed separately elsewhere

a note on generalized hessenberg matrices two different generalizations of the concept of hessenberg matrices have appeared recently in the literature in this note we study their connections in more detail and give a general structure theorem for these classes of matrices

a parallel shared memory implementation of a high order accurate solution technique for variable coefficient helmholtz problems abstract the recently developed hierarchical poincare steklov hps method is a high order discretization technique that comes with a direct solver results from previous papers demonstrate the method s ability to solve helmholtz problems to high accuracy without the so called pollution effect while the asymptotic scaling of the direct solver s computational cost is the same as the nested dissection method serial implementations of the solution technique are not practical for large scale numerical simulations this manuscript presents the first parallel implementation of the hps method specifically we introduce an approach for a shared memory implementation of the solution technique utilizing parallel linear algebra this approach is the foundation for future large scale simulations on supercomputers and clusters with large memory nodes performance results on a desktop computer resembling a large memory node are presented

a note on the nullity theorem in this paper we take a closer look at the nullity theorem as formulated by markham and fiedler in 1986 the theorem is a valuable tool in the computations with structured rank matrices it connects ranks of subblocks of an invertible matrix a with ranks of other subblocks in his inverse a 1 a little earlier barrett and feinsilver 1981 proved a theorem very close to the nullity theorem but restricted to semiseparable and tridiagonal matrices which are each others inverses we will adapt the ideas of barrett and feinsilver to come to a new alternative proof of the nullity theorem based on determinantal formulas in the second part of the paper we extend the nullity theorem to make it suitable for two types of decompositions namely the lu and the qr decomposition these theorems relate the ranks of subblocks of the factors l u and q to the ranks of subblocks of the factored matrix it is shown that a combination of the nullity theorem and his extended versions is suitable to predict in an easy manner the structure of decompositions and or of inverses of structured rank matrices e g higher order band higher order semiseparable hessenberg and many other types of matrices as examples to show the power of the nullity theorem and the related theorems we apply them to semiseparable and related matrices

completing a matrix when certain entries of its inverse are specified abstract suppose a a 11 a 12 a 21 a 22 is an n n matrix where a 11 has order p q if a is nonsingular let a 1 have the transposed partitioning b 11 b 12 b 21 b 22 first we show that the nullities of a 11 and b 22 are equal the remainder of our paper deals with the following completion problem suppose m 1 m 2 n 1 n 2 are nonnegative integers such that m 1 m 2 n 1 n 2 n 0

limit theorems for convex hulls it is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally after rescaling to a strongly mixing markov process as the sample size tends to infinity the structure of the limiting markov process is determined explicitly and from this a central limit theorem for the number of vertices of the convex hull is derived similar results are given for uniform samples from the unit disk

superfast divide and conquer method and perturbation analysis for structured eigenvalue solutions we present a superfast divide and conquer method for finding all the eigenvalues as well as all the eigenvectors in a structured form of a class of symmetric matrices with off diagonal ranks or n

intrinsic volumes and f vectors of random polytopes on appelle polytope aleatoire k n l enveloppe convexe de n points choisis de facon aleatoire dans un corps convexe k on s interesse a l approximation stochastique de k par k n plus particulierement on obtient des resultats d approximation pour les esperances de v s k v s k n ou v s est le s ieme volume intrinseque du nombre de faces s dimensionnelles de k n et de la distance de hausdorff entre k et k n

linking the tpr1 dpr1 and arrow head matrix structures some recent polynomial root finders rely on effective solution of the eigenproblem for special matrices such as dpr1 that is diagonal plus rank one and arrow head matrices we examine the correlation between these two classes and their links to the frobenius companion matrix and we show a gauss similarity transform of a tpr1 that is triangular plus rank one matrix into dpr1 and arrow head matrices theoretically the known unitary similarity transforms of a general matrix into a block triangular matrix with tpr1 diagonal blocks enable the extension of the cited effective eigen solvers from dpr1 and arrow head matrices to general matrices practically however the numerical stability problems with these transforms may limit their value to some special classes of input matrices

on the convergence properties of the orthogonal similarity transformations to tridiagonal and semiseparable plus diagonal form in this paper we will compare the convergence properties of three basic reduction methods by placing them in a general framework it covers the reduction to tridiagonal semiseparable and semiseparable plus diagonal form these reductions are often used as the first step in the computation of the eigenvalues and or eigenvectors of arbitrary matrices in this way the calculation of the eigenvalues using for example the qr algorithm reduces in complexity first we will investigate the convergence properties of these three reduction algorithms it will be shown that for the partially reduced matrices at step k of any of these reduction algorithms the lower right k k already reduced sub block will have the lanczos ritz values w r t a certain starting vector it will also be shown that the reductions to semiseparable and to semiseparable plus diagonal form have an extra convergence behavior a special type of subspace iteration is performed on the lower right k k submatrix which contains these ritz values secondly we look in more detail at the behavior of the involved subspace iteration it will be shown that the reduction method can be interpreted as a nested type of multi shift iteration theoretical results will be presented making it possible to predict the convergence behavior of these reduction algorithms also a theoretical bound on the convergence rate is presented finally we illustrate by means of numerical examples how it is possible to tune the convergence behavior such that it can become a powerful tool for certain applications

two fast algorithms for solving diagonal plus semiseparable linear systems in this paper we discuss the structure of the factors of a qr and a urv factorization of a diagonal plus semiseparable matrix the q factor of a qr factorization has the diagonal plus semiseparable structure the ut and v factor of a urv factorization are semiseparable lower hessenberg orthogonal matrices the strictly upper triangular part of the r factor of a qr and of a urv factorization is the strictly upper triangular part of a rank 2 matrix this latter fact provides a tool to construct a fast qr solver and a fast urv solver for linear systems of the form d s x b

