use of iterative refinement in the solution of sparse linear systems it is well known that if gaussian elimination with iterative refinement ir is used in the solution of systems of linear algebraic equations ax b whose matrices are dense then the accuracy of the results will usually be greater than the accuracy obtained by the use of gaussian elimination without iterative refinement ds however both more storage about 100 because a copy of matrix a is needed and more computing time some extra time is needed to perform the iterative process must be used with ir normally when the matrix is sparse the accuracy of the solution computed by some sparse matrix technique and ir will still be greater in this paper it is verified by many numerical experiments that the use of sparse matrix techniques with ir may also result in a reduction of both the computing time and the storage requirements this will never happen when ir is applied for dense matrices two parameters a drop tolerance t geqq 0 and a stability factor u 1 are introduced in the effo

characterizations of classes of stable matrices abstract this paper extends some results on the structure of subsets of the set of stable matrices for these subsets different characterizations are obtained using the set product defined in this paper as well as inertia and algebraic characterizations for low dimensions 2 2 and 3 3 matrices some inclusion relations that hold for these classes of matrices are proved and some open questions mentioned

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

technical communique some geometric properties of lyapunov equation and lti system while many algebraic properties of lyapunov equation and lti system are well known few geometric properties which may be used to analyze control system properties are studied in this paper some new necessary and sufficient conditions based on geometry are derived for properties of the lyapunov equation and lti system

a note on colored homfly polynomials for hyperbolic knots from wzw models using the correspondence between chern simons theories and wess zumino witten models we present the necessary tools to calculate colored homfly polynomials for hyperbolic knots for two bridge hyperbolic knots we derive the colored homfly invariants in terms of crossing matrices of the underlying wess zumino witten model our analysis extends previous works by incorporating non trivial multiplicities for the primaries appearing in the crossing matrices so as to describe colorings of homfly invariants beyond the totally symmetric or anti symmetric representations of su n the crossing matrices directly relate to 6j symbols of the quantum group mathcal u q su n we present powerful methods to calculate such quantum 6j symbols for general n this allows us to determine previously unknown colored homfly polynomials for two bridge hyperbolic knots we give explicitly the homfly polynomials colored by the representation 2 1 for two bridge hyperbolic knots with up to eight crossings yet the scope of application of our techniques goes beyond knot theory e g our findings can be used to study correlators in wess zumino witten conformal field theories or in the limit to classical groups to determine color factors for yang mills amplitudes

experiences with sparse matrix solvers in parallel ode software abstract the use of implicit methods for numerically solving stiff systems of differential equations requires the solution of systems of nonlinear equations normally these are solved by a newtontype process in which we have to solve systems of linear equations the jacobian of the derivative function determines the structure of the matrices of these linear systems since it often occurs that the components of the derivative function only depend on a small number of variables the system can be considerably sparse hence it can be worth the effort to use a sparse matrix solver instead of a dense lu decomposition this paper reports on experiences with the direct sparse matrix solvers ma28 by duff 1 y12m by zlatev et al 2 and one special purpose matrix solver all embedded in the parallel ode solver psode by sommeijer 3

closed form solutions to the nonhomogeneous yakubovich conjugate matrix equation some closed form solutions are provided for the nonhomogeneous yakubovich conjugate matrix equation email protected f by r with x and y being unknown matrices the presented solutions can offer all the degrees of freedom which is represented by an arbitrarily chosen parameter matrix the primary feature of the solutions is that the matrices f and r are not restricted to be in any canonical form or may be even unknown a priori one of the solutions is neatly expressed in terms of controllability matrices and observability matrices

the smoothed spectral abscissa for robust stability optimization this paper concerns the stability optimization of parameterized matrices a x a problem typically arising in the design of fixed order or fixed structured feedback controllers it is well known that the minimization of the spectral abscissa function alpha a gives rise to very difficult optimization problems since alpha a is not everywhere differentiable and even not everywhere lipschitz we therefore propose a new stability measure namely the smoothed spectral abscissa tilde alpha epsilon a which is based on the inversion of a relaxed h 2 type cost function the regularization parameter epsilon allows tuning the degree of smoothness for epsilon approaching zero the smoothed spectral abscissa converges towards the nonsmooth spectral abscissa from above so that tilde alpha epsilon a leq0 guarantees asymptotic stability evaluation of the smoothed spectral abscissa and its derivatives w r t matrix parameters x can be performed at the cost of solving a primal dual lyapunov equation pair allowing for an efficient integration into a derivative based optimization framework two optimization problems are considered on the one hand the minimization of the smoothed spectral abscissa tilde alpha epsilon a x as a function of the matrix parameters for a fixed value of epsilon and on the other hand the maximization of epsilon such that the stability requirement tilde alpha epsilon a x leq0 is still satisfied the latter problem can be interpreted as an h 2 norm minimization problem and its solution additionally implies an upper bound on the corresponding h infty norm or a lower bound on the distance to instability in both cases additional equality and inequality constraints on the variables can be naturally taken into account in the optimization problem

