restarted gmres preconditioned by deflation this paper presents a new preconditioning technique for the restarted gmres algorithm it is based on an invariant subspace approximation which is updated at each cycle numerical examples show that this deflation technique gives a more robust scheme than the restarted algorithm at a low cost of operations and memory

on the construction of deflation based preconditioners in this article we introduce new bounds on the effective condition number of deflated and preconditioned deflated symmetric positive definite linear systems for the case of a subdomain deflation such as that of nicolaides siam j numer anal 24 1987 pp 355 365 these theorems can provide direction in choosing a proper decomposition into subdomains if grid refinement is performed keeping the subdomain grid resolution fixed the condition number is insensitive to the grid size subdomain deflation is very easy to implement and has been parallelized on a distributed memory system with only a small amount of additional communication numerical experiments for a steady state convection diffusion problem are included

additive and multiplicative two level spectral preconditioning for general linear systems in this paper we introduce new preconditioning techniques for the solution of general symmetric and unsymmetric linear systems a x b these approaches borrow some ideas of the multigrid philosophy designed for the solution of linear systems arising from the discretization of elliptic partial differential equations we attempt to improve the convergence rate of a prescribed preconditioner m 1 in a two grid framework this preconditioner is viewed as a smoother and the coarse space is spanned by the eigenvectors associated with the smallest eigenvalues of m 1a we derive both additive and multiplicative variants of the resulting iterated two level preconditioners for unsymmetric linear systems that can also be adapted for hermitian positive definite problems we show that these two level preconditioners shift the smallest eigenvalues to one and tend to better cluster around one those eigenvalues that m 1 already succeeded in moving into the neighborhood of one we illustrate the behavior of our method through extensive numerical experiments on a set of general linear systems finally we show the effectiveness of these approaches on two challenging real applications the first comes from a nonoverlapping domain decomposition method in semiconductor device modeling the second from industrial electromagnetism applications

accelerated gcro dr method for solving sequences of systems of linear equations by recycling some error approximations generated during a previous cycle of iterations we present a technique for improving the convergence of gcro dr the scheme is able to mitigate the occurrence of small skip angles occurring in gcro dr and thus considerably accelerates the convergence the alternative phenomenon of the residual vectors is examined numerically and the effectiveness of the new method is illustrated by several examples from practical applications

the parallel subdomain levelset deflation method in reservoir simulation extreme and isolated eigenvalues are known to be harmful to the convergence of an iterative solver these eigenvalues can be produced by strong heterogeneity in the underlying physics we can improve the quality of the spectrum by deflating the harmful eigenvalues in this work deflation is applied to linear systems in reservoir simulation in particular large sudden differences in the permeability produce extreme eigenvalues the number and magnitude of these eigenvalues is linked to the number and magnitude of the permeability jumps two deflation methods are discussed firstly we state that harmonic ritz eigenvector deflation which computes the deflation vectors from the information produced by the linear solver is unfeasible in modern reservoir simulation due to high costs and lack of parallelism secondly we test a physics based subdomain levelset deflation algorithm that constructs the deflation vectors a priori numerical experiments show that both methods can improve the performance of the linear solver we highlight the fact that subdomain levelset deflation is particularly suitable for a parallel implementation for cases with well defined permeability jumps of a factor 104 or higher parallel physics based deflation has potential in commercial applications in particular the good scalability of parallel subdomain levelset deflation combined with the robust parallel preconditioner for deflated system suggests the use of this method as an alternative for amg

shannon like inequalities for f connections of positive linear maps and positive operators abstract in this paper we investigate a map t b t f t b called an f connection induced by an operator convex concave function f where t denotes a reflexive generalized inverse of a positive linear map t between unital c algebras a and b of hilbert space operators and b b some special cases of f connection with invertible t are related to geometric operator mean relative operator entropy and csiszar operator f divergence we formulate conditions under which the inequality f 1 i k 1 n t k f t k b k holds where t k a b are positive linear maps and b k ran t k in particular the following shannon like inequality 0 k 1 n t k f t k b k is shown for an operator convex function f with f 1 0 the obtained results are specified for csiszar operator f divergence some recent results by isa et al 2013 15 are recovered