fast and stable eigendecomposition of symmetric banded plus semi separable matrices abstract a new fast and stable algorithm to reduce a symmetric banded plus semi separable matrix to tridiagonal form via orthogonal similarity transformations is presented

a constraint generation algorithm for large scale linear programs using multiple points separation in order to solve linear programs with a large number of constraints constraint generation techniques are often used in these algorithms a relaxation of the formulation containing only a subset of the constraints is first solved then a separation procedure is called which adds to the relaxation any inequality of the formulation that is violated by the current solution the process is iterated until no violated inequality can be found in this paper we present a separation procedure that uses several points to generate violated constraints the complexity of this separation procedure and of some related problems is studied also preliminary computational results about the advantages of using multiple points separation procedures over traditional separation procedures are given for random linear programs and survivable network design they illustrate that for some specific families of linear programs multiple points separation can be computationally effective

generalized totally nonnegative matrices we define a new class of generalized totally nonnegative matrices shortly gtn matrices over a noncommutative ring with identity and a positive part the class of gtn matrices is closed under multiplication every submatrix of a gtn matrix is a gtn matrix as well we also define a standard form of a gtn matrix we then study the invertible gtn matrices of a fixed order and classify them the classes form a partially ordered set can be noncommutatively multiplied and multiplication preserves ordering

a unification of unitary similarity transforms to compressed representations in this paper a new framework for transforming arbitrary matrices to compressed representations is presented the framework provides a generic way of transforming a matrix via unitary similarity transformations to e g hessenberg hessenberg like form and combinations of both the new algorithms are deduced based on the qr factorization of the original matrix relying on manipulations with rotations all the algorithms consist of eliminating the correct set of rotations resulting in a matrix obeying the desired structural constraints based on this new reduction procedure we investigate further correspondences such as irreducibility uniqueness of the reduction procedure and the link with rational krylov methods the unitary similarity transform to hessenberg like form as presented here differs significantly from the one presented in earlier work not only does it use less rotations to obtain the desired structure also the convergence to rational ritz values is not observed in the conventional approach

limit theorems for functionals of convex hulls in 4 a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of a convex polygon is derived this is done by approximating the process of vertices of the convex hull by the process of extreme points of a poisson point process and by considering the latter process of extreme points as a markov process for a particular parametrization we show that this method can also be applied to derive limit theorems for the boundary length and for the area of the convex hull this extents results of renyi and sulanke 1963 and buchta 1984 and shows that the boundary length and the area have a strikingly different probabilistic behavior

the qr algorithm for unitary hessenberg matrices abstract let h be an n n unitary right hessenberg matrix with positive subdiagonal elements using what we call the schur parameterization of h we show how one step of the shifted qr algorithm for h can be carried out in o n arithmetic operations coupled with the shift strategy of eberlein and huang 3 this will permit computation of the spectrum of h to machine precision in o n 2 operations one potential application is the computation of gauss szego quadrature formulas 12 given the associated schur parameters 7 the weights can also be computed by direct analogy with 6

an implicit multishift qr algorithm for hermitian plus low rank matrices hermitian plus possibly non hermitian low rank matrices can be efficiently reduced into hessenberg form the resulting hessenberg matrix can still be written as the sum of a hermitian plus low rank matrix in this paper we develop a new implicit multishift qr algorithm for hessenberg matrices which are the sum of a hermitian plus a possibly non hermitian low rank correction the proposed algorithm exploits both the symmetry and low rank structure to obtain a qr step involving only mathcal o n floating point operations instead of the standard mathcal o n 2 operations needed for performing a qr step on a hessenberg matrix the algorithm is based on a suitable mathcal o n representation of the hessenberg matrix the low rank parts present in both the hermitian and low rank part of the sum are compactly stored by a sequence of givens transformations and a few vectors due to the new representation we cannot apply classical deflation techniques for hessenberg matrices a new efficient technique is developed to overcome this problem some numerical experiments based on matrices arising in applications are performed the experiments illustrate effectiveness and accuracy of both the qr algorithm and the newly developed deflation technique

on the variance of random polytopes abstract a random polytope is the convex hull of uniformly distributed random points in a convex body k a general lower bound on the variance of the volume and f vector of random polytopes is proved also an upper bound in the case when k is a polytope is given for polytopes as for smooth convex bodies the upper and lower bounds are of the same order of magnitude the results imply a law of large numbers for the volume and f vector of random polytopes when k is a polytope

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

a levinson like algorithm for symmetric strongly nonsingular higher order semiseparable plus band matrices in this paper we will derive a solver for a symmetric strongly nonsingular higher order generator representable semiseparable plus band matrix the solver we will derive is based on the levinson algorithm which is used for solving strongly nonsingular toeplitz systems in the first part an o p2n solver for a semiseparable matrix of semiseparability rank p is derived and in a second part we derive an o l2n solver for a band matrix with bandwidth 2l 1 both solvers are constructed in a similar way firstly a yule walker like equation needs to be solved and secondly this solution is used for solving a linear equation with an arbitrary right hand side finally a combination of the above methods is presented to solve linear systems with semiseparable plus band coefficient matrices the overall complexity of this solver is 6 l p 2n plus lower order terms in the final section numerical experiments are performed attention is paid to the timing and the accuracy of the described methods