numerical range and functional calculus in hilbert space we prove an inequality related to polynomial functions of a square matrix involving the numerical range of the matrix we also show extensions valid for bounded and also unbounded operators in hilbert spaces which allow the development of a functional calculus

ramanujan s series for 1 π arising from his cubic and quartic theories of elliptic functions using certain representations for eisenstein series we derive several of ramanujan s series for 1 arising from his cubic and quartic theories of elliptic functions

on the rapid computation of various polylogarithmic constants we give algorithms for the computation of the d th digit of certain transcendental numbers in various bases these algorithms can be easily implemented multiple precision arithmetic is not needed require virtually no memory and feature run times that scale nearly linearly with the order of the digit desired they make it feasible to compute for example the billionth binary digit of log 2 or on a modest work station in a few hours run time

convergence analysis of the extended krylov subspace method for the lyapunov equation the extended krylov subspace method has recently arisen as a competitive method for solving large scale lyapunov equations using the theoretical framework of orthogonal rational functions in this paper we provide a general a priori error estimate when the known term has rank one special cases such as symmetric coefficient matrix are also treated numerical experiments confirm the proved theoretical assertions

computing real low rank solutions of sylvester equations by the factored adi method abstract we investigate the factored alternating directions implicit adi iteration for large and sparse sylvester equations a novel low rank expression for the associated sylvester residual is established which enables cheap computations of the residual norm along the iteration and which yields a reformulated factored adi iteration the application to generalized sylvester equations is considered as well we also discuss the efficient handling of complex shift parameters and reveal interconnections between the adi iterates w r t those complex shifts this yields a further modification of the factored adi iteration which employs only an absolutely necessary amount of complex arithmetic operations and storage and which produces low rank solution factors consisting of entirely real data certain linear matrix equations such as e g cross gramian sylvester and stein equations are in fact special cases of generalized sylvester equations and we show how specially tailored low rank adi iterations can be deduced from the generalized factored adi 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

analysis of the rational krylov subspace and adi methods for solving the lyapunov equation for large scale problems an effective approach for solving the algebraic lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation although krylov subspaces have been used for a long time only more recent developments have shown that rational krylov subspaces can be a competitive alternative to the classical and very popular alternating direction implicit adi recurrence in this paper we develop a convergence analysis of the rational krylov subspace method rksm based on the kronecker product formulation and on potential theory moreover we propose new enlightening relations between this approach and the adi method our results provide solid theoretical ground for recent numerical evidence of the superiority of rksm over adi when the involved parameters cannot be computed optimally as is the case in many practical application problems

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

on the singular values of matrices with high displacement rank abstract we introduce a new adi based low rank solver for a x x b f where f has rapidly decaying singular values our approach results in both theoretical and practical gains including 1 the derivation of new bounds on singular values for classes of matrices with high displacement rank 2 a practical algorithm for solving certain lyapunov and sylvester matrix equations with high rank right hand sides and 3 a collection of low rank poisson solvers that achieve spectral accuracy and optimal computational complexity

adaptive rational krylov subspaces for large scale dynamical systems the rational krylov space is recognized as a powerful tool within model order reduction techniques for linear dynamical systems however its success has been hindered by the lack of a parameter free procedure which would effectively generate the sequence of shifts used to build the space in this paper we propose an adaptive computation of these shifts the whole procedure only requires us to inject some initial rough estimate of the spectral region of the matrix while further information is automatically generated during the process the approach is a full generalization to the nonsymmetric case of the idea first proposed in druskin et al 2010 18 and it is used for two important problems in control the approximation of the transfer function and the numerical solution of large lyapunov equations the procedure can be naturally extended to other related problems such as the solution of the sylvester equation and parametric or higher order systems several numerical experiments are proposed to assess the quality of the rational projection space over its most natural competitors

class number three ramanujan type series for 1 p abstract there is a class of remarkable series for 1 of the form c 3 n 0 a nb c 3n 6n 3n n 3 where a b c are certain algebraic numbers these were first examined by ramanujan these examples arise by computing singular invariants for j and the constants involved will have degree equal to the class number of the associated imaginary quadratic field our intention is compute all such class number three examples the largest such example with discriminant 907 adds 37 additional digit accuracy per term a class number four example with discriminant 1555 gives 50 digits per term

samuel multiplicity and fredholm theory in this note we combine methods from commutative algebra and complex analytic geometry to calculate the generic values of the cohomology dimensions of a commuting multioperator on its fredholm domain more precisely we prove that for a given fredholm tuple t t 1 t n of commuting bounded operators on a complex banach space x the limits c p t rm lim k rightarrow infty rm dim h p t k x k n exist and calculate the generic dimension of the cohomology groups h p z t x of the koszul complex of t near z 0 to deduce this result we show that the above limits coincide with the samuel multiplicities of the stalks of the cohomology sheaves h p z t mathcal o x mathbb c n of the associated complex of analytic sheaves at z 0

some properties of the arnoldi based methods for linear ill posed problems in this paper we study some properties of the classical arnoldi based methods for solving infinite dimensional linear equations involving compact operators these problems are intrinsically ill posed since a compact operator does not admit a bounded inverse we study the convergence properties and the ability of these algorithms to estimate the dominant singular values of the operator

