a fast and accurate algorithm for solving bernstein vandermonde linear systems abstract a fast and accurate algorithm for solving a bernstein vandermonde linear system is presented the algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of neville elimination the use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate

a study of the performance of neville elimination using two kinds of partitioning techniques abstract in this paper a method to compute the solution of a system of linear equations by means of neville elimination is described using two kinds of partitioning techniques block and block striped this type of approach is especially suited to the case of totally positive linear systems which is present in different fields of application although neville elimination carried out more floating point operations than gaussian elimination in some cases in this study we confirm that these advantages disappear when we use multiprocessor systems on the other hand the overall parallel run time of neville elimination is better than gauss time as neville elimination uses a lower cost communication model

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

legendre polynomials legendre stirling numbers and the left definite spectral analysis of the legendre differential expression in this paper we develop the left definite spectral theory associated with the self adjoint operator a k in l2 1 1 generated from the classical second order legendre differential equation ll k y t 1 t2 y ky y t 1 1 that has the legendre polynomials pm t m 0 as eigenfunctions here k is a fixed nonnegative constant more specifically for k 0 we explicitly determine the unique left definite hilbert sobolev space wn k and its associated inner product n k for each n n moreover for each n n we determine the corresponding unique left definite self adjoint operator an k in wn k and characterize its domain in terms of another left definite space the key to determining these spaces and inner products is in finding the explicit lagrangian symmetric form of the integral composite powers of ll k in turn the key to determining these powers is a remarkable new identity involving a double sequence of numbers which we call legendre stirling numbers

decompositions of strictly sign regular matrices abstract a matrix is called strictly sign regular if for each k all its k k minors are nonzero and have the same sign we study several decompositions of strictly sign regular matrices including qr decomposition schur decomposition svd and symmetric triangular decomposition several characterizations are provided

full rank factorization in echelon form of totally nonpositive negative rectangular matrices abstract an n m real matrix a is said to be totally nonpositive negative if every minor is nonpositive negative in this paper we study the full rank factorization in echelon form of a totally nonpositive negative matrix this characterization allows us to significantly reduce the number of minors to be checked in order to decide the total negativity of a matrix

two step runge kutta methods and hyperbolic partial differential equations the purpose of this study is the design of efficient methods for the solution of an ordinary differential system of equations arising from the semidiscretization of a hyperbolic partial differential equation jameson recently introduced the use of one step runge kutta methods for the numerical solution of the euler equations improvements in efficiency up to 80 may be achieved by using two step runge kutta methods instead of the classical onestep methods these two step runge kutta methods were first introduced by byrne and lambert in 1966 they are designed to have the same number of function evaluations as the equivalent one step schemes and thus they are potentially more efficient by solving a nonlinear programming problem which is specified by stability requirements optimal two step schemes are designed the optimization technique is applicable for stability regions of any shape

neville elimination for rank structured matrices abstract in this paper it is shown that neville elimination is suited to exploit the rank structure of an order r quasiseparable matrix a c n n by providing a condensed decomposition of a as product of unit bidiagonal matrices all together specified by o nr parameters at the cost of o nr 3 flops an application of this result for eigenvalue computation of totally positive rank structured matrices is also presented

explicit factorization of the vandermonde matrix the lu factorization of the vandermonde matrix is obtained using complete symmetric functions and the lower and upper triangular matrices are in turn factorized into 1 banded matrices thus expressing the vandermonde matrix as a product of 1 banded matrices

fast and accurate algorithms for jacobi stirling matrices abstract jacobi stirling numbers of first and second kind and their corresponding associated matrices play an important role in combinatorics this paper presents an efficient method that guarantees that the computation of the singular values and the inverses of these matrices can be performed with high relative accuracy numerical examples are included

corner cutting algorithms for the bezier representation of free form curves abstract we give a general treatment of the problem of converting a curve expressed in a b spline expansion into its bezier form we show that corner cutting of the b spline control polygon leads to the bezier points exactly when the bezier matrix is totally positive also for spline spaces used to model space curves with curvature continuity we give an algorithm for determining the bezier representation

characterizations and stable tests for the routh hurwitz conditions and for total positivity given a polynomial of degree n a test of o n2 elementary operations and growth factor 1 is presented in order to check the routh hurwitz conditions this optimal growth factor guarantees that the test presents better stability properties than other known tests we also present a test of o n3 elementary operations and growth factor 1 in order to check if a matrix is strictly totally positive finally totally positive matrices are characterized by their symmetric triangular decompositions

