an iterative method for nonsymmetric systems with multiple right hand sides we propose a method for the solution of linear systems ax b where a is a large possibly sparse nonsymmetric matrix of order n and b is an arbitrary rectangular matrix of order n times s with s of moderate size the method uses a single krylov subspace per step as a generator of approximations a projection process and a richardson acceleration technique it thus combines the advantages of recent hybrid methods with those for solving symmetric systems with multiple right hand sides numerical experiments indicate that in several cases the method has better practical performance and significantly lower memory requirements than block versions of nonsymmetric solvers and other proposed methods for the solution of systems with multiple right hand sides

bifurcation diagrams in estimated parameter space using a pruned extreme learning machine we propose an algorithm to estimate parameter spaces by using a pruned extreme learning machine but without using principal component analysis and we plot bifurcation diagrams in the estimated parameter spaces to visualize changes in system patterns the estimation of parameter spaces can predict changes in the behavior of a system when its parameters are changed it can be very helpful to adjust the optimal parameters of unknown systems in this paper we estimate the dimension of parameter spaces using the singular values of trained synaptic weights and the parameter spaces based on the method proposed by bagarinao et al using a pruned extreme learning machine we motivate this use of a pruned extreme learning machine through numerical experiments with the estimation of the dimension of parameter spaces for various systems and we show that the proposed method can successfully plot bifurcation diagrams in the estimated parameter spaces in addition we show the results of a bifurcation diagram in the estimated parameter space for maps that are nonlinear in parameters since bagarinao et al limited their method to linear in parameters maps in particular we estimate the parameter spaces for a mathematical model of induction motor drives and for a model of vegetation biomass in ecosystems which are both nonlinear in parameter maps corresponding to real world systems

an efficient ensemble algorithm for numerical approximation of stochastic stokes darcy equations abstract we propose and analyze an efficient ensemble algorithm for fast computation of multiple realizations of the stochastic stokes darcy model with a random hydraulic conductivity tensor the algorithm results in a common coefficient matrix for all realizations at each time step making solving the linear systems much less expensive while maintaining comparable accuracy to traditional methods that compute each realization separately moreover it decouples the stokes darcy system into two smaller sub physics problems which reduces the size of the linear systems and allows parallel computation of the two sub physics problems we prove the ensemble method is long time stable and first order in time convergent under a time step condition and two parameter conditions numerical examples are presented to support the theoretical results and illustrate the application of the algorithm

rapid solution of parameter dependent linear systems for electromagnetic problems in the frequency domain the minimal residual interpolation method reduces the number of iterations in an iterative method for multiple right hand sides it uses computed solutions to initialize an iterative solver with an accurate guess this work demonstrates the efficiency of the method for frequency sweeps and solving scattering problems by plane waves incident from multiple angles a bound on the number of solutions required for plane wave scattering before the remaining solutions obtained by minimal residual interpolation only is given we discuss the performance of the method compared to iterative seed techniques in a numerical example a reduction factor of 60 is obtained on the number of matrix vector multiplications

flexible variants of block restarted gmres methods with application to geophysics in a wide number of applications in computational science and engineering the solution of large linear systems of equations with several right hand sides given at once is required direct methods based on gaussian elimination are known to be especially appealing in that setting nevertheless when the dimension of the problem is very large preconditioned block krylov space solvers are often considered as the method of choice the purpose of this paper is thus to present iterative methods based on block restarted gmres that allow variable preconditioning for the solution of linear systems with multiple right hand sides the use of flexible methods is especially of interest when approximate possibly iterative solvers are considered in the preconditioning phase first we introduce a new variant of block flexible restarted gmres that includes a strategy for detecting when a linear combination of the systems has approximately converged this explicit block size reduction is often called deflation we analyze the main properties of this flexible method based on deflation and notably prove that the frobenius norm of the block residual is always nonincreasing we also present a flexible variant based on both deflation and truncation to especially be used in case of limited memory finally we illustrate the numerical behavior of these flexible block methods for large industrial simulations arising in geophysics where indefinite linear systems of size up to 1 billion unknowns with multiple right hand sides have been successfully solved in a parallel distributed memory environment