large n duality lagrangian cycles and algebraic knots we consider knot invariants in the context of large n transitions of topological strings in particular we consider aspects of lagrangian cycles associated to knots in the conifold geometry we show how these can be explicitly constructed in the case of algebraic knots we use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with homfly polynomials furthermore for torus knots using the explicit construction of the lagrangian cycle we also give a direct a model computation and recover the homfly polynomial for this case

optimization of the solution of the parameter dependent sylvester equation and applications this paper deals with an efficient algorithm for optimization of the solution of the parameter dependent sylvester equation a 0 vc 1c 2 t x v x v b 0 vd 1d 2 t e where a 0 b 0 are mxm and nxn matrices respectively further c 1 and c 2 are mxr 1 d 1 and d 2 are nxr 2 and x e are mxn matrices while v is real parameter for optimization we use the following two optimization criteria tr x v min and x v f min we present an efficient algorithm based on derived formulas for the trace and for the frobenius norm of the solution x as functions v tr x v and v x v f as well as for derivatives of these functions that ensures fast optimization of these functions via standard optimization methods like newton s method a special case of this problem is a very important problem of damper viscosity optimization in mechanical systems

an optimization approach for minimum norm and robust partial quadratic eigenvalue assignment problems for vibrating structures the partial quadratic eigenvalue assignment problem pqevap concerns the reassignment of a small number of undesirable eigenvalues of a quadratic matrix pencil while leaving the remaining large number of eigenvalues and the corresponding eigenvectors unchanged the problem arises in controlling undesirable resonance in vibrating structures and in stabilizing control systems the solution of this problem requires computations of a pair of feedback matrices for practical effectiveness these feedback matrices must be computed in such a way that their norms and the condition number of the closed loop eigenvector matrix are as small as possible these considerations give rise to the minimum norm partial quadratic eigenvalue assignment problem mnpqevap and the robust partial quadratic eigenvalue assignment problem rpqevap respectively in this paper we propose new optimization based algorithms for solving these problems the problems are solved directly in a second order setting without resorting to a standard first order formulation so as to avoid the inversion of a possibly ill conditioned matrix and the loss of exploitable structures of the original model the algorithms require the knowledge of only the open loop eigenvalues to be replaced and their corresponding eigenvectors the remaining open loop eigenvalues and their corresponding eigenvectors are kept unchanged the invariance of the large number of eigenvalues and eigenvectors under feedback is guaranteed by a proven mathematical result furthermore the gradient formulas needed to solve the problems by using the quasi newton optimization technique employed are computed in terms of the known quantities only above all the proposed methods do not require the reduction of the model order or the order of the controller even when the underlying finite element model has a very large degree of freedom these attractive features coupled with minimal computational requirements such as solutions of small diagonal sylvester equations make the proposed algorithms ideally suited for application to large real life structures numerical results show significant improvement in feedback norms and in the condition number of the closed loop system also the closed loop eigenvalues have acceptable accuracy

curve counting via stable pairs in the derived category for a nonsingular projective 3 fold x we define integer invariants virtually enumerating pairs c d where c x is an embedded curve and d c is a divisor a virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of x the resulting invariants are conjecturally equivalent after universal transformations to both the gromov witten and dt theories of x for calabi yau 3 folds the latter equivalence should be viewed as a wall crossing formula in the derived category

all loop topological string amplitudes from chern simons theory we demonstrate the equivalence of all loop closed topological string amplitudes on toric local calabi yau threefolds with computations of certain knot invariants for chern simons theory we use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera in particular we explicitly compute the topological string amplitudes for open image in new window 2 up to degree 12 and open image in new window 1 open image in new window 1 up to total degree 10 to all genera this also leads to certain novel large n dualities in the context of ordinary superstrings involving duals of type ii superstrings on local calabi yau three folds without any fluxes