accurate bidiagonal decomposition of totally positive cauchy vandermonde matrices and applications abstract cauchy vandermonde matrices play a fundamental role in rational interpolation theory and in other fields when all their corresponding nodes are different and positive and all poles are different and negative and follow adequate orderings these matrices are totally positive in this paper we provide fast algorithms for computing bidiagonal factorizations of these matrices and their inverses with high relative accuracy these algorithms can be used to solve with high relative accuracy other algebraic problems such as the computation of all singular values all eigenvalues or the solution of certain linear systems the error analysis of the algorithm for computing the bidiagonal factorization and the corresponding perturbation theory are also performed

growth factors of pivoting strategies associated with neville elimination neville elimination is a direct method for solving linear systems several pivoting strategies for neville elimination including pairwise pivoting are analyzed bounds for two different kinds of growth factors are provided finally an approximation of the average normalized growth factor associated with several pivoting strategies is computed and analyzed using random matrices

accurate computations with collocation matrices of rational bases the rational bernstein basis plays a crucial role in computer aided geometric design the collocation matrices of this basis are called rational bernstein vandermonde matrices in this paper we provide algorithms for computing the bidiagonal decomposition of rational bernstein vandermonde matrices and their inverses with high relative accuracy similar results are obtained for the collocation matrices of another important rational basis the rational said ball basis it is also shown that these algorithms can be used to perform accurately some computations with these matrices such us the calculation of their inverses their eigenvalues or their singular values numerical experiments illustrate the results

a supersymmetric extension of the kadomtsev petviashvili hierarchy an extension of the kadomtsev petviashvili hierarchy by odd variables is given conservation laws and formal integrability are proved

accurate bidiagonal decomposition of totally positive h bernstein vandermonde matrices and applications abstract an algorithm for the computation of the bidiagonal decomposition of strictly totally positive h bernstein vandermonde matrices is presented these matrices are collocation matrices of h bernstein bases a generalization of the bernstein basis of the space of polynomials of degree less than or equal to n the algorithm is fast its computational complexity is o n 2 and has high relative accuracy the computation of this bidiagonal decomposition is used as the initial stage for the accurate and efficient solution of several linear algebra problems with these matrices linear system solving eigenvalue computation and inverse computation numerical experiments are also included

on the structure of l algebra l cvir the scenery of the theory of groups their representations and homogeneous spaces in view of the intensive interaction with the mathematical physics deeply depends on changes in it thus formation of the modern quantum field theory string field theory conformal field theory etc provoked the elaboration of a new block including the infinite dimensional geometry harmonic analysis on infinite dimensional manifolds and the theory of representations of the infinite dimensional algebras and groups at the same time methods of the quantum field theory penetrated into abstract mathematics topological quantum field theory l 23 there exist some arguments for the hypothesis that the conformal quantum field theory and its generalisations 3 91 would be useful in the singularity theory one should suppose that the version of the operator formalism of the quantum field theory based on the local field algebras would play the role analogous to one of the coliomology theory in the topology all these circumstances explain the appearance of the papers io 1 l devoted to the algebraic formulation of the described version based on the infinite dimensional geometry in these papers we constructed the l algebra l cvir in which the model of the verma modules over the virasoro algebra is realised the local held algebras are constructed from the following data

elimination technique from extrapolation to totally positive matrices and cagd in this survey we will show some connections between several mathematical problems such as extrapolation linear systems totally positive matrices and computer aided geometric design with elimination techniques as the common tool to deal with all of them

factorizations of cauchy vandermonde matrices abstract this paper analyzes the factorization of the inverse of a cauchy vandermonde matrix as a product of bidiagonal matrices factorizations of cauchy vandermonde matrices are also considered the results are applied to derive a fast algorithm for solving linear systems whose coefficient matrices are cauchy vandermonde matrices

generalization of the algebra of the matrix pseudodifferential operators we introduce the new nonhomogeneous graded matrix algebra matn r over the ring r of the pseudodifferential operators of the arbitrary orders this algebra forms the generalization of the algebra of the standard matrix pseudodifferential operators we show that the matn r algebra admits the nontrivial central extension the case n 2 is considered in some detail as the example we consider the mat2 w1 mat2 virasoro and mat2 sl 2 c algebras