global simpler gmres for nonsymmetric systems with multiple right hand sides a new version of the simpler gmres for solving nonsymmetric linear system of equations with multiple right hand sides is presented the main advantage over the simpler block gmres is that the new method requires less storage and computational work per cycle preconditioning techniques can also be used to further accelerate the convergence of the new method finally numerical examples are given to illustrate the effectiveness of the proposed method and its some preconditioned versions

quasi kernel polynomials and their use in non hermitian matrix iterations abstract some of the most efficient iterative algorithms for large sparse hermitian matrix computations are based on orthogonal or kernel polynomials for the case of non hermitian matrices methods based on orthogonal or kernel polynomials are less satisfactory in that the resulting algorithms involve long recurrences consequently it is usually too expensive to run the full algorithms and restarts are necessary a typical example is the generalized minimal residual method gmres for solving non hermitian linear systems where work and storage per iteration grow linearly with the iteration number recently two quasi minimal residual methods qmr for solving non hermitian linear systems have been proposed which unlike gmres are based on short recurrences and hence can be used as true iterative schemes without restarts in this paper the concept of quasi kernel polynomials is introduced some general theory for quasi kernel polynomials is developed such as recurrence relations and a characterization of roots of quasi kernel polynomials as generalized eigenvalues it is pointed out that the qmr approaches are based on two particular instances of quasi kernel polynomials also the use of quasi kernel polynomials for approximating eigenvalues or pseudospectra of large sparse non hermitian matrices is briefly discussed

a hybrid block gmres method for nonsymmetric systems with multiple right hand sides the block gmres bgmres method is a natural generalization of the gmres algorithm for solving large non symmetric systems with multiple right hand sides unfortunately its cost increases significantly per iteration frequently rendering the method impractical in this paper we propose a hybrid block gmres method which offers significant performance improvements over bgmres this method uses the matrix polynomial obtained in the course of a bgmres step and combines the advantages of the block approach with those of successful hybrid methods we discuss the properties and several implementation variants of the method and report results from numerical experiments we also describe how to use these techniques in order to solve multiply shifted systems with multiple right hand sides

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

on the rate of entropy production for the boltzmann equation we show that there exists a wide class of distribution functions with moments of any order as close to their equilibrium values as we like which can provide an abnormally low rate of entropy production the result is valid for the boltzmann equation with any cross section v satisfying a mild restriction the functions are constructed in an explicit form and we discuss some applications of our results

the block cmrh method for solving nonsymmetric linear systems with multiple right hand sides abstract cmrh method changing minimal residual with hessenberg process is an iterative method for solving nonsymmetric linear systems this method is similar to qmr method but based on the hessenberg process instead of the lanczos process on dense matrices the cmrh method is less expensive and requires less storage than other krylov methods this paper presents a block version of the cmrh algorithm for solving linear systems with multiple right hand sides the new algorithm is based on the block hessenberg process and the iterates are characterized by a block version of the quasi minimization property we analyze its main properties and show that under the condition of full rank of block residual the block cmrh method cannot break down finally some numerical examples are presented to show the efficiency of the new method in comparison with the traditional cmrh method and a comparison with the block gmres method is also provided