matrix free continuation of limit cycles for bifurcation analysis of large thermoacoustic systems in order to define the nonlinear behaviour of a thermoacoustic system it is important to find the regions of parameter space where limit cycles exist continuation methods find limit cycles numerically in the time domain with no additional assumptions other than those used to form the governing equations once the limit cycles are found these continuation methods track them as the operating condition of the system changes most continuation methods are impractical for finding limit cycles in large thermoacoustic systems because the methods require too much computational time and memory in the literature there are therefore only a few applications of continuation methods to thermoacoustics all with low order models matrix free shooting methods efficiently calculate the limit cycles of dissipative systems and have been demonstrated recently in fluid dynamics but are as yet unused in thermoacoustics these matrix free methods are shown to converge quickly to limit cycles by implicitly using a reduced order model property this is because the methods preferentially use the influential bulk motions of the system whilst ignoring the features that are quickly dissipated in time the matrix free methods are demonstrated on a model of a ducted 2d diffusion flame and the stability limits are calculated as a function of the peclet number and the heat release parameter both subcritical and supercritical hopf bifurcations are found physical information about the flame acoustic interaction is found from the limit cycles and floquet modes invariant subspace preconditioning higher order prediction techniques and multiple shooting techniques are all shown to reduce the time required to generate bifurcation surfaces two types of shooting are compared and two types of matrix free evaluation are compared

deflated gmres for systems with multiple shifts and multiple right hand sides abstract we consider solution of multiply shifted systems of nonsymmetric linear equations possibly also with multiple right hand sides first for a single right hand side the matrix is shifted by several multiples of the identity such problems arise in a number of applications including lattice quantum chromodynamics where the matrices are complex and non hermitian some krylov iterative methods such as gmres and bicgstab have been used to solve multiply shifted systems for about the cost of solving just one system restarted gmres can be improved by deflating eigenvalues for matrices that have a few small eigenvalues we show that a particular deflated method gmres dr can be applied to multiply shifted systems in quantum chromodynamics it is common to have multiple right hand sides with multiple shifts for each right hand side we develop a method that efficiently solves the multiple right hand sides by using a deflated version of gmres and yet keeps costs for all of the multiply shifted systems close to those for one shift an example is given showing this can be extremely effective with a quantum chromodynamics matrix

a deflated version of the conjugate gradient algorithm we present a deflated version of the conjugate gradient algorithm for solving linear systems the new algorithm can be useful in cases when a small number of eigenvalues of the iteration matrix are very close to the origin it can also be useful when solving linear systems with multiple right hand sides since the eigenvalue information gathered from solving one linear system can be recycled for solving the next systems and then updated

contiguous relations of hypergeometric series the 15 gauss contiguous relations for 2f1 hypergeometric series imply that any three 2f1 series whose corresponding parameters differ by integers are linearly related over the field of rational functions in the parameters we prove several properties of coefficients of these general contiguous relations and use the results to propose effective ways to compute contiguous relations we also discuss contiguous relations of generalized and basic hypergeometric functions and several applications of them

deflated preconditioned conjugate gradient solvers for the pressure poisson equation a deflated preconditioned conjugate gradient technique has been developed for the solution of the pressure poisson equation within an incompressible flow solver the deflation is done using a region based decomposition of the unknowns making it extremely simple to implement the procedure has shown a considerable reduction in the number of iterations for grids with large graph depth the savings exceed an order of magnitude furthermore the technique has shown a remarkable insensitivity to the number of groups regions chosen and to the way the groups are formed

polynomial preconditioned gmres and gmres dr we look at solving large nonsymmetric systems of linear equations using polynomial preconditioned krylov methods we give a simple way to find the polynomial it is shown that polynomial preconditioning can significantly improve restarted gmres for difficult problems and the reasons for this are examined stability is discussed and algorithms are given for increased stability next we apply polynomial preconditioning to gmres with deflated restarting it is shown that this is worthwhile for sparse matrices and for problems with many small eigenvalues multiple right hand sides are also considered