running relative error for the evaluation of polynomials running relative error for evaluating polynomials is analyzed and applied to different evaluation algorithms horner vs de casteljau and clenshaw algorithms numerical experiments show that the derived running relative error bounds are sharp and that the de casteljau algorithm presents better stability properties than the remaining algorithms

natural continuous extensions of runge kutta methods the present paper develops a theory of natural continuous extensions nces for the discrete approximate solution of an ode given by a runge kutta process these nces are defined in such a way that the continuous solutions furnished by the one step collocation methods are included

two step runge kutta methods implicit two step runge kutta methods are studied it will be shown that these methods require fewer stages to achieve the same order as one step runge kutta methods which means the two step methods are potentially more efficient than one step methods order conditions are derived and examples of two step one stage methods of order 2 and two step two stage methods of order 4 are presented stability properties of these methods with respect to y ay are studied and a stable two step methods of order 2 are characterized two step two stage methods of order 4 which are a stable are found by an extensive computer search semi implicit two stage methods of order 4 were also constructed this is in contrast to the situation encountered in the runge kutta theory where the unique two stage method of order 4 is not semi implicit

order barriers for continuous explicit runge kutta methods in this paper we deal with continuous numerical methods for solving initial value problems for ordinary differential equations the need for which occurs frequently in applications whereas most of the commonly used multistep methods provide continuous extensions by means of an interpolant which is available without making extra function evaluations this is not always the case for one step methods we consider the class of explicit runge kutta methods and provide theorems used to obtain lower bounds for the number of stages required to construct methods of a given unifonn order p these bounds are similar to the butcher barriers known for the discrete case and are derived up to order p 5 as far as we know the examples we present of 8 stage continuous runge kutta methods of uniform order 5 are the first of their kind

stability analysis of two step runge kutta methods for delay differential equations abstract we investigate stability properties of two step runge kutta methods with respect to the linear test equation y t ay t by t t 0 y t g t t 0 where a and b are complex parameters it is known that the solution y t to this equation tends to zero as t if b a we will show that under some conditions this property is inherited by any a stable two step runge kutta method applied on a constrained mesh to delay differential equations i e that the corresponding method is p stable

accurate computations with said ball vandermonde matrices abstract a generalization of the vandermonde matrices which arise when the power basis is replaced by the said ball basis is considered when the nodes are inside the interval 0 1 then those matrices are strictly totally positive an algorithm for computing the bidiagonal decomposition of those said ball vandermonde matrices is presented which allows us to use known algorithms for totally positive matrices represented by their bidiagonal decomposition the algorithm is shown to be fast and to guarantee high relative accuracy some numerical experiments which illustrate the good behaviour of the algorithm are included

the left definite spectral theory for the classical hermite differential equation abstract in this paper we develop the left definite spectral theory associated with the self adjoint operator a in l2 exp t2 generated from the classic second order hermite differential equation l h y t y 2ty ky y t that has the hermite polynomials hm t m 0 as eigenfunctions more specifically for each n n we explicitly determine the unique left definite hilbert sobolev space wn and associated inner product n which is generated from the nth integral power lhn of lh moreover for each n n we determine the corresponding unique left definite self adjoint operator an in wn and characterize its domain in terms of another left definite space as a consequence of this we explicitly determine the domain of each integral power of a and in particular we obtain a new characterization of the domain of the classical right definite operator a

a test and bidiagonal factorization for certain sign regular matrices abstract it is well known that a totally nonnegative matrix can be characterized in terms of its bidiagonal factorization which is critical to perform accurate computations however there is no known such factorization of other sign regular matrices in this paper we provide a characterization and test for certain sign regular matrices with the same signature sequence as that of nonsingular jacobi sign regular matrices consequently a bidiagonal factorization is derived for such sign regular matrix which in turn provides an effective way to generate these sign regular matrices

total positivity and neville elimination abstract neville elimination is described in terms of schur complements of matrices and used to improve some well known characterizations of totally positive and strictly totally positive matrices

integrability of the frobenius algebra valued kadomtsev petviashvili hierarchy we introduce a frobenius algebra valued kadomtsev petviashvili kp hierarchy and show the existence of frobenius algebra valued function for this hierarchy in addition we construct its hamiltonian structures by using the adler dickey gelfand method as a byproduct of these constructions we show that the coupled kp hierarchy defined by casati and ortenzi j geom phys 56 418 449 2006 has at least n basic different local bi hamiltonian structures finally via the construction of the second hamiltonian structures we obtain some local matrix or frobenius algebra valued generalizations of classical w algebras

