the extended krylov subspace method and orthogonal laurent polynomials abstract the need to evaluate expressions of the form f a v where a is a large sparse or structured symmetric matrix v is a vector and f is a nonlinear function arises in many applications the extended krylov subspace method can be an attractive scheme for computing approximations of such expressions this method projects the approximation problem onto an extended krylov subspace k l m a span a l 1 v a 1 v v a v a m 1 v of fairly small dimension and then solves the small approximation problem so obtained we review available results for the extended krylov subspace method and relate them to properties of laurent polynomials the structure of the projected problem receives particular attention we are concerned with the situations when m l and m 2 l

orthogonal laurent polynomials and strong moment theory a survey these topics are found in many parts of 20th century mathematics and its applications in mathematical physics chemistry statistics and engineering historically the analytic theory of continued fractions has played a central role in both the origin and the development of the other topics continued fractions are intimately related to pad e approximants and special functions emphasis is given to the development of strong moment theory and orthogonal laurent polynomials and to the related continued fractions quadrature formulas integral transforms and linear functionals by a strong moment problem we mean the following for a given bisequence n n of real numbers does there exist a distribution function such that

probabilistic bounds for the matrix condition number with extended lanczos bidiagonalization reliable estimates for the condition number of a large sparse real matrix a are important in many applications to get an approximation for the condition number kappa a an approximation for the smallest singular value is needed standard krylov subspaces are usually unsuitable for finding a good approximation to the smallest singular value therefore we study extended krylov subspaces which turn out to be ideal for the simultaneous approximation of both the smallest and largest singular value of a matrix first we develop a new extended lanczos bidiagonalization method with this method we obtain a lower bound for the condition number moreover the method also yields probabilistic upper bounds for kappa a the user can select the probability with which the upper bound holds as well as the ratio of the probabilistic upper bound and the lower bound

a new iterative method for solving large scale lyapunov matrix equations in this paper we propose a new projection method to solve large scale continuous time lyapunov matrix equations the new approach projects the problem onto a much smaller approximation space generated as a combination of krylov subspaces in a and a 1 the reduced problem is then solved by means of a direct lyapunov scheme based on matrix factorizations the reported numerical results show the competitiveness of the new method compared to a state of the art approach based on the factorized alternating direction implicit iteration

solution of large scale evolutionary problems using rational krylov subspaces with optimized shifts we consider the computation of u t exp ta varphi using rational krylov subspace reduction for 0 le t infty where u t varphi in mathbf r n and 0 a a in mathbf r n times n the objective of this work is the optimization of the shifts for the rational krylov subspace rks we consider this problem in the frequency domain and reduce it to a classical zolotaryov problem the latter yields an asymtotically optimal solution with real shifts we also construct an infinite sequence of shifts yielding a nested sequence of the rkss with the same optimal cauchy hadamard convergence rate the effectiveness of the developed approach is demonstrated on an example of the three dimensional diffusion problem for maxwell s equation arising in geophysical exploration

a branch and cut and price algorithm for one dimensional stock cutting and two dimensional two stage cutting abstract the one dimensional cutting stock problem 1d csp and the two dimensional two stage guillotine constrained cutting problem 2d 2cp are considered in this paper the gilmore gomory models of these problems have very strong continuous relaxations providing a good bound in an lp based solution approach in recent years there have been several efforts to attack the one dimensional problem by lp based branch and bound with column generation called branch and price and by general purpose c hv a tal g omory cutting planes in this paper we investigate a combination of both approaches i e the lp relaxation at each branch and price node is strengthened by c hv a tal g omory and g omory mixed integer cuts the branching rule is that of branching on variables of the gilmore gomory formulation tests show that for 1d csp general purpose cuts are useful only in exceptional cases however for 2d 2cp their combination with branching is more effective than either approach alone and mostly better than other methods from the literature

bidimensional packing by bilinear programming we consider geometric problems in which rectangles have to be packed into identical squares these problems turn out to be very hard in practice and ilp formulations in which variables specify the coordinates in the packing perform very poorly while most methods developed so far are based on simple geometric considerations a recent landmark result of fekete and schepers suggests to put these geometric aspects aside and use the most advanced tools for the one dimensional case in this paper we make additional progress in this direction especially on the basic question does a given set of rectangles fit into a square which turns out to be the bottleneck of all the approaches known given a set of rectangles and the associated convex hull of rectangle subsets that fit into a square we derive a wide class of valid inequalities for this convex hull from a complete description of the two knapsack polytopes associated with the widths and the heights of the rectangles respectively in addition we illustrate how to solve the associated separation problem as a bilinear program for which we develop a solution method that turns out to be fast in practice and show that the integer solutions that satisfy all these constraints generally correspond to vertices of the original convex hull for the benchmark instances in the literature the same tools are used to derive lower bounds for the two dimensional bin packing problem corresponding to the determination of an optimal pair of so called dual feasible functions these lower bounds in many cases equal those obtained by the customary set covering formulation for which column generation is very hard but are computable within a time that is smaller by some orders of magnitude this allows us to close a few of the benchmark instances in the literature

approximate mean value theorem for upper subderivatives the purpose of this paper is to give a mean value theorem for a lower semicontinuous function f on a banach space using the upper subderivatives defined by rockafellar l as will be shown by example section 4 it may happen that the subdifferential af x is empty for all x in the closed line segment a 61 where a and b belong to x therefore we cannot formulate the mean value theorem in such a form as that obtained in 2 and 3 for other generalized derivatives we prove among other factors that there exists a sequence xk converging to some point x e a 61 x b such that the sets df xj are nonempty and

lp models for bin packing and cutting stock problems abstract we review several linear programming lp formulations for the one dimensional cutting stock and bin packing problems namely the models of kantorovich gilmore gomory onecut models as in the dyckhoff stadtler approach position indexed models and a model derived from the vehicle routing literature we analyse some relations between the corresponding lp relaxations and their relative strengths and refer how to derive branching schemes that can be used in the exact solution of these problems using branch and price

