a graph theoretic model of symmetric givens operations and its implications abstract symmetric givens operations a gag t g a givens rotation matrix are a basic tool in many matrix computations especially for eigenvalue eigenvector computations a graph theoretic model of these operations is given for symmetric matrices analogous to the graph theoretic model of cholesky factorization using this model it is shown that unless there is accidental cancellation it is impossible to reduce a range of different matrix classes to tridiagonal form in o n 2 givens operations these classes include arrowhead matrices pentadiagonal matrices and cyclic tridiagonal matrices

discounted dynamic programming with unbounded returns application to economic models in this paper we study discounted markov decision processes on an uncountable state space we allow a utility reward function to be unbounded both from above and below a new feature in our approach is an easily verifiable rate of growth condition introduced for a positive part of the utility function this assumption in turn enables us to prove the convergence of a value iteration algorithm to a solution to the bellman equation moreover by virtue of the optimality equation we show the existence of an optimal stationary policy

uniform fatou s lemma abstract fatou s lemma is a classic fact in real analysis stating that the limit inferior of integrals of functions is greater than or equal to the integral of the inferior limit this paper introduces a stronger inequality that holds uniformly for integrals on measurable subsets of a measurable space the necessary and sufficient condition under which this inequality holds for a sequence of finite measures converging in total variation is provided this statement is called the uniform fatou lemma and it holds under the minor assumption that all the integrals in the inequality are well defined the uniform fatou lemma improves the classic fatou lemma in the following directions the uniform fatou lemma states a more precise inequality it provides the necessary and sufficient condition and it deals with variable measures various corollaries of the uniform fatou lemma are formulated the examples in this paper demonstrate that a the uniform fatou lemma may indeed provide a more accurate inequality than the classic fatou lemma b the uniform fatou lemma does not hold if convergence of measures in total variation is relaxed to setwise convergence

explicit preconditioned domain decomposition schemes for solving nonlinear boundary value problems abstract a new class of inner outer iterative procedures in conjunction with picard newton methods based on explicit preconditioning iterative methods for solving nonlinear systems is presented explicit preconditioned iterative schemes based on the explicit computation of a class of domain decomposition generalized approximate inverse matrix techniques are presented for the efficient solution of nonlinear boundary value problems on multiprocessor systems applications of the new composite scheme on characteristic nonlinear boundary value problems are discussed and numerical results are given

fast and stable qr eigenvalue algorithms for generalized companion matrices and secular equations we introduce a class of n n structured matrices which includes three well known classes of generalized companion matrices tridiagonal plus rank one matrices comrade matrices diagonal plus rank one matrices and arrowhead matrices relying on the structure properties of we show that if a then a rq where a qr is the qr decomposition of a this allows one to implement the qr iteration for computing the eigenvalues and the eigenvectors of any a with o n arithmetic operations per iteration and with o n memory storage this iteration applied to generalized companion matrices provides new o n2 flops algorithms for computing polynomial zeros and for solving the associated rational secular equations numerical experiments confirm the effectiveness and the robustness of our approach

nonstationary discrete time deterministic and stochastic control systems bounded and unbounded cases this paper is about nonstationary nonlinear discrete time deterministic and stochastic control systems with borel state and control spaces with either bounded or unbounded costs the control problem is to minimize an infinite horizon total cost performance index using dynamic programming arguments we show that under suitable assumptions the optimal cost functions satisfy optimality equations which in turn give a procedure to find optimal control policies we also prove the convergence of value iteration or successive approximations functions several examples illustrate our results under different sets of assumptions

factoring matrices with a tree structured sparsity pattern abstract let a be a matrix whose sparsity pattern is a tree with maximal degree d max we show that if the columns of a are ordered using minimum degree on a a then factoring a using a sparse lu with partial pivoting algorithm generates only o d max n fill requires only o d max n operations and is much more stable than lu with partial pivoting on a general matrix we also propose an even more efficient and just as stable algorithm called sibling dominant pivoting this algorithm is a strict partial pivoting algorithm that modifies the column preordering locally to minimize fill and work it leads to only o n work and fill more conventional column pre ordering methods that are based usually implicitly on the sparsity pattern of a a are not as efficient as the approaches that we propose in this paper