accurate computations with lupa matrices lupa q analogues of the bernstein functions play an important role in approximation theory and computer aided geometric design their collocation matrices are called lupa matrices in this paper we provide algorithms for computing the bidiagonal decomposition of these matrices and their inverses to high relative accuracy it is also shown that these algorithms can be used to perform to high relative accuracy several algebraic calculations with these matrices such as the calculation of their inverses their eigenvalues or their singular values numerical experiments are included

n 2 super w algebras and generalized n 2 super kdv heirarchies based on lie superalgebras the authors propose a hierarchy of super lax equations based on an affine lie superalgebra a n n 1 as the generalized n 2 super kdv hierarchy and show that the associated super gel fand dikii bracket defines an n 2 super w algebra for a 2 2 1 they obtain the n 2 super boussiensque equation and the n 2 super w2 algebra containing additional currents of spins 2 5 2 3

on nonsingular sign regular matrices abstract an m n matrix a is sign regular if for each k 1 k min m n all k k submatrices of a have determinant with the same nonstrict sign the zero pattern of nonsingular sign regular matrices is analyzed it is proved that the number of zero entries which can appear in a nonsingular sign regular matrix depends on its signature a matrix is totally nonpositive if all its minors are nonpositive a test for recognizing nonsingular totally nonpositive matrices is also provided

rank structure properties of rectangular matrices admitting bidiagonal type factorizations abstract in this paper we investigate the class of rectangular matrices that admit bidiagonal type factorizations by neville elimination without exchanges we provide a complete characterization for a rectangular matrix to be factored as a product of bidiagonal factors and a banded factor in terms of rank structure properties consequently we give a complete characterization of the class of rectangular matrices that have neville factorizations

bidiagonal decompositions of oscillating systems of vectors abstract we establish necessary and sufficient conditions in the language of bidiagonal decompositions for a matrix v to be an eigenvector matrix of a totally positive matrix namely this is the case if and only if v and v t are lowerly totally positive these conditions translate into easy positivity requirements on the parameters in the bidiagonal decompositions of v and v t using these decompositions we give elementary proofs of the oscillating properties of v in particular the fact that the jth column of v has j 1 changes of sign our new results include the fact that the q matrix in a qr decomposition of a totally positive matrix belongs to the above class and thus has the same oscillating properties

full rank factorization in quasi ldu form of totally nonpositive rectangular matrices abstract let a a i j r n m be a totally nonpositive matrix with rank a r min n m and a 11 0 in this paper we obtain a characterization in terms of the full rank factorization in quasi ldu form that is a l d u where l r n r is a block lower echelon matrix u r r m is a unit upper echelon totally positive matrix and d r r r is a diagonal matrix with rank l rank u rank d r we use this quasi ldu decomposition to construct the quasi bidiagonal factorization of a moreover some properties about these matrices are studied

non local matrix generalizations of w algebras there is a standard way to define two symplectic hamiltonian structures the first and second gelfand dikii brackets on the space of ordinarymth order linear differential opeatorsl d m u1dm 1 u2dm 2 u m in this paper i consider in detail the case where theu k aren n matrix valued functions with particular emphasis on the more interesting second gelfand dikii bracket of particular interest is the reduction to the symplectic submanifoldu1 0 this reduction gives rise to matrix generalizations of the classical version of thenon linear w m algebras calledv n m algebras the non commutativity of the matrices leads tonon local terms in thesev n m algebra s i show that these algebras contain a conformal virasoro subalgebra and that combinationsw k of theu k can be formed that aren n matrices of conformally primary fields of spink in analogy with the scalar casen 1 in general however thev m n algebras have a much richer structure than thew m algebras as can be seen on the examples of thenon linear andnon local poisson brackets u2 ab u2 cd u2 ab w3 cd and w3 ab w3 cd which i work out explicitly for allm andn a matrix miura transformations is derived mapping these complicated second gelfand dikii brackets of theu k to a set of much simpler poisson brackets providing the analogoue of the free field representation of thew m algebras