contiguous relations basic hypergeometric functions and orthogonal polynomials iii associated continuous dual q hahn polynomials explicit solutions for the three term recurrence satisfied by associated continuous dual q hahn polynomials are obtained a minimal solution is identified and an explicit expression for the related continued fraction is derived the absolutely continuous component of the spectral measure is obtained eleven limit cases are discussed in some detail these include associated big q laguerre associated wall associated al salam chihara associated al salam carlitz i and associated continuous q hermite polynomials

deflated and restarted symmetric lanczos methods for eigenvalues and linear equations with multiple right hand sides a deflated restarted lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors the restarting limits the storage so that finding eigenvectors is practical meanwhile the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting some reorthogonalization is necessary to control roundoff error and several approaches are discussed the eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right hand sides experiments are given with large matrices from quantum chromodynamics that have many right hand sides

restarted weighted full orthogonalization method for shifted linear systems it is known that the restarted full orthogonalization method fom outperforms the restarted generalized minimum residual method gmres in several circumstances for solving shifted linear systems when the shifts are handled simultaneously on the basis of the weighted arnoldi process a weighted version of the restarted shifted fom is proposed which can provide accelerating convergence rate with respect to the number of restarts in the cases where our hybrid algorithm needs less enough number of restarts to converge than the restarted shifted fom the associated cpu consuming time is also reduced as shown by the numerical experiments moreover our algorithm is able to solve certain shifted systems which the restarted shifted fom cannot handle sometimes

adaptively preconditioned gmres algorithms the restarted gmres algorithm proposed by saad and schultz siam j sci statist comput 7 1986 pp 856 869 is one of the most popular iterative methods for the solution of large linear systems of equations ax b with a nonsymmetric and sparse matrix this algorithm is particularly attractive when a good preconditioner is available the present paper describes two new methods for determining preconditioners from spectral information gathered by the arnoldi process during iterations by the restarted gmres algorithm these methods seek to determine an invariant subspace of the matrix a associated with eigenvalues close to the origin and to move these eigenvalues so that a higher rate of convergence of the iterative methods is achieved

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

contiguous relations basic hypergeometric functions and orthogonal polynomials i abstract the second order q difference equation of a 2 1 and the contiguous relations of 2 1 s are used to derive continued fraction expansions for quotients of 2 1 s we identify the cases when these continued fractions are positive definite j fractions i e the cases when the denominators of the approximants of the j fractions are orthogonal polynomials the measures with respect to which the denominator polynomials are orthogonal are found and the orthogonal polynomials are studied in some detail

an efficient block preconditioner for jacobian free global local multiscale methods in this paper we develop a block preconditioner for jacobian free global local multiscale methods in which the explicit computation of the jacobian may be circumvented at the macroscale by using a newton krylov process effective preconditioning is necessary for the krylov subspace iterations e g gmres to enhance computational efficiency this is however challenging since no explicit information regarding the jacobian matrix is available the block preconditioning technique developed in this paper circumvents this problem by effectively deflating the spectrum of the jacobian matrix at the current newton step using information about only the krylov subspaces corresponding to the jacobian matrices in the previous newton steps and their representations on those subspaces this approach is optimal and results in exponential convergence of the gmres iterations within each newton step thus minimizing expensive microscale computations without requiring explicit jacobian formation in any step in terms of both computational cost and storage requirements the action of a single block of the preconditioner per gmres step scales linearly as the number of degrees of freedom of the macroscale problem as well as the dimension of the invariant subspace of the preconditioned jacobian matrix copyright 2011 john wiley sons ltd

deflated preconditioned conjugate gradient solvers for the pressure poisson equation extensions and improvements extensions and improvements to a deflated preconditioned conjugate gradient technique for the solution of the pressure poisson equation within an incompressible flow solver are described in particular the use of the technique for embedded grids for cases where volume of fluid or level set schemes are required and its implementation on parallel machines are considered several examples are included that demonstrate a considerable reduction in the number of iterations and a remarkable insensitivity to the number of groups regions chosen and or to the way the groups are formed copyright 2010 john wiley sons ltd