block minimum perturbation algorithm based on block arnoldi process for nonsymmetric linear systems with multiple right hand sides abstract in many applications in scientific computing and engineering one has to solve huge sparse linear systems of equations with several right hand sides customarily one uses the residual error as a stopping condition but small residue does not imply accurate approximate solutions therefore minimizing quantities other than the residual norm may be more adequate in this context based on the above consideration a block minimum perturbation krylov subspace method bminpet for the solution xm satisfying a a x m b b at each step of nonsymmetric linear systems a x b is shown such that the normwise backward error meets some optimality condition with minimizing the block joint backward perturbation norm a b f of the matrix a b and a stopping criterion is given for consistent system with multiple right hand sides to get a more robust way to stop the iterations this process is a generalization of minimum perturbation algorithm minpert by kasenally and simoncini 1997 analysis of a minimum perturbation algorithm for nonsymmetric linear systems siam j numer anal 34 48 66 to multiple right hand sides for the benefit of the amount of calculation we give lower and upper bounds about min a b f that are easier to compute as a by product we come up with a conception of function f g which is a generalization of function that is a symmetric gauge sg function the algorithm is used to analyze the performance of bgmres method by evaluating the proximity of their solutions to block joint backward perturbation optimality the relationships between bminpet method and relative methods are investigated numerical examples show us that bminpert shows its superiority compared with bfgmres s m pf gsgmres bl bicg rq bgmres and barnoldi in solving the large sparse ill conditioned problems

a block gcrot m k method for linear systems with multiple right hand sides this study is mainly focused on iterative solution to multiple linear systems with several right hand sides for solving such systems efficiently we explore a new block gcrot m k bgcrot m k method which is derived by extending gcrot m k method jason e hicken david w zingg a simplified and flexible variant of gcrot for solving nonsymmetric linear systems siam j sci comput 32 2010 1672 1694 we analyze its main properties it is shown that under the condition of full rank of block residual the frobenius norm of the block residual generated by the proposed method is always nonincreasing moreover we also present its block flexible version bfgcrot m k finally numerical examples demonstrate that the bgcrot m k method and its flexible variant can achieve a smoothed residual and can be more competitive than some other block solvers

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

global scd algorithm for real positive definite linear systems with multiple right hand sides abstract in the present paper we present the left conjugate direction lcd method for linear systems with multiple right hand sides we define the method by using left and right conjugate direction matrices and petrov galerkin condition the method has no breakdown for real positive definite systems the method reduces to the usual cg type method when a is symmetric positive definite we also show how to apply these methods for solving the lyapunov matrix equation finally a numerical experiment is given to illustrate the result

tracing the buckling of a rectangular plate with the block gmres method we investigate bifurcation scenario of the von karman equations with partially clamped boundary conditions defined in rectangular domains first we study how the preconditioned block gmres method can be used in the context of continuation methods for tracing solution curves of large systems of nonlinear equations next we discuss linear stabilities of the von karman equations with partially clamped boundary conditions in particular we calculate the first seven eigenvalues and its associated eigenfunctions of the linearized von karman equations via computer algebra the block gmres method is used to solve linear systems and to detect singularity along solution paths of the discrete problem sample numerical results are reported

a new implementation of the lanczos method in linear problems the lanczos algorithm has proved to be a powerful solution method not only for finding the eigenvalues but for solving linear systems of equations in this work a new implementation of the algorithm is presented for solving linear systems of equations with a sequence of right hand sides the versions of the method proposed in the past treat the right hand side vectors successively by keeping the tridiagonal matrix and the orthonormal basis in fast or secondary storage the new technique handles all approximations to the solution vectors simultaneously without the necessity for keeping the tridiagonal matrix or the orthonormal basis in fast or secondary storage thus when the first solution vector has converged to a required accuracy good approximations to the remaining solution vectors have simultaneously been obtained it then takes fewer iterations to reach the final accuracy by working separately on each of the remaining vectors

convergence properties of some block krylov subspace methods for multiple linear systems in the present paper we give some convergence results of the global minimal residual methods and the global orthogonal residual methods for multiple linear systems using the schur complement formulae and a new matrix product we give expressions of the approximate solutions and the corresponding residuals we also derive some useful relations between the norm of the residuals

block arnoldi and davidson methods for unsymmetric large eigenvalue problems we present two methods for computing the leading eigenpairs of large sparse unsymmetric matrices namely the block arnoldi method and an adaptation of the davidson method to unsymmetric matrices we give some theoretical results concerning the convergence and discuss implementation aspects of the two methods finally some results of numerical tests on a variety of matrices in which we compare these two methods are reported