a parallel divide and conquer algorithm for the symmetric eigenvalue problem on distributed memory architectures we present a new parallel implementation of a divide and conquer algorithm for computing the spectral decomposition of a symmetric tridiagonal matrix on distributed memory architectures the implementation we develop differs from other implementations in that we use a two dimensional block cyclic distribution of the data we use the lowner theorem approach to compute orthogonal eigenvectors and we introduce permutations before the back transformation of each rank one update in order to make good use of deflation this algorithm yields the first scalable portable and numerically stable parallel divide and conquer eigensolver numerical results confirm the effectiveness of our algorithm we compare performance of the algorithm with that of the qr algorithm and of bisection followed by inverse iteration on an ibm sp2 and a cluster of pentium piis

relative perturbation bound for invariant subspaces of graded indefinite hermitian matrices abstract we give a bound for the perturbations of invariant subspaces of graded indefinite hermitian matrix h d ad which is perturbed into h h d a a d such relative perturbations include an important case where h is given with an element wise relative error application of our bounds requires only the knowledge of the size of relative perturbation a and not the perturbation a itself this typically occurs when data are given with relative uncertainties when the matrix is being stored into computer memory and when analyzing some numerical algorithms subspace perturbations are measured in terms of perturbations of angles between subspaces and our bound is therefore a relative variant of the well known davis kahan sin theorem our bounds generalize some of the recent relative perturbation results

kernel based covariate functional balancing for observational studies covariate balance is often advocated for objective causal inference since it mimics randomization in observational data unlike methods that balance specific moments of covariates our proposal attains uniform approximate balance for covariate functions in a reproducing kernel hilbert space the corresponding infinite dimensional optimization problem is shown to have a finite dimensional representation in terms of an eigenvalue optimization problem large sample results are studied and numerical examples show that the proposed method achieves better balance with smaller sampling variability than existing methods

markov decision processes with iterated coherent risk measures this paper considers a markov decision process in borel state and action spaces with the aggregated or say iterated coherent risk measure to be minimised for this problem we establish the bellman optimality equation as well as the value and policy iteration algorithms and show the existence of a deterministic stationary optimal policy the cost function while being allowed to be unbounded from below in the sense that its negative part needs be bounded by some nonnegative real valued possibly unbounded weight function can be arbitrarily unbounded from above and possibly infinitely valued

computing singular values of diagonally dominant matrices to high relative accuracy for a row diagonally dominant matrix if all of its off diagonal entries and its diagonally dominant parts which are defined for each row as the absolute value of the diagonal entry subtracted by the sum of the absolute values of off diagonal entries in that row are accurately known we develop an algorithm that computes all the singular values including zero ones if any with relative errors in the order of the machine precision when the matrix is also symmetric with positive diagonals i e a symmetric positive semi definite diagonally dominant matrix our algorithm computes all eigenvalues to high relative accuracy rounding error analysis will be given and numerical examples will be presented to demonstrate the high relative accuracy of the algorithm

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

conjugacy growth of commutators abstract for the free group f r on r 1 generators respectively the free product g 1 g 2 of two nontrivial finite groups g 1 and g 2 we obtain the asymptotic for the number of conjugacy classes of commutators in f r respectively g 1 g 2 with a given word length in a fixed set of free generators respectively the set of generators given by the nontrivial elements of g 1 and g 2 our result is proven by using the classification of commutators in free groups and in free products by wicks and builds on the works of rivin and sharp who asymptotically counted the conjugacy classes of commutator subgroup elements in f r with a given word length

highly accurate symmetric eigenvalue decomposition and hyperbolic svd let g be a m n real matrix with full column rank and let j be a n n diagonal matrix of signs jii 1 1 the hyperbolic singular value decomposition hsvd of the pair g j is defined as g u v 1 where u is orthogonal is positive definite diagonal and v is j orthogonal matrix vtjv j we analyze when it is possible to compute the hsvd with high relative accuracy this essentially means that each computed hyperbolic singular value is guaranteed to have some correct digits even if they have widely varying magnitudes we show that one sided j orthogonal jacobi method method computes the hsvd with high relative accuracy more precisely let b gd 1 where d is diagonal such that the columns of b have unit norms essentially we show that the computed hyperbolic singular values of the pair g j will have log10 e min b correct decimal digits where e is machine precision we give the necessary relative perturbation bounds and error analysis of the algorithm our numerical tests confirmed all theoretical results for the symmetric non singular eigenvalue problem hx x we analyze the two step algorithm which consists of factorization h gjgt followed by the computation of the hsvd of the pair g j here g is square and non singular let b dg where d is diagonal such that the rows of b have unit norms and let b be defined as above essentially we show that the computed eigenvalues of h will have log10 e 2min b e min b correct decimal digits this accuracy can be much higher then the one obtained by the classical qr and jacobi methods applied to h where the accuracy depends on the spectral condition number of h particularly if the matrices b and b are well conditioned and we are interested in the accurate computation of tiny eigenvalues again we give the perturbation and error bounds and our theoretical predictions are confirmed by a series of numerical experiments we also give the corresponding results for eigenvectors and hyperbolic singular vectors