a general left definite theory for certain self adjoint operators with applications to differential equations we show that any self adjoint operator a bounded or unbounded in a hilbert space h v that is bounded below generates a continuum of hilbert spaces hr r 0 and a continuum of self adjoint operators ar r 0 for reasons originating in the theory of differential operators we call each hr the rth left definite space and each ar the rth left definite operator associated with h a each space hr can be seen as the closure of the domain d ar of the self adjoint operator ar in the topology generated from the inner product arx y x y d ar furthermore each ar is a unique self adjoint restriction of a in hr we show that the spectrum of each ar agrees with the spectrum of a and the domain of each ar is characterized in terms of another left definite space the hilbert space spectral theorem plays a fundamental role in these constructions we apply these results to two examples including the classical laguerre differential expression l in which we explicitly find the left definite spaces and left definite operators associated with a the self adjoint operator generated by l in l2 0 t e t having the laguerre polynomials as eigenfunctions

explicit two step peer methods we present a new class of explicit two step peer methods for the solution of nonstiff differential systems a construction principle for methods of order p s s the number of stages with optimal zero stability is given two methods of order p 6 found by numerical search are tested in matlab on several representative nonstiff problems the comparison with ode45 confirms the high potential of the new class of methods

bernstein bezoutian matrices and curve implicitization a new application of bernstein bezoutian matrices a type of resultant matrices constructed when the polynomials are given in the bernstein basis is presented in particular the approach to curve implicitization through sylvester and bezout resultant matrices and bivariate interpolation in the usual power basis is extended to the case in which the polynomials appearing in the rational parametric equations of the curve are expressed in the bernstein basis avoiding the basis conversion from the bernstein to the power basis the coefficients of the implicit equation are computed in the bivariate tensor product bernstein basis and their computation involves the bidiagonal factorization of the inverses of certain totally positive matrices

polynomial least squares fitting in the bernstein basis the problem of polynomial least squares fitting in which the usual monomial basis is replaced by the bernstein basis is considered the coefficient matrix of the overdetermined system to be solved in the least squares sense is then a rectangular bernstein vandermonde matrix in order to use the method based on the qr decomposition of a the first stage consists of computing the bidiagonal decomposition of the coefficient matrix a starting from that bidiagonal decomposition an algorithm for obtaining the qr decomposition of a is the applied finally a triangular system is solved by using the bidiagonal decomposition of the r factor of a some numerical experiments showing the behavior of this approach are included

a test for strict sign regularity a characterization of strict sign regular matrices is obtained it is given in an algorithmic form which allows the easy construction of a test to check if a matrix is strictly sign regular the test is based on the neville elimination of several submatrices of a and requires less computational work than those which can be derived from other previously known characterizations of these matrices

accurate computation of the moore penrose inverse of strictly totally positive matrices abstract the computation of the moore penrose inverse of structured strictly totally positive matrices is addressed since these matrices are usually very ill conditioned standard algorithms fail to provide accurate results an algorithm based on the q r factorization and which takes advantage of the special structure and the totally positive character of these matrices is presented the first stage of the algorithm consists of the accurate computation of the bidiagonal decomposition of the matrix numerical experiments illustrating the good behavior of our approach are included

order conditions for general linear methods we describe the derivation of order conditions without restrictions on stage order for general linear methods for ordinary differential equations this derivation is based on the extension of the albrecht approach proposed in the context of runge kutta and composite and linear cyclic methods this approach was generalized by jackiewicz and tracogna to two step runge kutta methods by jackiewicz and vermiglio to general linear methods with external stages of different orders and by garrappa to some classes of runge kutta methods for volterra integral equations with weakly singular kernels this leads to general order conditions for many special cases of general linear methods such as diagonally implicit multistage integration methods nordsieck methods and general linear methods with inherent runge kutta stability exact coefficients for several low order methods with some desirable stability properties are presented for illustration

on the classical algebra we consider the classical algebra from the integrable system viewpoint the integrable evolution equations associated with the algebra are constructed and the miura maps and consequent modifications are presented modifying the miura maps we supply a free field realization for the classical algebra we also construct the corresponding toda type integrable systems

consecutive column and row properties of matrices and the loewner neville factorization abstract we say that a square matrix over a ring with identity has the consecutive column property if for all k all its relevant submatrices having k consecutive rows and the first k columns are invertible similarly the consecutive row cr is defined we show that analogously to the commutative case for totally positive matrices a matrix has both cc and cr properties if and only if it admits a certain loewner neville type factorization with invertible entries this factorization is unique since the result is proved for matrices in such generality it holds also for block matrices over a field with all blocks square explicit both ways formulae are found between two sets of parameters the loewner neville coefficients in the factorization and the schur complements of relevant submatrices in relevant submatrices larger by one we also show that for lower triangular matrices the cc property is preserved by inversion