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

iterative exponential filtering for large discrete ill posed problems we describe a new iterative method for the solution of large very ill conditioned linear systems of equations that arise when discretizing linear ill posed problems the right hand side vector represents the given data and is assumed to be contaminated by measurement errors our method applies a filter function of the form varphi beta t 1 exp beta t 2 with the purpose of reducing the influence of the errors in the right hand side vector on the computed approximate solution of the linear system here beta is a regularization parameter the iterative method is derived by expanding varphi beta t in terms of chebyshev polynomials the method requires only little computer memory and is well suited for the solution of large scale problems we also show how a value of beta and an associated approximate solution that satisfies the morozov discrepancy principle can be computed efficiently an application to image restoration illustrates the performance of the method

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

worst case analyses linear programming and the bin packing problem in this paper we consider the familiar bin packing problem and its associated set partitioning formulation we show that the optimal solution to the bin packing problem can be no larger than 4 3 zlp wherezlp is the optimal solution value of the linear programming relaxation of the set partitioning formulation an example is provided to show that the bound is tight a by product of our analysis is a new worst case bound on the performance of the well studied first fit decreasing and best fit decreasing heuristics

error estimates and evaluation of matrix functions via the faber transform the need to evaluate expressions of the form f a or f a b where f is a nonlinear function a is a large sparse n times n matrix and b is an n vector arises in many applications this paper describes how the faber transform applied to the field of values of a can be used to determine improved error bounds for popular polynomial approximation methods based on the arnoldi process applications of the faber transform to rational approximation methods and in particular to the rational arnoldi process also are discussed

an iterative solution method for solving f a x b using krylov subspace information obtained for the symmetric positive definite matrix a abstract the conjugate gradients method generates successive approximations x i for the solution of the linear system ax b where a is symmetric positive definite and usually sparse it will be shown how intermediate information obtained by the conjugate gradients cg algorithm or by the closely related lanczos algorithm can be used to solve f a x b iteratively in an efficient way for suitable functions f the special case f a a 2 is discussed in particular we also consider the problem of solving ax b for different right hand sides b a variant on a well known algorithm for that problem is proposed which does not seem to suffer from the usual loss of orthogonality in the standard cg and lanczos algorithms

orthogonal laurent polynomials and the strong hamburger moment problem abstract this paper is concerned with the strong hamburger moment problem shmp for a given double sequence of real numbers c c n does there exist a real valued bounded non decreasing function on with infinitely many points of increase such that for every integer n c n t n d t necessary and sufficient conditions for the existence of such a function are given in terms of the positivity of certain hankel determinants associated with c our approach is made through the study of orthogonal and quasi orthogonal laurent polynomials referred to here as l polynomials and closely related gaussian type quadrature formulas in the proof of sufficiency an inner product for l polynomials is defined in terms of the given double sequence c since orthogonal l polynomials are believed to be of interest in themselves some examples of specific systems are considered

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

iterative aggregation and disaggregation algorithm for pseudo polynomial network flow models with side constraints this paper develops a general solution framework based on aggregation techniques to solve np hard problems that can be formulated as a circulation model with specific side constraints the size of the extended mixed integer linear programming formulation is generally pseudo polynomial to efficiently solve exactly these large scale models we propose a new iterative aggregation and disaggregation algorithm at each iteration it projects the original model onto an aggregated one producing an approximate model the process iterates to refine the current aggregated model until the opti mality is proved the computational experiments on two hard optimization problems a variant of the vehicle routing problem and the cutting stock problem show that a generic implementation of the proposed framework allows us to out perform previous known methods

friendly bin packing instances without integer round up property it is well known that the gap between the optimal values of bin packing and fractional bin packing if the latter is rounded up to the closest integer is almost always null known counterexamples to this for integer input values involve fairly large numbers specifically the first one was derived in 1986 and involved a bin capacity of the order of a billion later in 1998 a counterexample with a bin capacity of the order of a million was found in this paper we show a large number of counterexamples with bin capacity of the order of a hundred showing that the gap may be positive even for numbers which arise in customary applications the associated instances are constructed starting from the petersen graph and using the fact that it is fractionally but not integrally 3 edge colorable

numerical approximation of the product of the square root of a matrix with a vector abstract given an n n symmetric positive definite matrix a and a vector c two numerical methods for approximating a 1 2 c are developed analyzed and computationally tested the first method applies a newton iteration to a specific nonlinear system to approximate a 1 2 c while the second method applies a step control method to numerically solve a specific initial value problem to approximate a 1 2 c assuming that a is first reduced to tridiagonal form the first method requires o n 2 operations per iteration while the second method requires o n operations per iteration in contrast numerical methods that first approximate a 1 2 and then compute a 1 2 c generally require o n 3 operations per iteration