probability metrics and uniqueness of the solution to the boltzmann equation for a maxwell gas we consider a metric for probability densities with finite variance on d and compare it with other metrics we use it for several applications both in probability and in kinetic theory the main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous boltzmann equation for a gas of true maxwell molecules

global lsmr gl lsmr method for solving general linear systems with several right hand sides abstract the global solvers are an attractive class of iterative solvers for solving linear systems with multiple right hand sides in this paper first a new global method for solving general linear systems with several right hand sides is presented this method is the global version of the lsmr algorithm presented by fong and saunders 2011 then some theoretical properties of the new method are discussed finally numerical experiments from real applications are used to confirm the effectiveness of the proposed method

on improving linear solver performance a block variant of gmres the increasing gap between processor performance and memory access time warrants the re examination of data movement in iterative linear solver algorithms for this reason we explore and establish the feasibility of modifying a standard iterative linear solver algorithm in a manner that reduces the movement of data through memory in particular we present an alternative to the restarted gmres algorithm for solving a single right hand side linear system ax b based on solving the block linear system ax b algorithm performance i e time to solution is improved by using the matrix a in operations on groups of vectors experimental results demonstrate the importance of implementation choices on data movement as well as the effectiveness of the new method on a variety of problems from different application areas

skew symmetric methods for nonsymmetric linear systems with multiple right hand sides by transforming nonsymmetric linear systems to the extended skew symmetric ones we present the skew symmetric methods for solving nonsymmetric linear systems with multiple right hand sides these methods are based on the block and global arnoldi algorithm which is formed by implementing orthogonal projections of the initial matrix residual onto a matrix krylov subspace the algorithms avoid the tediously long arnoldi process and highly reduce expensive storage numerical experiments show that these algorithms are effective and give better practical performances than global gmres for solving nonsymmetric linear systems with multiple right hand sides

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

the block grade of a block krylov space abstract the aim of the paper is to compile and compare basic theoretical facts on krylov subspaces and block krylov subspaces many krylov sub space methods for solving a linear system ax b have the property that in exact computer arithmetic the true solution is found after iterations where is the dimension of the largest krylov subspace generated by a from r 0 the residual of the initial approximation x 0 this dimension is called the grade of r 0 with respect to a though the structure of block krylov subspaces is more complicated than that of ordinary krylov subspaces we introduce here a block grade for which an analogous statement holds when block krylov space methods are applied to linear systems with multiple say s right hand sides in this case the s initial residuals are bundled into a matrix r 0 with s columns the possibility of linear dependence among columns of the block krylov matrix r 0 ar 0 a 1 r 0 which in practical algorithms calls for the deletion or deflation of some columns requires extra care relations between grade and block grade are also established as well as relations to the corresponding notions of a minimal polynomial and its companion matrix

a block idr s method for nonsymmetric linear systems with multiple right hand sides the idr s based on the induced dimension reduction idr theorem is a new class of efficient algorithms for large nonsymmetric linear systems idr 1 is mathematically equivalent to bicgstab at the even idr 1 residuals and idr s with s 1 is competitive with most bi cg based methods for these reasons we extend the idr s to solve large nonsymmetric linear systems with multiple right hand sides in this paper a variant of the idr theorem is given at first then the block idr s an extension of idr s based on the variant idr s theorem is proposed by analysis the upper bound on the number of matrix vector products of block idr s is the same as that of the idr s for a single right hand side in generic case i e the total number of matrix vector products of idr s may be m times that of of block idr s where m is the number of right hand sides numerical experiments are presented to show the effectiveness of our proposed method

on the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds we prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails in particular boltzmann type equations with smoothed soft potentials we compensate the lack of uniform in time estimates by the use of precise logarithmic sobolev type inequalities and the assumption that the initial datum decays rapidly at large velocities our method not only gives explicit results on the times of convergence but is also able to cover situations in which compactness arguments apparently do not apply even mere convergence to equilibrium was an open problem for soft potentials