a plu factorization of rectangular matrices by the neville elimination in this paper we prove that neville elimination can be matricially described by elementary matrices a plu factorization is obtained for any n m matrix where p is a permutation matrix l is a lower triangular matrix product of bidiagonal factors and u is an upper triangular matrix this result generalizes the neville factorization usually applied to characterize the totally positive matrices we prove that this elimination procedure is an alternative to gaussian elimination and sometimes provides a lower computational cost

jacobi stirling numbers jacobi polynomials and the left definite analysis of the classical jacobi differential expression we develop the left definite analysis associated with the self adjoint jacobi operator generated from the classical second order jacobi differential expression in the hilbert space where w t 1 t 1 t that has the jacobi polynomials as eigenfunctions here 1 and k is a fixed non negative constant more specifically for each we explicitly determine the unique left definite hilbert sobolev space and the corresponding unique left definite self adjoint operator in associated with the pair the jacobi polynomials form a complete orthogonal set in each left definite space and are the eigenfunctions of each moreover in this paper we explicitly determine the domain of each as well as each integral power of the key to determining these spaces and operators is in finding the explicit lagrangian symmetric form of the integral composite powers of l k in turn the key to determining these powers is a double sequence of numbers which we introduce in this paper as the jacobi stirling numbers some properties of these numbers which in some ways behave like the classical stirling numbers of the second kind are established including a remarkable and yet somewhat mysterious identity involving these numbers and the eigenvalues of

accurate solutions of product linear systems associated with rank structured matrices abstract in this paper we consider how to accurately solve linear systems associated with a wide class of rank structured matrices containing the well known vandermonde and cauchy matrices i e consecutive rank descending crd matrices we provide a mechanism to guarantee that the inverse of any product of crd matrices is generated in a subtraction free manner with the mechanism the solutions of linear systems associated with such products are accurately determined by the parameters of crd factors and we then accurately compute the solutions as warranted by these parameters in particular linear systems associated with products of vandermonde and cauchy matrices whose nodes satisfy certain positive or negative properties are solved to high relative accuracy error analysis and numerical experiments are provided to confirm the high accuracy

higher spin fields and the gelfand dickey algebra we show that in 2 dimensional field theory higher spin algebras are contained in the algebra of formal pseudodifferential operators introduced by gelfand and dickey to describe integrable nonlinear differential equations in lax form the spin 2 and 3 algebras are discussed in detail and the generalization to all higher spins is outlined this provides a conformal field theory approach to the representation theory of gelfand dickey algebras

on the gelfand dickey algebra gd sln and the wn symmetry i the bosonic case the algebra of nonlinear local and nonlocal differential operators acting on the ring of analytic functions is studied it is shown in particular that this space splits into 3 2 special subalgebras jr j 0 1 r 1 each subalgebra is completely specified by quantum numbers s and p q describing the conformal spin and the lowest and the highest degrees respectively the algebra and its dual of local pure nonlocal differential operators is used to calculate the general expression of the gelfand dickey bracket and the wn symmetry poisson one in terms of a set of spin j canonical fields uj 2 j n and a nonlinear u cubic dependent differential operator d n i j z u the explicit form of this operator is worked out other remarkable features are also discussed

a general class of two step runge kutta methods for ordinary differential equations a general class of two step runge kutta methods that depend on stage values at two consecutive steps is studied these methods are special cases of general linear methods introduced by butcher and are quite efficient with respect to the number of function evaluations required for a given order general order conditions are derived using the approach proposed recently by albrecht and examples of methods are given up to the order 5 these methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments

classical lattice w algebras and integrable systems several aspects of the lattice algebra are studied motivated by the fact that the lotka volterra model can be written in terms of a current of the lattice virasoro algebra the faddeev takhtajan volkov algebra integrable dynamical models on the lattice have been formulated as a model associated with the lattice algebra

componentwise backward error analysis of neville elimination abstract in this paper we perform a backward error analysis of neville elimination whenever there need row exchanges componentwise backward error bounds are presented for this elimination procedure applied to any nonsingular matrices consequently it is shown that neville elimination with two determinant pivoting proposed by cortes and pena 2007 5 is an excellent method for the triangularization of nonsingular sign regular matrices including totally nonnegative and totally nonpositive matrices which has a pleasantly small componentwise relative backward error in particular a small componentwise relative error bound is also provided for the bidiagonal factorization of totally nonnegative matrices