torus knots and the rational daha author s gorsky e oblomkov a rasmussen j shende v abstract 2014 we conjecturally extract the triply graded khovanov rozansky homology of the m n torus knot from the unique finite dimensional simple representation of the rational daha of type a rank n 1 and central character m n the conjectural differentials of gukov dunfield and the third author receive an explicit algebraic expression in this picture yielding a prescription for the doubly graded khovanov rozansky homologies we match our conjecture to previous conjectures of the first author relating knot homology to q t catalan numbers and to previous conjectures of the last three authors relating knot homology to hilbert schemes on singular curves

symplectic conifold transitions we introduce a symplectic surgery in six dimensions which collapses lagrangian three spheres and replaces them by symplectic two spheres under mirror symmetry it corresponds to an operation on complex 3 folds studied by clemens friedman and tian we describe several examples which show that there are either many more calabi yau manifolds e g rigid ones than previously thought or there exist symplectic calabi yaus non kahler symplectic 6 folds with c1 0 the analogous surgery in four dimensions with a generalisation to ade trees of lagrangians implies that the canonical class of a minimal complex surface contains symplectic forms if and only if it has positive square

legendrian knots and constructible sheaves we study the unwrapped fukaya category of lagrangian branes ending on a legendrian knot our knots live at contact infinity in the cotangent bundle of a surface the fukaya category of which is equivalent to the category of constructible sheaves on the surface itself consequently our category can be described as constructible sheaves with singular support controlled by the front projection of the knot we use a theorem of guillermou kashiwara schapira to show that the resulting category is invariant under legendrian isotopies a subsequent article establishes its equivalence to a category of representations of the chekanov eliashberg differential graded algebra we also find two connections to topological knot theory first drawing a positive braid closure on the annulus the moduli space of rank n objects maps to the space of local systems on a circle the second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the colored by n triply graded khovanov rozansky homology second drawing a positive braid closure in the plane the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in a of the homfly polynomial of the braid closure

the re nonnegative definite and re positive definite solutions to the matrix equation axb d in this paper the re nonnegative definite re nnd and re positive definite re pd solutions of the matrix equation axb d are considered the necessary and sufficient conditions for the existence of re nnd and re pd solutions to the equation are provided and the explicit representations of the general re nnd and re pd solutions are given when it is solvable

the topological vertex we construct a cubic field theory which provides all genus amplitudes of the topological a model for all non compact toric calabi yau threefolds the topology of a given feynman diagram encodes the topology of a fixed calabi yau with schwinger parameters playing the role of kahler classes of the threefold we interpret this result as an operatorial computation of the amplitudes in the b model mirror which is the quantum kodaira spencer theory the only degree of freedom of this theory is an unconventional chiral scalar on a riemann surface in this setup we identify the b branes on the mirror riemann surface as fermions related to the chiral boson by bosonization

localization and gluing of topological amplitudes we develop a gluing algorithm for gromov witten invariants of toric calabi yau threefolds based on localization and gluing graphs the main building block of this algorithm is a generating function of cubic hodge integrals of special form we conjecture a precise relation between this generating function and the topological vertex at fractional framing

the sylvester equation and approximate balanced reduction abstract the purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems the resulting approach is completely automatic once an error tolerance is specified and also yields an error bound this is to be contrasted with existing projection methods namely pade via lanczos pvl and rational krylov which do not satisfy these properties our approach is based on the computation and approximation of the cross gramian of the system the cross gramian is the solution of a sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights our method produces a low rank approximation to this gramian in factored form and thus directly provides a reduced order model and a reduced basis for the original system it is well suited to large scale problems because there are no matrix factorizations of the large sparse system matrix only matrix vector products are required

iterative algorithm for the first passage time distribution in a jump diffusion model with regime switching and its applications for a regime switching model with a finite number of regimes and double phase type jumps jiang and pistorius 2008 derived matrix equations with real parameters for the wiener hopf factorization the laplace transform of the first passage time distribution is expressed in terms of the solution of the matrix equations in this paper we provide an iterative algorithm for solving the matrix equations of jiang and pistorius 2008 with complex parameters this makes it possible to obtain numeric values of the laplace transform with complex parameters for the first passage time distribution the laplace transform with complex parameters can be inverted by numerical inversion algorithms such as the euler method as an application we compute the prices of defaultable bonds under a structural model with regime switching and double phase type jumps