robust optimal control using conditional risk mappings in infinite horizon abstract we use one step conditional risk mappings to formulate a risk averse version of a total cost problem on a controlled markov process in discrete time infinite horizon the nonnegative one step costs are assumed to be lower semi continuous but not necessarily bounded we derive the conditions for the existence of the optimal strategies and solve the problem explicitly by giving the robust dynamic programming equations under very mild conditions we further give an optimal approximation to the solution and illustrate our algorithm in two examples of optimal investment and lq regulator problems

berge s theorem for noncompact image sets abstract for an upper semi continuous set valued mapping from one topological space to another and for a lower semi continuous function defined on the product of these spaces berge s theorem states lower semi continuity of the minimum of this function taken over the image sets it assumes that the image sets are compact for hausdorff topological spaces this paper extends berge s theorem to set valued mappings with possible noncompact image sets and studies relevant properties of minima

a fast and reliable numerical solver for general bordered k tridiagonal matrix linear equations to improve on the shortcomings observed in symbolic algorithms introduced recently for related matrices a reliable numerical solver is proposed for computing the solution of the matrix linear equation a x b the n n matrix coefficient a is a nonsingular bordered k tridiagonal matrix the particular structure of a is exploited through an incomplete or full givens reduction depending on the singularity of its associated k tridiagonal matrix then adapted back substitution and sherman morrison s formula can be applied specially the inverse of the matrix a is computed moreover for a wide range of matrices a the solution of the vector linear equation a x b can be computed in o n time numerical comparisons illustrate the results

computable eigenvalue bounds for rank k perturbations we investigate lower bounds for the eigenvalues of perturbations of matrices in the footsteps of weyl and ipsen nadler we develop approximating matrices whose eigenvalues are lower bounds for the eigenvalues of the perturbed matrix the number of available eigenvalues and eigenvectors of the original matrix determines how close those approximations can be and if the perturbation is of low rank such bounds are relatively inexpensive to obtain moreover because the process need not be restricted to the eigenvalues of perturbed matrices lower bounds for eigenvalues of bordered diagonal matrices as well as for singular values of rank k perturbations and other updates of n m matrices are given

entrywise perturbation theory for diagonally dominant m matrices with applications this paper introduces a new perspective on the study of computational problems related to diagonally dominant m matrices by establishing some new entrywise perturbation results if a diagonally dominant m matrix is perturbed in a way that each off diagonal entry and its row sums i e the quantities of diagonal dominance have small relative errors we show that its determinant cofactors each entry of the inverse and the smallest eigenvalue all have small relative errors the error bounds are given and they do not depend on any condition number applying this result to the studies of electrical circuits and tail probabilities of a queue whose embedded markov chains is of gi m 1 type we discuss the relative sensitivity of the operating speed of circuits and of the percentile of the queue length respectively

least squares solutions of the matrix equation axb cyd e with the least norm for symmetric arrowhead matrices in this paper the least squares solution of the matrix equation axb cyd e for symmetric arrowhead matrices with the least norm is discussed by using moore penrose inverse and the kronecker product the general expression of the solution to this problem is derived a corresponding numerical algorithm and an example are also given

entrywise perturbation theory and error analysis for markov chains grassmann taksar and heyman introduced a variant of gaussian climination for computing the steady state vector of a markov chain in this paper we prove that their algorithm is stable and that the problem itself is well conditioned in the sense of entrywise relative error thus the algorithm computes each entry of the steady state vector with low relative error even the small steady state probabilities are computed accurately the key to our analysis is to focus on entrywise relative error in both the data and the computed solution rather than making the standard assessments of error based on norms our conclusions do not depend on any condition numbers for the problem

generalized bounded variation and applications to piecewise monotonic transformations we prove the quasi compactness of the perron frobenius operator of piecewise monotonic transformations when the inverse of the derivative is holder continuous or more generally of bounded p variation