ml k bicgstab a bicgstab variant based on multiple lanczos starting vectors we present a variant of the popular bicgstab method for solving nonsymmetric linear systems the method which we denote by ml k bicgstab is derived from a variant of the bicg method based on a lanczos process using multiple k 1 starting left lanczos vectors compared with the original bicgstab method our new method produces a residual polynomial which is of lower degree after the same number of steps but which also requires fewer matrix vector products to generate on average requiring only 1 1 k matvecs per step empirically it also seems to be more stable and more quickly convergent the new method can be implemented as a k term recurrence and can be viewed as a bridge connecting the arnoldi based fom gmres methods and the lanczos based bicgstab methods

contiguous relations and their computations for f12 hypergeometric series the hypergeometric function f12 a 1 a 2 a 3 z plays an important role in mathematical analysis and its application gauss defined two hypergeometric functions to be contiguous if they have the same power series variable if two of the parameters are pairwise equal and if the third pair differs by 1 he showed that a hypergeometric function and any two other contiguous to it are linearly related in this paper we present an interesting formula as a linear relation of three shifted gauss polynomials in the three parameters a 1 a 2 and a 3 more precisely we obtained a recurrence relation including f12 a email protected 1 a 2 a 3 z f12 a 1 a email protected 2 a 3 z andf12 a 1 a 2 a email protected 3 z for any arbitrary integers a 1 a 2 and a 3

some results on matrix monotone functions abstract let a and b be hermitian matrices we say that a b if a b is nonnegative definite a function 0 r is said to be matrix monotone m m if a b 0 implies that a b matrix convex and concave functions can be similarly defined in this paper several known results some in sharper forms are given alternative proofs these include the following a a positive function t is m m iff t t is m m b if t is m m then both t t and t t are matrix convex and are almost always strictly convex on the other hand t is always concave and almost always strictly concave the concavity of m m functions and an inequality of hansen are consequences of a more general inequality on m m functions some complementary inequalities are also established we then prove an extension of lyapunov s theorem let a and p be positive definite matrices and either or 1 be a m m function then the unique matrix solution x of the matrix equation ax xa a p p a is positive definite a similar result holds for a related equation most of the results are obtained by exploiting the integral representation of m m functions

parametric extensions of shannon inequality and its reverse one in hilbert space operators we shall state the following parametric extensions of shannon inequality and its reverse one in hilbert space operators let p 0 1 and also let a1 a2 an and b1 b2 bn be two sequences of strictly positive operators on a hilbert space h such that j 1naj pbj i then j 1nsp 1 aj bj j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj j 1nsp aj bj log j 1n aj p 1bj i j 1naj pbj j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj j 1nsp 1 aj bj where sq a b a12a 12ba 12qloga 12ba 12a12 for a 0 b 0 and any real number q and a qb a12a 12ba 12qa12 for a 0 b 0 and any real number q in particular if j 1naj j 1nbj i then j 1ns2 aj bj j 1nbjaj 1bjlog j 1nbjaj 1bj log j 1nbjaj 1bj j 1ns1 aj bj 0 j 1ns aj bj log j 1najbj 1aj j 1najbj 1ajlog j 1najbj 1aj j 1ns 1 aj bj where s a b s0 a b a12loga 12ba 12a12 which is the relative operator entropy of a 0 and b 0 our results can be considered as parametric extensions of the following celebrated shannon inequality ann math statistics 22 1951 79 bull syst tech j 27 1948 379 pitman monographs and surveys in pure and applied mathematics 97 addison wesley longman 1998 p 233 which is very useful and so famous in information theory let a1 a2 an and b1 b2 bn be two probability vectors then 0 j 1najlogbj j 1najlogaj see inequalities 2 4 of corollary 2 4