a new family of global methods for linear systems with multiple right hand sides the global bi conjugate gradient gl bcg method is an attractive matrix krylov subspace method for solving nonsymmetric linear systems with multiple right hand sides but it often show irregular convergence behavior in many applications in this paper we present a new family of global a biorthogonal methods by using short two term recurrences and formal orthogonal polynomials which contain the global bi conjugate residual gl bcr algorithm and its improved version finally numerical experiments illustrate that the proposed methods are highly competitive and often superior to originals

computing the determinants of matrix pade approximation this paper is devoted to a computational problem of two special determinants which appear in the construction of generalized inverse matrix pade approximants of type n 2k for the given power series with matrix coefficients the main tools to be used are well known schur complement theorem and arnoldi process for skew symmetric systems

exact and inexact breakdowns in the block gmres method this paper addresses the issue of breakdowns in the block gmres method for solving linear systems with multiple right hand sides of the form ax b an exact inexact breakdown occurs at iteration j of this method when the block krylov matrix b ab aj 1b is singular almost singular exact breakdowns are the sign that a part of the exact solution is in the range of the krylov matrix they are primarily of theoretical interest from a computational point of view inexact breakdowns are most likely to occur in such cases the underlying block arnoldi process that is used to build the block krylov space should not be continued as usual a natural way to continue the process is the use of deflation however as shown by langou j langou iterative methods for solving linear systems with multiple right hand sides ph d dissertation th pa 03 24 cerfacs france 2003 deflation in block gmres may lead to a loss of information that slows down the convergence in this paper instead of deflating the directions associated with almost converged solutions these are kept and reintroduced in next iterations if necessary two criteria to detect inexact breakdowns are presented one is based on the numerical rank of the generated block krylov basis the second on the numerical rank of the residual associated to approximate solutions these criteria are analyzed and compared implementation details are discussed numerical results are reported

projection methods for large lyapunov matrix equations abstract in the present paper we propose krylov subspace methods for solving large lyapunov matrix equations of the form ax xat bbt 0 where a and b are real n n and n s matrices respectively with s n such problems appear in many areas of control theory such as the computation of hankel singular values model reduction algorithms and others the proposed methods are based on the arnoldi process we show how to extract low rank approximate solutions to lyapunov matrix equations and give some theoretical results finally some numerical tests will be reported to illustrate the effectiveness of the proposed method

the lanczos arnoldi algorithm and controllability we present methods for computing the controllable space of a linear dynamic system the methods are based on orthogonalizing krylov sequences and can use much less storage than other methods with little or no loss of numerical stability

on short recurrence krylov type methods for linear systems with many right hand sides block and global krylov subspace methods have been proposed as methods adapted to the situation where one iteratively solves systems with the same matrix and several right hand sides these methods are advantageous since they allow to cast the major part of the arithmetic in terms of matrix block vector products and since in the block case they take their iterates from a potentially richer subspace in this paper we consider the most established krylov subspace methods which rely on short recurrences i e bicg qmr and bicgstab we propose modifications of their block variants which increase numerical stability thus at least partly curing a problem previously observed by several authors moreover we develop modifications of the global variants which almost halve the number of matrix vector multiplications we present a discussion as well as numerical evidence which both indicate that the additional work present in the block methods can be substantial and that the new economic versions of the global bicg and qmr method can be considered as good alternatives to the bicgstab variants

implementation of an adaptive algorithm for richardson s method abstract we discuss the implementation of an adaptive algorithm proposed by one of us the algorithm is a hybrid of the gmres method and richardson s method richardson s method rm depends on a set of parameters that are computed by minimizing the l 2 norm of a polynomial over the convex hull of eigenvalues execution of gmres yields not only an approximate solution but also the approximate convex hull rm is used to avoid storing and working with a large number of vectors as gmres often requires this method is also advantageous for the solution of large problems we consider several test problems and compare our algorithm primarily with the conjugate gradient squared algorithm but also with gmres and to cg applied to the normal equations for many test problems our algorithm takes roughly 50 percent more work than the conjugate gradient squared algorithm although if the matrix is either preconditioned or indefinite our algorithm is more efficient however our algorithm currently imposes an undesirable burden on the user who is invited to consider a variety of numerical parameters to manipulate such as the number of steps of rm in order to enhance performance the values we suggest are only empirical