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

regenerative analysis and steady state distributions for markov chains we apply regenerative theory to derive certain relations between steady state probabilities of a markov chain these relations are then used to develop a numerical algorithm to find these probabilities the algorithm is a modification of the gauss jordan method in which all elements used in numerical computations are nonnegative as a consequence the algorithm is numerically stable

accurate singular values and differential qd algorithms in september 1991 j w demmel and w m kahan were awarded the second siam prize in numerical linear algebra for their paper accurate singular values of bidiagonal matrices 1 referred to as dk hereafter among several valuable results was the observation that the standard bidiagonal qr algorithm used in linpack 2 and in many other svd programs can be simplified when the shift is zero and of greater importance no subtractions occur the last feature permits very small singular values to be found with almost all the accuracy permitted by the data and at no extra cost

accurate eigenvalues of certain sign regular matrices we present a new o n 3 algorithm for computing all eigenvalues of certain sign regular matrices to high relative accuracy in floating point arithmetic the accuracy and cost are unaffected by the conventional eigenvalue condition numbers a matrix is called sign regular when the signs of its nonzero minors depend only of the order of the minors the sign regular matrices we consider are the ones which are nonsingular and whose kth order nonzero minors are of sign 1 k k 1 2 for all k this class of matrices can also be characterized as nonsingular totally nonnegative matrices with columns in reverse order we exploit a characterization of these particular sign regular matrices as products of nonnegative bidiagonals and the reverse identity we arrange the computations in such a way that no subtractive cancellation is encountered thus guaranteeing high relative forward accuracy

relative error bounds for the lu decomposition via the gth algorithm recently the author showed that the grassmann taksar heyman gth algorithm computes the steady state distribution of a finite state markov chain with low relative error here it is shown that the lu decomposition computed in the course of the gth algorithm also has low relative error the proof requires a refinement of the methods used in the earlier paper

nonstationary discrete time deterministic and stochastic control systems with infinite horizon this article is about nonstationary nonlinear discrete time deterministic and stochastic control systems with borel state and control spaces possibly noncompact control constraint sets and unbounded costs the control problem is to minimise an infinite horizon total cost performance index using dynamic programming arguments we show that under suitable assumptions the optimal cost functions satisfy optimality equations which in turn give a procedure to find optimal control policies

on computing accurate singular values and eigenvalues of matrices with acyclic graphs abstract it is known that small relative perturbations in the entries of a bidiagonal matrix only cause small relative perturbations in its singular values independentof the values of the matrix entries in this paper we show that a matrix has this property if and only if its associated bipartite graph is acyclic we also show how to compute the singular values of such a matrix to high relative accuracy the same algorithm can compute eigenvalues of symmetric matrices with acyclic graphs with tiny componentwise relative backward error this class includes tridiagonal matrices arrow matrices and exponentially many others

symbolic dynamics for geodesic floes the subject of symbolic dynamics is of central importance in the modern theory of dynamical systems the use of symbolic sequences to study dynamical properties of geodesies originates in the work of koebe 21 22 and morse 24 25 and is already foreshadowed in hadamard 15 and jordan 19 the method of koebe and of morse is to code a geodesic on a surface m of negative curvature by recording the order in which it traverses a given set of labelled curves on m the treatment of morse allows variable curvature but assumes at least two boundary components whereas koebe assumes constant curvature but allows infinite connectivity and nonorientability and treats the more difficult case of a closed surface this last case is handled by recording crossings of a fixed pants decomposition anticipating the thurston parameterization of simple curves as described in 11 both koebe and morse used their codings to demonstrate the existence of countably many closed geodesies and of everywhere dense transitive geodesies morse further constructed the first nonsynthetic example of a recurrent nonperiodic discontinuous motion in modern terminology a minimal nowhere dense set later 26 morse treated the special closed surfaces of genus g associated to tesselations of the disc d by regular 4 gons each edge of a region in such a tesselation may be labelled by the isometry which glues it to another side of the same region in forming the quotient sin face m the set of isometries appearing as labels generate tti m thus any geodesic is coded as a doubly infinite sequence of generators of iri m the same method applies quite generally to tesselations associated to any fuchsian group we call the sequences thus obtained cutting sequences and refer to generating sets of this kind as geometric the difficulty of course is to determine precisely the class of sequences which occur for a surface with boundary one obtains exactly reduced sequences in the generators in generalthe problem4s gompligated1 henge the difficulties encountered in 26 see also theorem 3 below there is another method of coding geodesies using certain boundary expansions for points at infinity in the universal cover of m for the modular surface h sl 2 2 the appropriate expansions are continued fractions and for the symmetrical genus g surfaces above they are the nielsen boundary expansions of 27