a collection of examples where neville elimination outperforms gaussian elimination neville elimination is an elimination procedure alternative to gaussian elimination it is very useful when dealing with totally positive matrices for which nice stability results are known here we include examples most of them test matrices used in matlab which are not totally positive matrices where neville elimination outperforms gaussian elimination

quantization and representation theory of finite w algebras in this paper we study the finitely generated algebras underlyingw algebras these so called finitew algebras are constructed as poisson reductions of kirillov poisson structures on simple lie algebras the inequivalent reductions are labeled by the inequivalent embeddings ofsl 2 into the simple lie algebra in question for arbitrary embeddings a coordinate free formula for the reduced poisson structure is derived we also prove that any finitew algebra can be embedded into the kirillov poisson algebra of a semi simple lie algebra generalized miura map furthermore it is shown that generalized finite toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finitew symmetry in the second part we brst quantize the finitew algebras the brst cohomology is calculated using a spectral sequence which is different from the one used by feigin and frenkel this allows us to quantize all finitew algebras in one stroke examples are given in the last part of the paper we study the representation theory of finitew algebras it is shown using a quantum version of the generalized miura transformation that the representations of finitew algebras can be constructed from the representations of a certain lie subalgebra of the original simple lie algebra as a byproduct of this we are able to construct the fock realizations of arbitrary finitew algebras

factorizations of confluent cauchy vandermonde matrices abstract the confluent cauchy vandermonde matrices are considered which were studied earlier by various authors in different way in this paper we analyze the factorization of the inverse of a special type of confluent cauchy vandermonde matrix as a product of block bidiagonal matrices by using block neville elimination factorizations of confluent cauchy vandermonde matrices are also considered which generalize the results of j j martinez j m pena factorizations of cauchy vandermonde matrices linear algebra appl 284 1998 229 237

neville elimination a study of the efficiency using checkerboard partitioning abstract it is well known that checkerboard partitioning can exploit more concurrency than striped partitioning because the matrix computation can be divided among more processors than in the case of striping in this work we analyze the performance of neville method when a checkerboard partitioning is used focusing on the special case of block cyclic checkerboard partitioning this method is an alternative to gaussian elimination and it has been proved to be very useful for some classes of matrices such as totally positive matrices the performance of this parallel system is measured in terms of the efficiency the fraction of time for which a processor is usefully employed which in our model is close to one when the optimum block size is used also we have executed our algorithms on a parallel pc cluster observing that both efficiencies theoretical and empirical are quite similar

a matricial description of neville elimination with applications to total positivity abstract the neville elimination process used by the authors in some previous papers in connection with totally positive matrices is studied in detail in the case of nonsingular matrices a wide class of matrices is found where neville elimination has a lower computational cost than gauss elimination finally some new characterizations are obtained for strictly totally positive and nonsingular totally positive matrices in terms of their neville elimination and that of their inverses

the relation between quantum w algebras and lie algebras by quantizing the generalized drinfeld sokolov reduction scheme for arbitrarysl2 embeddings we show that a large set open image in new window of quantumw algebras can be viewed as brst cohomologies of affine lie algebras the set open image in new window contains many knownw algebras such asw n andw 3 2 our formalism yields a completely algorithmic method for calculating thew algebra generators and their operator product expansions replacing the cumbersome construction ofw algebras as commutants of screening operators by generalizing and quantizing the miura transformation we show that anyw algebra in open image in new window can be embedded into the universal enveloping algebra of a semisimple affine lie algebra which is up to shifts in level isomorphic to a subalgebra of the original affine algebra thereforeany realization of this semisimple affine lie algebra leads to a realization of thew algebra in particular one obtains in this way a general and explicit method for constructing the free field realizations and fock resolusions for all algebras in open image in new window some examples are explicitly worked out

bidiagonal decomposition of rectangular totally positive said ball vandermonde matrices error analysis perturbation theory and applications abstract an algorithm for computing the bidiagonal decomposition of said ball vandermonde matrices in the rectangular case is presented which allows one to use known algorithms for totally positive matrices represented by their bidiagonal decomposition the algorithm is fast and possesses high relative accuracy the error analysis of the algorithm is also addressed along with the perturbation theory for the bidiagonal factorization of totally positive said ball vandermonde matrices some numerical experiments showing the good behavior of the algorithm are included