an efficient preconditioned cg method for the solution of a class of layered problems with extreme contrasts in the coefficients knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs a mathematical model for the prediction of fluid pressures is given by a time dependent diffusion equation application of the finite element method leads to a system of linear equations a complication is that the underground consists of layers with very large differences in permeability this implies that the symmetric and positive definite coefficient matrix has a very large condition number bad convergence behavior of the cg method has been observed moreover a classical termination criterion is not valid in this problem after diagonal scaling of the matrix the number of extreme eigenvalues is reduced and it is proved to be equal to the number of layers with a high permeability for the ic preconditioner the same behavior is observed to annihilate the effect of the extreme eigenvalues a deflated cg method is used the convergence rate improves considerably and the termination criterion becomes again reliable finally a cheap approximation of the eigenvectors is proposed

on the contiguous relations of hypergeometric series contiguous relations are a fundamental concept within the theory of hypergeometric series and orthogonal polynomials their study goes back to gauss who gave a list of 15 fundamental relations for the 2f1 hypergeometric series in this paper we will prove some consequences of contiguous relations of 2f1

transverse electric scattering on inhomogeneous objects spectrum of integral operator and preconditioning the domain integral equation method with its fft based matrix vector products is a viable alternative to local methods in free space scattering problems however it often suffers from the extremely slow convergence of iterative methods especially in the transverse electric te case with large or negative permittivity we identify very dense line segments in the spectrum as being partly responsible for this behavior and the main reason why a normally efficient deflating preconditioner does not work we solve this problem by applying an explicit multiplicative regularizing operator which on the operator level transforms the system to the form identity plus compact on the matrix level this regularization reduces the length of the dense spectral segments roughly by a factor of four while preserving the ability to calculate the matrix vector products using the fft algorithm such a regularized system is then further preconditioned by deflating an apparently stable set of eigenvalues with largest magnitudes which results in a robust acceleration of the restarted gmres under constraint memory conditions

a robust gmres based adaptive polynomial preconditioning algorithm for nonsymmetric linear systems in this study a hybrid generalized minimal residual gmres polynomial preconditioning algorithm for solving nonsymmetric systems of linear equations is defined the algorithm uses the results from cycles of restarted gmres to form an effective polynomial preconditioner typically resulting in decreased work requirements the algorithm has the advantage over other hybrid algorithms in that its convergence behavior is well understood the new algorithm converges for all starting vectors if and only if restarted gmres converges the results of numerical experiments with the algorithm are presented

fast computational methods for locating fold points for the power flow equations voltage collapse in a power system can occur following a progressive decline in voltage magnitude at the system buses and mathematically this phenomenon is associated with a fold bifurcation point occurring in the nonlinear algebraic equations used to model the power system in this paper we first discuss some of the methods used to speed up the process of detecting a fold bifurcation focussing on designing test function methods to predict a performance index we then discuss some iterative methods that can be used to improve the newton iterations in particular we present new and efficient preconditioners of the two level type for the jacobian matrix numerical results are given using standard ieee test bus systems

a deflation technique for linear systems of equations iterative methods for solving linear systems of equations can be very efficient if the structure of the coefficient matrix can be exploited to accelerate the convergence of the iterative process however for classes of problems for which suitable preconditioners cannot be found or for which the iteration scheme does not converge iterative techniques may be inappropriate this paper proposes a technique for deflating the eigenvalues and associated eigenvectors of the iteration matrix which either slow down convergence or cause divergence this process is completely general and works by approximating the eigenspace bbb p corresponding to the unstable or slowly converging modes and then applying a coupled iteration scheme on bbb p and its orthogonal complement bbb q

flexible inner outer krylov subspace methods flexible krylov methods refers to a class of methods which accept preconditioning that can change from one step to the next given a krylov subspace method such as cg gmres qmr etc for the solution of a linear system ax b instead of having a fixed preconditioner m and the right preconditioned equation am 1 y b mx y one may have a different matrix say mk at each step in this paper the case where the preconditioner itself is a krylov subspace method is studied there are several papers in the literature where such a situation is presented and numerical examples given a general theory is provided encompassing many of these cases including truncated methods the overall space where the solution is approximated is no longer a krylov subspace but a subspace of a larger krylov space we show how this subspace keeps growing as the outer iteration progresses thus providing a convergence theory for these inner outer methods numerical tests illustrate some important implementation aspects that make the discussed inner outer methods very appealing in practical circumstances