convergence analysis of projection methods for the numerical solution of large lyapunov equations the numerical solution of large scale continuous time lyapunov matrix equations is of great importance in many application areas assuming that the coefficient matrix is positive definite but not necessarily symmetric in this paper we analyze the convergence of projection type methods for approximating the solution matrix under suitable hypotheses on the coefficient matrix we provide new asymptotic estimates for the error matrix when a galerkin method is used in a krylov subspace numerical experiments confirm the good behavior of our upper bounds when linear convergence of the solver is observed

domb s numbers and ramanujan sato type series for 1 π abstract in this article we construct a general series for 1 we indicate that ramanujan s 1 series are all special cases of this general series and we end the paper with a new class of 1 series our work is motivated by series recently discovered by takeshi sato

rational krylov and adi iteration for infinite size quasi toeplitz matrix equations abstract we consider a class of linear matrix equations involving semi infinite matrices which have a quasi toeplitz structure these equations arise in different settings mostly connected with pdes or the study of markov chains such as random walks on bidimensional lattices we present the theory justifying the existence of the solution in an appropriate banach algebra which is computationally treatable and we propose several methods for computing them we show how to adapt the adi iteration to this particular infinite dimensional setting and how to construct rational krylov methods convergence theory is discussed and numerical experiments validate the proposed approaches

series expansions for 1 πm and πm abstract in terms of the telescoping method we establish a general identity then it is used to derive several interesting series expansions for 1 m and m with m n

numerical computation and new output bounds for time limited balanced truncation of discrete time systems abstract in this paper balancing based model order reduction mor for large scale linear discrete time time invariant systems in prescribed finite time intervals is studied the first main topic is the development of error bounds regarding the approximated output vector within the time limits the influence of different components in the established bounds will be highlighted after that the second part of the article proposes strategies that enable an efficient numerical execution of time limited balanced truncation for large scale systems numerical experiments illustrate the performance of the proposed techniques

an error analysis for rational galerkin projection applied to the sylvester equation in this paper we suggest a new formula for the residual of galerkin projection onto rational krylov spaces applied to a sylvester equation and establish a relation to three different underlying extremal problems for rational functions these extremal problems enable us to compare the size of the residual for the above method with that obtained by adi in addition we deduce several new a priori error estimates for galerkin projection onto rational krylov spaces both for the sylvester and for the lyapunov equation

low rank methods for a class of generalized lyapunov equations and related issues in this paper we study possible low rank solution methods for generalized lyapunov equations arising in bilinear and stochastic control we show that under certain assumptions one can expect a strong singular value decay in the solution matrix allowing for low rank approximations since the theoretical tools strongly make use of a connection to the standard linear lyapunov equation we can even extend the result to the d dimensional case described by a tensorized linear system of equations we further provide some reasonable extensions of some of the most frequently used linear low rank solution techniques such as the alternating directions implicit adi iteration and the krylov plus inverted krylov k pik method by means of some standard numerical examples used in the area of bilinear model order reduction we will show the efficiency of the new methods

the dirac operator of a commuting d tuple abstract given a commuting d tuple t t1 td of otherwise arbitrary operators on a hilbert space there is an associated dirac operator dt significant attributes of the d tuple are best expressed in terms of dt including the taylor spectrum and the notion of fredholmness in fact all properties of t derive from its dirac operator we introduce a general notion of dirac operator in dimension d 1 2 that is appropriate for multivariable operator theory we show that every abstract dirac operator is associated with a commuting d tuple and that two dirac operators are isomorphic iff their associated operator d tuples are unitarily equivalent by relating the curvature invariant introduced in a previous paper to the index of a dirac operator we establish a stability result for the curvature invariant for pure d contractions of finite rank it is shown that for the subcategory of all such t that are a fredholm and and b graded the curvature invariant k t is stable under compact perturbations we do not know if this stability persists when t is fredholm but ungraded although there is concrete evidence that it does

samuel multiplicity and the structure of semi fredholm operators abstract two numerical invariants refining the fredholm index are introduced for any semi fredholm operator in such a way that their difference calculates the fredholm index these two invariants are inspired by samuel multiplicity in commutative algebra and can be regarded as the stabilized dimension of the kernel and cokernel a geometric interpretation of these invariants leads naturally to a 4 4 uptriangular matrix model for any semi fredholm operator on a separable hilbert space this model can be regarded as a refined local version of the apostol s 3 3 triangular representation for arbitrary operators some classical results such as gohberg s punctured neighborhood theorem can be read off directly from our matrix model banach space operators are also considered