a proof of a conjecture of marino vafa on hodge integrals where i c1 li is the first chern class of li and j cj e is the j th chern class of the hodge bundle hodge integrals arise in the calculations of gromov witten invariants by localization techniques 14 7 the explicit evaluation of hodge integrals is a difficult problem the hodge integrals involving only classes can be computed recursively by witten s conjecture 26 proven by kontsevich 13 algorithms of computing hodge integrals are described in 2 in 23 m marino and c vafa obtained a closed formula for a generating function of certain open gromov witten invariants some of which has been reduced to hodge integrals by localization techniques which are not fully clarified mathematically this leads to a conjectural formula of hodge integrals to state this formula we introduce some notation following 28 let

an efficient algorithm for damper optimization for linear vibrating systems using lyapunov equation we consider a second order damped vibration equation mx d e x kx 0 where m d e k are real symmetric matrices of order n the damping matrix d e is defined by d e c u c e where c u presents internal damping and rank c e r where e is dampers viscosity we present an algorithm which derives a formula for the trace of the solution x of the lyapunov equation atx xa b as a function e tr zx e where a a e is a 2n 2n matrix obtained from m d e k such that the eigenvalue problem ay y is equivalent with the quadratic eigenvalue problem 2m d e k x 0 b and z are suitably chosen positive semidefinite matrices moreover our algorithm provides the first and the second derivative of the function e tr zx e almost for free the optimal dampers viscosity is derived as e opt argmin tr zx e if r is small our algorithm allows a sensibly more efficient optimization than standard methods based on the bartels stewart s lyapunov solver

stable pairs and the homfly polynomial given a planar curve singularity we prove a conjecture of oblomkov shende relating the geometry of its hilbert scheme of points to the homfly polynomial of the associated algebraic link more generally we prove an extension of this conjecture due to diaconescu hua soibelman relating stable pair invariants on the conifold to the colored homfly polynomial of the algebraic link our proof uses wall crossing techniques to prove a blowup identity on the algebro geometric side we prove a matching identity for the colored homfly polynomials of a link using skein theoretic techniques

the alexander polynomial of a plane curve singularity and the ring of functions on it we prove two formulae that express the alexander polynomial delta sp c of several variables of a plane curve singularity c in terms of the ring mathscr o sb c of germs of analytic functions on the curve one of them expresses delta sp c in terms of dimensions of some factors corresponding to a multi indexed filtration on the ring mathscr o sb c the other one gives the coefficients of the alexander polynomial delta sp c as euler characteristics of some explicitly described spaces complements to arrangements of projective hyperplanes

implementation of sparse matrix algorithms in an advection diffusion chemistry module a two dimensional advection diffusion chemistry module of a large scale environmental model is taken the module is described mathematically by a system of partial differential equations sequential splitting is used in the numerical treatment the non linear chemistry is most time consuming part and it is handled by six implicit algorithms for solving ordinary differential equations this leads to the solution of very long sequences of systems of linear algebraic equations it is crucial to solve these systems efficiently this is achieved by applying four different algorithms the numerical results indicate that the algorithms based on a preconditioned sparse matrix technique and on a specially designed algorithm for the particular problem under consideration perform best

lyapunov optimization of a damped system our aim is to optimize the damping of a linear vibrating system the penalty function is the average total energy which is equal to the trace of the corresponding lyapunov solution we prove the existence and the uniqueness of the global minimum if the damping varies over the set of all possible positive definite matrices the minimum is shown to be taken on the so called modal critical damping thus confirming a long existing conjecture we also give some preliminary results concerning dampings which depend linearly on the viscosity parameters whereas the damper positions are kept fixed we produce physical examples on which the minimum is taken on a negative viscosity or which have several local minima both phenomena seem to be a consequence of a bad choice of the damper positions