eigenvalues of a symmetric tridiagonal matrix a divide and conquer approach the symmetric tridiagonal eigenproblem has been the topic of some recent work many methods have been advanced for the computation of the eigenvalues of such a matrix in this paper we present a divide and conquer approach to the computation of the eigenvalues of a symmetric tridiagonal matrix via the evaluation of the characteristic polynomial the problem of evaluation of the characteristic polynomial is partitioned into smaller parts which are solved and these solutions are then combined to form the solution to the original problem we give the update equations for the characteristic polynomial and certain auxiliary polynomials used in the computation furthermore this set of recursions can be implemented on a regulartree structure if the concurrency exhibited by this algorithm is exploited it can be shown that thetime for computation of all the eigenvalues becomeso nlogn instead ofo n2 as is the case for the approach where the order is increased by only one at every step we address the numerical problems associated with the use of the characteristic polynomial and present a numerically stable technique for the eigenvalue computation

computing the singular value decomposition with high relative accuracy we analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy this means that each computed singular value is guaranteed to have some correct digits even if the singular values have widely varying magnitudes this is in contrast to the absolute accuracy provided by conventional backward stable algorithms which in general only guarantee correct digits in the singular values with large enough magniturds it is of interest to compute the tiniest singular values with several correct digits because in some cases such as finite element problems and quantum mechanics it is the smallest singular values that have physical meaning and should be determined accurately by the data many recent papers have identified special classes of matrices where high relative accuracy is possible since it is not possible in general the perturbation theory and algorithms for these matrix classes have been quite different motivating us to seek a common perturbation theory and common algorithm we provide these in this paper and show that high relative accuracy is possible in many new cases as well the briefest way to describe our results is that we can compute the svd to high relative accuracy provided we can compute a high accuracy pivoted ldu decomposition we provide many examples of matrix classes permitting such an ldu decomposition

accurate svds of polynomial vandermonde matrices involving orthonormal polynomials abstract we present a new o n3 algorithm for computing the svd of an n n polynomial vandermonde matrix vp pi 1 xj to high relative accuracy in o n3 time the pi are orthonormal polynomials deg pi i and xj are complex nodes the small singular values of vp can be arbitrarily smaller than the largest ones so that traditional algorithms typically compute them with no relative accuracy at all we show that the singular values even the tiniest ones are usually well conditioned functions of the data xj justifying this computation we also explain how this theory can be extended to other polynomial vandermonde matrices involving polynomials that are not orthonormal or even orthogonal

accurate svds of weakly diagonally dominant m matrices we present a new o n3 algorithm which computes the svd of a weakly diagonally dominant m matrix to high relative accuracy the algorithm takes as an input the offdiagonal entries of the matrix and its row sums

relative perturbation theory iv sin 2θ theorems abstract the double angle theorems of davis and kahan bound the change in an invariant subspace when a hermitian matrix a is subject to an additive perturbation a a a a this paper supplies analogous results when a is subject to a congruential or multiplicative perturbation a a d ad the relative gaps that appear in the bounds involve the spectrum of only one matrix either a or a in contrast to the gaps that appear in the single angle bounds the double angle theorems do not directly bound the difference between the old invariant subspace s and the new one s but instead bound the difference between s and its reflection j s where the mirror is s and j reverses s the orthogonal complement of s the double angle bounds are proportional to the departure from the identity and from orthogonality of the matrix d def d 1 jdj note that d is invariant under the transformation d d for 0 whereas the single angle theorems give bounds proportional to d s departure from the identity and from orthogonality the corresponding results for the singular value problem when a nonsquare matrix b is perturbed to b d 1 bd 2 are also presented

on computing the points and weights for gauss legendre quadrature three methods are compared for computing n points xi and weights w xi for gauss legendre quadrature the eigensystem method computes the points as the eigenvalues of a symmetric tridiagonal matrix and the weights are computed from its eigenvectors the resulting error is o n for the points and o n 2 for the weights using an inverse taylor series computation for the points yakimiw j comput phys 129 1996 pp 406 430 reduced the point error growth to o 1 he also reduced the error in the weights to o n and obtained uniform pointwise relative accuracy however the points xi cluster quadratically near the end points x pm 1 where for some n nmax they coalesce or even exchange to within machine accuracy and computation fails although this has not been a problem in the past it could soon become so because weather models are now approaching resolutions with n 1000 here we compare these methods to the fourier newton method which consists of newton s method for computing the transformed points theta i arccos x i as the zeros of the fourier representations of the legendre polynomials p n theta the theta i are near equally spaced particularly in the neighborhood of x 0 for a fixed arithmetic precision this extends the range of computable points and weights to n n max 2 where nmax is the limit for methods based on xi both the accuracy and speed of the three methods are tabulated