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

superlinear convergence of conjugate gradients we give a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the conjugate gradient method or other krylov subspace methods we present a new bound on the relative error after n iterations this bound is valid in an asymptotic sense when the size n of the system grows together with the number of iterations the bound depends on the asymptotic eigenvalue distribution and on the ratio n n under appropriate conditions we show that the bound is asymptotically sharp our findings are related to some recent results concerning asymptotics of discrete orthogonal polynomials an important tool in our investigations is a constrained energy problem in logarithmic potential theory the new asymptotic bounds for the rate of convergence are illustrated by discussing toeplitz systems as well as a model problem stemming from the discretization of the poisson equation

fredholm and invertible tuples of operators the deformation problem using j l taylor s definition of joint spectrum we study fredholm and invertible tuples of operators on a hilbert space we give the foundations for a several variables theory including a natural generalization of atkinson s theorem and an index which well behaves we obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length we then consider the deformation problem and solve it for the class of almost doubly commuting fredholm pairs with a semi fredholm coordinate 1 introduction 1 let t be a bounded linear operator on a banach space t is said to be invertible if there exists an operator s on such that ts st 1 the identity operator on 9c by the open mapping theorem this is equivalent to ker t 0 and r t range of t the last formulation does not rely upon the existence of an inverse for t but rather on the action of the operator t when t is replaced by an tuple of commuting operators several definitions of nonsingular ity exist j l taylor 19 has obtained one which reflects the actions of the operators by considering the koszul complex associated with the tuple 2 in this paper we develop a general several variables theory on the basis of taylor s work and study commuting and almost commuting commuting mod ulo the compacts tuples of operators on a hilbert space we obtain a characterization of joint invertibility in terms of the invertibility of a single operator which is essential for our approach from that we get a number of corollaries which generalize nicely the known elementary results in one variable at the same time the referred characterization allows us to define a continuous invariant under compact perturbations integer valued index on the class of fred holm tuples those almost commuting tuples which are invertible in the calkin algebra this index extends the classical one for fredholm operators we prove that an almost commuting tuple of essentially normal operators with all commu tators in trace class has index zero 2 and that a natural generalization of atkinson s theorem holds for tuples

approximation of functions of large matrices with kronecker structure we consider the numerical approximation of f mathcal a b f a b where b in mathbb r n b rn and mathcal a a is the sum of kronecker products that is mathcal a m 2 otimes i i otimes m 1 in mathbb r n times n a m2 i i m1 rn n here f is a regular function such that f mathcal a f a is well defined we derive a computational strategy that significantly lowers the memory requirements and computational efforts of the standard approximations with special emphasis on the exponential and on completely monotonic functions for which the new procedure becomes particularly advantageous our findings are illustrated by numerical experiments with typical functions used in applications

estimates for hilbertian koszul homology abstract the objective of this paper is to give new kind of estimates for hilbertian koszul homology inspired by commutative algebra in multivariable fredholm theory

an error analysis of galerkin projection methods for linear systems with tensor product structure recent results on the convergence of a galerkin projection method for the sylvester equation are extended to more general linear systems with tensor product structure in the hermitian positive definite case explicit convergence bounds are derived for galerkin projection based on tensor products of rational krylov subspaces the results can be used to optimize the choice of shifts for these methods numerical experiments demonstrate that the convergence rates predicted by our bounds appear to be sharp

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

π formulas with free parameters abstract in terms of the hypergeometric method we establish ten general formulas with free parameters which include several known results as special cases

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

a summation formula and ramanujan type series abstract using some properties of the general rising shifted factorial and the gamma function we derive a variant form of dougall s f 4 5 summation for the classical hypergeometric functions this variant form allows us to derive easily many ramanujan type series for 1 and ramanujan type series for some other constants

hypergeometric series summations and π formulas abstract by two slater s hypergeometric series identities we establish two general summation formulas with six free parameters specializing certain parameters in these formulas we obtain a list of ramanujan type series formulas for 2 and 3

the adi iteration for lyapunov equations implicitly performs h2 pseudo optimal model order reduction abstracttwo approaches for approximating the solution of large scale lyapunov equations are considered the alternating direction implicit adi iteration and projective methods by krylov subspaces we show that they are linked in the way that the adi iteration can always be identified by a petrov galerkin projection with rational block krylov subspaces therefore a unique krylov projected dynamical system can be associated with the adi iteration which is proven to be an h2 pseudo optimal approximation this includes the generalisation of previous results on h2 pseudo optimality to the multivariable case additionally a low rank formulation of the residual in the lyapunov equation is presented which is well suited for implementation and which yields a measure of the obliqueness that the adi iteration is associated with

on the adi method for sylvester equations this paper is concerned with the numerical solution of large scale sylvester equations ax xb c lyapunov equations as a special case in particular included with c having very small rank for stable lyapunov equations penzl 2000 22 and li and white 2002 20 demonstrated that the so called cholesky factor adi method with decent shift parameters can be very effective in this paper we present a generalization of the cholesky factor adi method for sylvester equations an easily implementable extension of penz s shift strategy for the lyapunov equation is presented for the current case it is demonstrated that galerkin projection via adi subspaces often produces much more accurate solutions than adi solutions

eigenvalue decay bounds for solutions of lyapunov equations the symmetric case abstract we present two new bounds for the eigenvalues of the solutions to a class of continuous and discrete time lyapunov equations these bounds hold for lyapunov equations with symmetric coefficient matrices and right hand side matrices of low rank they reflect the fast decay of the nonincreasingly ordered eigenvalues of the solution matrix