on operator inequalities of some relative operator entropies abstract in our recent paper we introduced the notions of relative operator entropy and tsallis relative operator entropy as a parameter extensions of relative operator entropy and tsallis relative operator entropy in this paper we give upper and lower bounds of these new notions according to operator geometric mean introduced in nikoufar et al 2013 14

simultaneous solution of large scale linear systems and eigenvalue problems with a parallel gmres method a method for simultaneous solution of large and sparse linearized equation sets and the corresponding eigenvalue problems is presented such problems arise from the discretization and the solution of nonlinear problems with the finite element method and newton iteration the method is based on a parallel version of the preconditioned gmres m by deflation the parallel code exploits the architecture of the computational clusters using the mpi message passing interface the convergence rate the parallel speedup and the memory requirements of the proposed method are reported and evaluated

block conjugate gradient algorithms for least squares problems abstract in this paper extensions for the conjugate gradient least squares cgls algorithm in block forms so called block conjugate gradient least squares bcgls are described block parameter matrices are designed to explore the block krylov subspace so that multiple right hand sides can be treated simultaneously while maintaining orthogonality and minimization properties along iterations search subspace is reduced adaptively in case of near rank deficiency to prevent breakdown a deflated form of bcgls is developed to accelerate convergence numerical examples demonstrate effectiveness in handling rank deficiency and efficiency in convergence accelerations in these bcgls forms

fundamental properties of tsallis relative entropy fundamental properties for the tsallis relative entropy in both classical and quantum systems are studied as one of our main results we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by hiai and petz the monotonicity of the quantum tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible the generalized tsallis relative entropy is defined and its subadditivity is shown by its joint convexity moreover the generalized peierls bogoliubov inequality is also proven

stabilization of unstable procedures the recursive projection method fixed point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes the recursive projection method rpm which stabilizes such procedures by computing a projection onto the unstable subspace is presented on this subspace a newton or special newton iteration is performed and the fixed point iteration is used on the complement as continuation in the parameter proceeds the projection is efficiently updated possibly increasing or decreasing the dimension of the unstable subspace the method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system convergence proofs are given and pseudo arclength continuation on the unstable subspace is introduced to allow continuation past folds examples are presented for an important application of the rpm in which a black box time integration scheme is stabilized enabling it to compute unstable steady states the rpm can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes

global fom and gmres algorithms for a class of complex matrix equations abstract in this paper a generalized global arnoldi process is given to produce an f orthonormal basis of complex matrix space then we propose the global full orthogonalization method gl fom and global generalized minimum residual gl gmres method for solving a class of complex matrix equations the new methods can be implemented by the original coefficient matrices by using schur complement we give the expressions of the approximate solutions and the corresponding residuals some convergence results of the proposed methods are also derived finally numerical examples are given to illustrate the effectiveness with comparison to other existing methods

using spectral low rank preconditioners for large electromagnetic calculations for solving large dense complex linear systems that arise in electromagnetic calculations we perform experiments using a general purpose spectral low rank update preconditioner in the context of the gmres method preconditioned by an approximate inverse preconditioner the goal of the spectral preconditioner is to improve the convergence properties by shifting by one the smallest eigenvalues of the original preconditioned system numerical experiments on parallel distributed memory computers are presented to illustrate the efficiency of this technique on large and challenging real life industrial problems copyright 2004 john wiley sons ltd

pole placement preconditioning the partial pole placement problem has received considerable attention in control theory where it is applied to stabilize time invariant linear control systems this paper discusses application of the partial pole placement problem to the construction of preconditioners for linear systems of equations numerical examples show that these preconditioners can improve the rate of convergence of the restarted gmres methods significantly