computing approximate eigenpairs of symmetric block tridiagonal matrices a divide and conquer method for computing approximate eigenvalues and eigenvectors of a block tridiagonal matrix is presented in contrast to a method described earlier w n gansterer r c ward and r p muller acm trans math software 28 2002 pp 45 58 the off diagonal blocks can have arbitrary ranks it is shown that lower rank approximations of the off diagonal blocks as well as relaxation of deflation criteria permit the computation of approximate eigenpairs with prescribed accuracy at significantly reduced computational cost compared to standard methods such as for example implemented in lapack

relative perturbation techniques for singular value problems a technique is presented for deriving bounds on the relative change in the singular values of a real matrix or the eigenvalues of a real symmetric matrix due to a perturbation as well as bounds on the angles between the unperturbed and perturbed singular vectors or eigenvectors the class of perturbations considered consists of all delta b for which b delta b d l bd r for some nonsingular matrices d l and d r this class includes componentwise relative perturbations of a bidiagonal or biacyclic matrix and perturbations that annihilate the off diagonal block in a block triangular matrix many existing relative perturbation and deflation bounds are derived from results for this general class of perturbations also some new relative perturbation and deflation results for the singular values and vectors of biacyclic triangular and shifted triangular matrices are presented

relative growth series in some hyperbolic groups we study certain groups of isometries of hyperbolic space including certain hyperbolic coxeter groups and normal subgroups with quotient z 1 we show that the associated relative growth series are not rational this extends results obtained by grigorchuk and de la harpe for free groups and pollicott and the author for surface groups

a mathematical biography of danny c sorensen abstract on the occasion of his 65th birthday we briefly recount dan sorensen s profound contributions to optimization numerical linear algebra and model order reduction for dynamical systems

surface subgroups from linear programming we show that a random endomorphism of a free group gives rise to an hnn extension which contains a closed surface subgroup with probability one a special case is the hnn extension associated to the endomorphism of a rank 2 free group considered by sapir sending a to ab and b to ba we further show that a group obtained by doubling a free group along any collection of subgroups of arbitrary rank has the property that every rational 2 dimensional homology class is virtually represented by surface subgroups and the unit ball in the gromov norm is a finite sided rational polyhedron these results are obtained by a mixture of combinatorial geometric and linear programming techniques we obtain other related results of a more technical nature concerning stable commutator length in free groups especially concerning the structure of the space of so called counting quasimorphisms

implicit standard jacobi gives high relative accuracy we prove that the jacobi algorithm applied implicitly on a decomposition a xdx t of the symmetric matrix a where d is diagonal and x is well conditioned computes all eigenvalues of a to high relative accuracy the relative error in every eigenvalue is bounded by o epsilon kappa x where epsilon is the machine precision and kappa x equiv x 2 cdot x 1 2 is the spectral condition number of x the eigenvectors are also computed accurately in the appropriate sense we believe that this is the first algorithm to compute accurate eigenvalues of symmetric indefinite matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations

perturbation theory for the ldu factorization and accurate computations for diagonally dominant matrices we present a structured perturbation theory for the ldu factorization of row diagonally dominant matrices and we use this theory to prove that a recent algorithm of ye math comp 77 264 2195 2230 2008 computes the l d and u factors of these matrices with relative errors less than 14n 3 u where u is the unit roundoff and n n is the size of the matrix the relative errors for d are componentwise and for l and u are normwise with respect the max norm a m max ij a ij these error bounds guarantee that for any diagonally dominant matrix a we can compute accurately its singular value decomposition and the solution of the linear system ax b for most vectors b independently of the magnitude of the traditional condition number of a and in o n 3 flops

more risk sensitive markov decision processes we investigate the problem of minimizing a certainty equivalent of the total or discounted cost over a finite and an infinite horizon that is generated by a markov decision process mdp in contrast to a risk neutral decision maker this optimization criterion takes the variability of the cost into account it contains as a special case the classical risk sensitive optimization criterion with an exponential utility we show that this optimization problem can be solved by an ordinary mdp with extended state space and give conditions under which an optimal policy exists in the case of an infinite time horizon we show that the minimal discounted cost can be obtained by value iteration and can be characterized as the unique solution of a fixed point equation using a sandwich argument interestingly it turns out that in the case of a power utility the problem simplifies and is of similar complexity than the exponential utility case however has not been treated in the literature so far we also establish the validity and convergence of the policy improvement method a simple numerical example namely the classical repeated casino game is considered to illustrate the influence of the certainty equivalent and its parameters finally the average cost problem is also investigated surprisingly it turns out that under suitable recurrence conditions on the mdp for convex power utility the minimal average cost does not depend on the parameter of the utility function and is equal to the risk neutral average cost this is in contrast to the classical risk sensitive criterion with exponential utility