learning based fast iterative convergence of 3 d mom via eigen agmres method three dimensional 3 d full wave electromagnetic simulation using method of moments mom under the framework of fast solver algorithms like fast multipole method fmm is often bottlenecked by the speed of convergence of the krylov subspace based iterative process this is primarily because the electric field integral equation efie matrix even with cutting edge preconditioning techniques often exhibits bad spectral properties arising from frequency or geometry based ill conditioning which render iterative solvers slow to converge or stagnate occasionally in this communication a novel technique to expedite the convergence of mom matrix solution at a specific frequency is proposed by extracting and applying eigen vectors from a previously solved neighboring frequency in an augmented generalized minimum residual agmres iterative framework this technique can be applied in unison with any preconditioner numerical results demonstrate up to 40 speed up in convergence using the proposed eigen agmres method

a block gmres method augmented with eigenvectors the gmres method augmented with some eigenvectors is extended to a block version for solving large nonsymmetric linear systems with multiple right hand sides residual bound is established which has an improved convergence rate on the standard block gmres method this improvement displays a good efficiency when restarting is used also an improved residual norm estimate is derived by using matrix valued polynomials numerical experiments show that the new algorithm is more efficient compared with the block gmres method

a new deflated block gcrot m k method for the solution of linear systems with multiple right hand sides linear systems with multiple right hand sides arise in many applications to solve such systems efficiently a new deflated block gcrot m k method is explored in this paper by exploiting a modified block arnoldi deflation incorporating this modified block arnoldi deflation the new algorithm can address the possible linear dependence at each iteration during the block arnoldi procedure and reduces expensive computational operations moreover as a block version of gcrot m k the new method inherits the property of easy operability finally some numerical examples also illustrate the effectiveness of the proposed method

convergence properties of block gmres and matrix polynomials abstract this paper studies convergence properties of the block gmres algorithm when applied to nonsymmetric systems with multiple right hand sides a convergence theory is developed based on a representation of the method using matrix valued polynomials relations between the roots of the residual polynomial for block gmres and the matrix e pseudospectrum are derived and illustrated with numerical experiments the role of invariant subspaces in the effectiveness of block methods is also discussed

generalized global conjugate gradient squared algorithm in this paper a generalized global conjugate gradient squared method for solving nonsymmetric linear systems with multiple right hand sides is presented the method can be derived by using products of two nearby global bicg polynomials and formal orthogonal polynomials of which global cgs and global bicgstab are just particular cases we also show to apply the method for solving the sylvester matrix equation finally numerical examples are given to illustrate the effectiveness of the proposed method

on vlasov manev equations ii local existence and uniqueness we prove that the initial value problem associated with the vlasov manev system a vlasov equation in which a correction of type e r 2 is added to the newtonian or coulomb potential has a local in time classical and unique solution for sufficiently regular initial data

on the block gmres method with deflated restarting abstract in this paper we consider the block gmres method with deflated restarting for solving nonsymmetric linear systems with multiple right hand sides we modify slightly the restarted block gmres method with deflation of eigenvalues proposed by morgan to obtain a new one it is shown that the modified method is mathematically equivalent to the morgan s original method however the error analysis shows the modified version minimizes the numerical errors during a restart and therefore is better suited if the linear systems have to be solved with high precision numerical experiments report the effectiveness of the modified method

stability and uniqueness for the spatially homogeneous boltzmann equation with long range interactions in this paper we prove some a priori stability estimates in weighted sobolev spaces for the spatially homogeneous boltzmann equation without angular cutoff covering all physical collision kernels these estimates are conditional on some regularity estimates on the solutions and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions we then prove such regularity bounds for a class of interactions including the so called non cutoff and non mollified hard potentials and moderately soft potentials in particular we obtain the first result of global existence and uniqueness for these long range interactions