an adaptive chebyshev iterative method for nonsymmetric linear systems based on modified moments large sparse nonsymmetric systems of linear equations with a matrix whose eigenvalues lie in the right half plane may be solved by an iterative method based on chebyshev polynomials for an interval in the complex plane knowledge of the convex hull of the spectrum of the matrix is required in order to choose parameters upon which the iteration depends adaptive chebyshev algorithms in which these parameters are determined by using eigenvalue estimates computed by the power method or modifications thereof have been described by manteuffel 1978 this paper presents adaptive chebyshev iterative methods in which eigenvalue estimates are computed from modified moments determined during the iterations the computation of eigenvalue estimates from modified moments requires less computer storage than when eigenvalue estimates are computed by a power method and yields faster convergence for many problems

the superlinear convergence behaviour of gmres abstract gmres is a rather popular iterative method for the solution of nonsingular nonsymmetric linear systems it is well known that gmres often has a so called superlinear convergence behaviour i e the rate of convergence seems to improve as the iteration proceeds for the conjugate gradients method this phenomenon has been related to a modest degree of convergence of the ritz values it has been observed in experiments that for gmres too changes in the convergence behaviour seem to be related to the convergence of ritz values in this paper we prove that as soon as eigenvalues of the original operator are sufficiently well approximated by ritz values gmres from then on converges at least as fast as for a related system in which these eigenvalues and their eigenvector components are missing

generalized shannon inequalities based on tsallis relative operator entropy abstract tsallis relative operator entropy is defined and then its properties are given shannon inequality and its reverse one in hilbert space operators derived by furuta linear algebra apll 381 2004 219 are extended in terms of the parameter of the tsallis relative operator entropy moreover the generalized tsallis relative operator entropy is introduced and then several operator inequalities are derived

a polynomial preconditioner for the gmres algorithm the major drawback of gmres is that the storage demands and the number of operations per iteration increase with the number of iterations it is important to avoid that so many iterations are needed that the work per iteration and the storage requirements become unacceptably high this paper describes a polynomial preconditioner with which this can be achieved efficiently the polynomial preconditioner is constructed so that it has a minimization property in an area of the complex plane a suitable area and hence the preconditioning polynomial can be obtained from eigenvalue estimates the polynomial preconditioner is very simple and easy to implement

two level algebraic domain decomposition preconditioners using jacobi schwarz smoother and adaptive coarse grid corrections we investigate two level preconditioners on the extended linear system arising from the domain decomposition method the additive schwarz method is used as a smoother and the coarse grid space is constructed by using the ritz vectors obtained in the arnoldi process the coarse grid space can be improved adaptively as the ritz vectors become a better approximation of the eigenvectors numerical tests on the model problem demonstrate the efficiency

gmres with deflated restarting a modification is given of the gmres iterative method for nonsymmetric systems of linear equations the new method deflates eigenvalues using wu and simon s thick restarting approach siam j matrix anal appl 22 2000 pp 602 616 it has the efficiency of implicit restarting but is simpler and does not have the same numerical concerns the deflation of small eigenvalues can greatly improve the convergence of restarted gmres also it is demonstrated that using harmonic ritz vectors is important because then the whole subspace is a krylov subspace that contains certain important smaller subspaces

recycling krylov subspaces for sequences of linear systems many problems in science and engineering require the solution of a long sequence of slowly changing linear systems we propose and analyze two methods that significantly reduce the total number of matrix vector products required to solve all systems we consider the general case where both the matrix and right hand side change and we make no assumptions regarding the change in the right hand sides furthermore we consider general nonsingular matrices and we do not assume that all matrices are pairwise close or that the sequence of matrices converges to a particular matrix our methods work well under these general assumptions and hence form a significant advancement with respect to related work in this area we can reduce the cost of solving subsequent systems in the sequence by recycling selected subspaces generated for previous systems we consider two approaches that allow for the continuous improvement of the recycled subspace at low cost we consider both hermitian and non hermitian problems and we analyze our algorithms both theoretically and numerically to illustrate the effects of subspace recycling we also demonstrate the effectiveness of our algorithms for a range of applications from computational mechanics materials science and computational physics