a numerical comparison of methods for solving secular equations numerical comparisons are carried out by the most prominent methods both second and higher order for solving secular equations these equations arise in several matrix problems and numerical linear algebra applications in addition to the comparisons we propose a new starting point our results show a clear advantage to higher order methods

accurate eigenvalue decomposition of real symmetric arrowhead matrices and applications abstract we present a new algorithm for solving the eigenvalue problem for an n n real symmetric arrowhead matrix the algorithm computes all eigenvalues and all components of the corresponding eigenvectors with high relative accuracy in o n 2 operations under certain circumstances the algorithm is based on a shift and invert approach only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision each eigenvalue and the corresponding eigenvector can be computed separately which makes the algorithm adaptable for parallel computing our results extend to hermitian arrowhead matrices real symmetric diagonal plus rank one matrices and singular value decomposition of real triangular arrowhead matrices

a parallel davidson type algorithm for several eigenvalues in this paper we propose a new parallelization of the davidson algorithm adapted for many eigenvalues in our parallelization we use a relationship between two consecutive subspaces which allows us to calculate eigenvalues in the subspace through an arrowhead matrix theoretical timing estimates for the parallel algorithm are developed and compared against our numerical results on the paragon finally our algorithm is compared against another recent parallel algorithm for multiple eigenvalues but based on arnoldi parpack

random equations in nilpotent groups abstract in this paper we study satisfiability of random equations in an infinite finitely generated nilpotent group g let sat g k be the set of all satisfiable equations over g in k variables for a free abelian group a m of rank m we show that the asymptotic density sat a m k of the set sat a m k in the whole space of all equations in k variables over g is equal to 0 for k 1 and is equal to k m k for k 2 more generally if g is a finitely generated nilpotent infinite group then again the asymptotic density of the set sat g 1 is 0 for k 2 we give robust estimates for the upper and lower asymptotic densities of the set sat g k namely we prove that these densities lie in the interval from 1 t g k h g k to k m k where h g is the hirsch length of g t g is the order of lower central torsion of g and m is the torsion free rank hirsch length of the abelianization of g this allows one to describe the asymptotic behavior of the set sat g k for different parameters k m h g t g

finite sample approximation results for principal component analysis a matrix perturbation approach principal component analysis pca is a standard tool for dimensional reduction of a set of n observations samples each with p variables in this paper using a matrix perturbation approach we study the nonasymptotic relation between the eigenvalues and eigenvectors of pca computed on a finite sample of size n and those of the limiting population pca as n oo as in machine learning we present a finite sample theorem which holds with high probability for the closeness between the leading eigenvalue and eigenvector of sample pca and population pca under a spiked covariance model in addition we also consider the relation between finite sample pca and the asymptotic results in the joint limit p n with p n c we present a matrix perturbation view of the phase transition phenomenon and a simple linear algebra based derivation of the eigenvalue and eigenvector overlap in this asymptotic limit moreover our analysis also applies for finite p n where we show that although there is no sharp phase transition as in the infinite case either as a function of noise level or as a function of sample size n the eigenvector of sample pca may exhibit a sharp loss of tracking suddenly losing its relation to the true eigenvector of the population pca matrix this occurs due to a crossover between the eigenvalue due to the signal and the largest eigenvalue due to noise whose eigenvector points in a random direction

accurate and efficient floating point summation we present and analyze several simple algorithms for accurately computing the sum of n floating point numbers using a wider accumulator let f and f be the number of significant bits in the summands and the accumulator respectively then assuming gradual underflow no overflow and round to nearest arithmetic up to approximately 2f f numbers can be added accurately by simply summing the terms in decreasing order of exponents yielding a sum correct to within about 1 5 units in the last place ulps we apply this result to the floating point formats in the ieee floating point standard for example a dot product of single precision vectors of length at most 33 computed using double precision and sorting is guaranteed correct to nearly 1 5 ulps if double extended precision is used the vector length can be as large as 65 537 we also investigate how the cost of sorting can be reduced or eliminated while retaining accuracy