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

knots bps states and algebraic curves we analyze relations between bps degeneracies related to labastida marino ooguri vafa lmov invariants and algebraic curves associated to knots we introduce a new class of such curves which we call extremal a polynomials discuss their special properties and determine exact and asymptotic formulas for the corresponding extremal bps degeneracies these formulas lead to nontrivial integrality statements in number theory as well as to an improved integrality conjecture which is stronger than the known m theory integrality predictions furthermore we determine the bps degeneracies encoded in augmentation polynomials and show their consistency with known colored homfly polynomials finally we consider refined bps degeneracies for knots determine them from the knowledge of super a polynomials and verify their integrality we illustrate our results with twist knots torus knots and various other knots with up to 10 crossings

a comparison of some gmres like methods abstract gmres and cgs are well known iterative methods for the solution of certain sparse linear systems with a nonsymmetric matrix these methods have been compared experimentally in many studies and specific observations on their convergence behavior have been reported a new iterative method to solve a nonsymmetric system has been proposed by eirola and nevanlinna the purpose of this paper is to investigate this method and to compare it with gmres we have seen problems for which this method is more efficient than gmres the original method has as drawbacks that it is not scaling invariant and that it may suffer from numerical instability but it is shown that these deficiencies can be repaired a method proposed by broyden 1969 seems related to the new method and is therefore included in the comparison

multiscale modeling using goal oriented adaptivity and numerical homogenization part i mathematical formulation and numerical results abstract this paper is the first in this series to develop a numerical homogenization method for heterogeneous media and integrate it with goal oriented finite element mesh adaptivity we describe the physical application step and flash imprint lithography in brief and present the mathematical ideas and numerical verification the method requires the moore penrose pseudoinverse of element stiffness matrices algorithms for efficiently computing the pseudoinverse of sparse matrices will be presented in the second paper the purpose of numerical homogenization is to reduce the number of degrees of freedom find locally optimal effective material properties and perform goal oriented mesh refinement traditionally a finite element mesh is designed after obtaining material properties in different regions the mesh has to resolve material discontinuities and rapid variations in the solution in our approach however we generate a sequence of coarse meshes possibly 1 irregular and homogenize material properties on each coarse mesh element using a locally posed constrained convex quadratic optimization problem this upscaling is done using the moore penrose pseudoinverse of the linearized fine scale element stiffness matrices and a material independent interpolation operator numerical verification is done using a two dimensional conductivity problem with known analytical limit finally we present results for two and three dimensional geometries the results show that this method uses orders of magnitude fewer degrees of freedom to give fast and approximate solutions of the original fine scale problem

a note on hurwitz stability of matrices this note proves that every hurwitz stable matrix can be expressed as the product of a symmetric positive definite matrix and a generalised negative definite matrix based on this it is further shown that the entire set of all hurwitz stable matrices of order n is the product of two convex open cones and itself forms a simply connected open cone in the parameter space with a vertex at the origin

a formula of two partition hodge integrals jmg n where tpi ci l is the first chern class of l and xj cj e is the j th chern class of the hodge bundle the study of hodge integrals is an important part of intersection theory on mg n hodge integrals also naturally arise when one computes gromov witten invariants by localization techniques for example the following generating series of hodge integrals arises when one computes local invariants of a toric fano surface in a calabi yau 3 fold by virtual localization 33

knot invariants from topological recursion on augmentation varieties using the duality between wilson loop expectation values of su n chern simons theory on s 3 and topological open string amplitudes on the local mirror of the resolved conifold we study knots on s 3 and their invariants encoded in colored homfly polynomials by means of topological recursion in the context of the local mirror calabi yau threefold of the resolved conifold we generalize the topological recursion of the remodelled b model in order to study branes beyond the class of toric harvey lawson special lagrangians as required for analyzing non trivial knots on s 3 the basic ingredients for the proposed recursion are the spectral curve given by the augmentation variety of the knot and the calibrated annulus kernel encoding the topological annulus amplitudes associated to the knot we present an explicit construction of the calibrated annulus kernel for torus knots and demonstrate the validity of the topological recursion we further argue that if an explicit form of the calibrated annulus kernel is provided for any other knot the proposed topological recursion should still be applicable we study the implications of our proposal for knot theory which exhibit interesting consequences for colored homfly polynomials of mutant knots

accelerating with rank one updates abstract consider the iteration xk 1 xk h b axk for solving ax b a is n x n nonsingular we discuss rank one updates to improve h as an approximation to a 1 during the iteration the update kills and reduces singular values of i ah and thus speeds up the convergence the algorithm terminates after at most n sweeps and if all n sweeps are needed then a 1 has been computed

bps states torus links and wild character varieties a string theoretic framework is constructed relating the cohomology of wild character varieties to refined stable pair theory and torus link invariants explicit conjectural formulas are derived for wild character varieties with a unique irregular point on the projective line for this case this leads to a conjectural colored generalization of existing results of hausel mereb and wong as well as shende treumann and zaslow