generalizations of operator shannon inequality based on tsallis and renyi relative entropies abstract recently we obtained relations among relative operator entropies of sequences for example for relative operator entropy s a b renyi relative operator entropy i t a b and tsallis relative operator entropy t t a b of sequences of strictly positive operators a a 1 a n and b b 1 b n such that i 1 n a i i 1 n b i i s a b i t a b t t a b 0 holds for 0 t 1 this is an extension of operator version of shannon inequality briefly operator shannon inequality discussed by furuta and yanagi kuriyama furuichi in this paper we shall obtain two generalizations of this inequality by considering generalizations of relative operator entropies of sequences

polynomial preconditioners for conjugate gradient calculations dubois greenbaum and rodrigue proposed using a truncated neumann series as an approximation to the inverse of a matrix a for the purpose of preconditioning conjugate gradient iterative approximations to ax b if we assume that a has been symmetrically scaled to have unit diagonal and is thus of the form i g then the neumann series is a power series in g with unit coefficients the incomplete inverse was thought of as a replacement of the incomplete cholesky decomposition suggested by meijerink and van der vorst in the family of methods iccg n the motivation for the replacement was the desire to have a preconditioned conjugate gradient method which only involved vector operations and which utilized long vectors we here suggest parameterizing the incomplete inverse to form a preconditioning matrix whose inverse is a polynomial in g we then show how to select the parameters to minimize the condition number of the product of the polynomial and i g theoretically the resulting algorithm

analysis of acceleration strategies for restarted minimal residual methods we provide an overview of existing strategies which compensate for the deterioration of convergence of minimum residual mr krylov subspace methods due to restarting we evaluate the popular practice of using nearly invariant subspaces to either augment krylov subspaces or to construct preconditioners which invert on these subspaces in the case where these spaces are exactly invariant the augmentation approach is shown to be superior we further show how a strategy recently introduced by de sturler for truncating the approximation space of an mr method can be interpreted as a controlled loosening of the condition for global mr approximation based on the canonical angles between subspaces for the special case of krylov subspace methods we give a concise derivation of the role of ritz and harmonic ritz values and vectors in the polynomial description of krylov spaces as well as of the use of the implicitly updated arnoldi method for manipulating krylov spaces

sharp power mean bounds for the gaussian hypergeometric function sharp inequalities are established between the gaussian hypergeometric function and the power mean these results extend known inequalities involving the complete elliptic integral and the hypergeometric mean

spectral two step preconditioning of multilevel fast multipole algorithm for the fast monostatic rcs calculation a new spectral two step preconditioning of multilevel fast multipole algorithm mlfma is proposed to solve large dense linear systems with multiple right hand sides arising in monostatic radar cross section rcs calculations the first system is solved with a deflated generalized minimal residual gmres method and the eigenvector information is generated at the same time based on this eigenvector information a spectral preconditioner is defined and combined with a previously constructed sparse approximate inverse sai preconditioner in a two step manner resulting in the proposed spectral two step preconditioner restarted gmres with the newly constructed spectral two step preconditioner is considered as the iterative method for solving subsequent systems and the mlfma is used to speed up the matrix vector product operations numerical experiments indicate that the new preconditioner is very effective with the mlfma and can reduce both the iteration number and the computational time significantly

the cycle convergence of restarted gmres for normal matrices is sublinear we prove that the cycle convergence of the restarted gmres applied to a system of linear equations with a normal coefficient matrix is sublinear

bifurcation detection with the un preconditioned gmres m abstract when iterative solvers are used for the solution of the linearized equations sets arising from newton s method detection of bifurcation points in the solution space can be done by solving several eigenvalue problems with the jacobian matrix during parameter continuation monitoring the condition number of appropriate matrices involved in the gmres method employed here replaces or significantly reduces the costly eigenvalue computations the effectiveness of the detection method is examined in connection with the usage of unpre and preconditioned iterative solvers