risk averse dynamic programming for markov decision processes we introduce the concept of a markov risk measure and we use it to formulate risk averse control problems for two markov decision models a finite horizon model and a discounted infinite horizon model for both models we derive risk averse dynamic programming equations and a value iteration method for the infinite horizon problem we develop a risk averse policy iteration method and we prove its convergence we also propose a version of the newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence finally we discuss relations to min max markov decision models

forward stable eigenvalue decomposition of rank one modifications of diagonal matrices abstract we present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rank one modification of a diagonal matrix the algorithm computes each eigenvalue and all components of the corresponding eigenvector with high relative accuracy in o n operations the algorithm is based on a shift and invert approach only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision each eigenvalue and the corresponding eigenvector can be computed separately which makes the algorithm adaptable for parallel computing our results extend to the complex hermitian case the algorithm is similar to the algorithm for solving the eigenvalue problem for real symmetric arrowhead matrices from n jakovcevic stor et al 2015 16

a divide and conquer implementation of the discrete variational dft method for large molecular and solid systems in this paper a novel density functional theory code is described that implements yang s divide and conquer approach in the framework of the discrete variational method the primary aim of the software is the rapid computation of approximate electron densities and density of states for a given arrangement of atoms by using moderately sized grids and compact basis and density fit function sets a high degree of efficiency is achieved through the use of the example of linear alkane chains it is demonstrated that the performance of the method scales linearly with respect to system size for up to more than 1000 atoms details of the implementation are given where emphasis is placed on the approximations made and how linear scaling is achieved finally calculations on some example structures will be presented to survey possible applications of the code

random walks with internal degrees of freedom between two absorbing barriers consider a random walk with a finite number of internal degrees of freedom and with zero drift by using a functional analytic approach based on the spectral theory of matrix polynomials the asymptotics of the first hitting probabilities is obtained when the distance of the barriers tends to infinity

accurate computation of the smallest eigenvalue of a diagonally dominant m matrix if each off diagonal entry and the sum of each row of a diagonally dominant m matrix are known to certain relative accuracy then its smallest eigenvalue and the entries of its inverse are known to the same order relative accuracy independent of any condition numbers in this paper we devise algorithms that compute these quantities with relative errors in the magnitude of the machine precision rounding error analysis and numerical examples are presented to demonstrate the numerical behaviour of the algorithms

a numerical solver for general bordered tridiagonal matrix equations abstract to overcome several limitations of symbolic algorithms introduced recently for matrices of large order a fast numerical solver is proposed for the matrix linear equation a x b where the n n coefficient matrix a is a general nonsingular bordered tridiagonal matrix its sparse structure is preserved through partial givens reduction in particular the matrix inverse of a can be computed for a wide range of bordered tridiagonal linear systems a x b the solution is computed in linear time using back substitution and sherman morrison s formula numerical comparisons illustrate the results

a parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix a parallel algorithm called polysection is presented for computing the eigenvalues of a symmetric tridiagonal matrix the method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level intervals that contain exactly one zero are determined by the zeros of polynomials at the previous level which ensures that different processors compute different zeros the signs of the polynomials at the interval endpoints are determined a priori and used to guarantee that all zeros are found the use of finite precision arithmetic may result in multiple zeros however in this case the intervals coalesce and their number determines exactly the multiplicity of the zero for an n x n matrix the eigenvalues can be determined in o log squared n time with n squared processors and o n time with n processors the method is compared with a parallel variant of bisection that requires o n squared time on a single processor o n time with n processors and o log n time with n squared processors

local limit theorems for free groups in this paper we obtain a local limit theorem for elements of a free group g under the abelianization map cdot g to g g g this is obtained via an analysis involving subshifts of finite type where we obtain a result of independent interest the case of fundamental groups of compact surfaces of genus geq 2 is also discussed

risk sensitive markov control processes we introduce a general framework for measuring risk in the context of markov control processes with risk maps on general borel spaces that generalize known concepts of risk measures in mathematical finance operations research and behavioral economics within the framework applying weighted norm spaces to incorporate unbounded costs also we study two types of infinite horizon risk sensitive criteria discounted total risk and average risk and solve the associated optimization problems by dynamic programming for the discounted case we propose a new discount scheme which is different from the conventional form but consistent with the existing literature while for the average risk criterion we state lyapunov like stability conditions that generalize known conditions for markov chains to ensure the existence of solutions to the optimality equation