lower bounds for the condition number of vandermonde matrices we derive lower bounds for the condition number of then n vandermonde matrixvn x in the cases where the node vectorxt x1 x2 xn has positive elements or real elements located symmetrically with respect to the origin the bounds obtained grow exponentially inn witho 2n ando 2n 2 respectively we also compute the optimal spectral condition numbers ofvn x for the two node configurations including the optimal nodes and compare them with the bounds obtained

transformations of matrix structures work again abstract matrices with the structures of toeplitz hankel vandermonde and cauchy types are omnipresent in modern computations in sciences engineering and signal and image processing these four matrix classes have distinct features but in our 1990 paper in mathematics of computation we showed that vandermonde and hankel multipliers can be applied to transform each class into the others and then we demonstrated how by using these transforms we can readily extend any successful matrix inversion algorithm from one of these classes to all the others the power of this approach was widely recognized later when novel numerically stable algorithms solved nonsingular toeplitz linear systems of equations in quadratic versus classical cubic arithmetic time based on transforming toeplitz into cauchy matrix structures more recent papers combined the same transformation with a link of the cauchy matrices to the hierarchical semiseparable matrix structure which is a specialization of matrix representations employed by the fast multipole method this produced numerically stable algorithms that approximated the solution of a nonsingular toeplitz linear system of equations in nearly linear arithmetic time we first revisit the successful method of structure transformation covering it comprehensively then we analyze the latter approximation algorithms for toeplitz linear systems and extend them to approximate multiplication of vandermonde and cauchy matrices by a vector and to approximate solution of vandermonde and cauchy linear systems of equations provided that they are nonsingular and well conditioned we decrease the arithmetic cost of the known numerical approximation algorithms for these tasks from quadratic to nearly linear and similarly for the computations with the matrices having structures that generalize the structures of vandermonde and cauchy matrices and for polynomial and rational evaluation and interpolation we also accelerate a little further the known numerical approximation algorithms for a nonsingular toeplitz or toeplitz like linear system by employing distinct transformations of matrix structures and we briefly comment on some natural research challenges particularly some promising applications of our techniques to high precision computations

new studies of randomized augmentation and additive preprocessing abstract a standard gaussian random matrix hereafter referred to just as gaussian matrix has full rank with probability 1 and is well conditioned with a probability quite close to 1 and converging to 1 fast as the matrix deviates from the square shape and becomes more rectangular if we append sufficiently many gaussian rows or columns to any normalized and possibly rank deficient or ill conditioned matrix then the augmented matrix has full rank with probability 1 and is well conditioned with a probability close to 1 we specify and prove these properties of augmentation and extend them to additive preprocessing that is to adding a product of two rectangular gaussian matrices by applying our randomization techniques to a matrix that has numerical rank r we accelerate the known algorithms for the approximation of its trailing singular spaces associated with all its positive singular values except for the r largest values our algorithms use much fewer random parameters and run much faster when various random sparse and structured preprocessors replace gaussian empirically the outputs of the resulting algorithms are as accurate as the outputs under gaussian preprocessing our novel duality techniques provides formal support so far missing for these empirical observations and opens door to de randomization of our preprocessing and to further acceleration and simplification of our algorithms by using more efficient sparse and structured preprocessors our techniques and our progress can be applied to various other fundamental matrix computations such as the celebrated low rank approximation of a matrix by means of random sampling

random multipliers numerically stabilize gaussian and block gaussian elimination proofs and an extension to low rank approximation abstract we study two applications of standard gaussian random multipliers at first we prove that with a probability close to 1 such a multiplier is expected to numerically stabilize gaussian elimination with no pivoting as well as block gaussian elimination then by extending our analysis we prove that such a multiplier is also expected to support low rank approximation of a matrix without customary oversampling our test results are in good accordance with this formal study the results remain similar when we replace gaussian multipliers with random circulant or toeplitz multipliers which involve fewer random parameters and enable faster multiplication we formally support the observed efficiency of random structured multipliers applied to approximation but we still continue our research in the case of elimination we specify a narrow class of unitary inputs for which gaussian elimination with no pivoting is numerically unstable and then prove that with a probability close to 1 a gaussian random circulant multiplier does not fix numerical stability problems for such inputs we also prove that the power of the random circulant preprocessing increases if we also include random permutations

graded artin algebras a graded artin algebra is a graded ring which neglecting the grading is an artin algebra although this definition has the merit of brevity it does not expose some of the basic properties of graded artin algebras for instance that there is a bound on absolute values of degrees of homogeneous elements indeed a different definition is given in section 1 and the discrepancy is rectified by theory 1 4 we admit that our chief interest in graded artin algebras lies in applications to the representation theory of artin algebras see 5 1 we feel nonetheless that graded artin algebras are interesting objects of study in their own right and some of our techniques especially in sections 5 and 6 are equally applicable to arbitrary graded rings thus this paper unlike its sequel 51 is designed to be read by a general ring theoretical audience it is in addition self contained from the standpoint of knowledge concerning graded rings required the major concern of the paper is the existence of gradings on given modules over a graded artin algebra we use the term gradable when a gradation exists our transcendent result along these lines is that direct summands of finitely generated gradable modules are gradable it follows immediately see section 3 that simple modules and projective modules over a graded artin algebra are gradable also inasmuch as we do not restrict ourselves to positively graded rings endomorphism rings of graded finitely generated modules over graded artin algebras are graded artin algebras in a similar vein duality and horn

real polynomial root finding by means of matrix and polynomial iterations frequently one seeks approximation to all r real roots of a polynomial of degree n with real coefficients which also has nonreal roots we split a polynomial into two factors one of which has degree r and has r real roots we approximate them at a low cost and then decrease the arithmetic time of the known algorithms for this popular problem by roughly a factor of n k if k iterations prepare splitting k is a small integer unless some nonreal roots lie close to the real axis but even if there nonreal roots near the real axis we substantially accelerate the known algorithms we also propose a dual algorithm operating with the associated structured matrices at the price of minor increase of the arithmetic time it facilitates numerical implementation our analysis and tests demonstrate the efficiency of our approach

the tame infinitesimal groups of odd characteristic abstract given an infinitesimal group g over an algebraically closed field k of characteristic p 3 we provide criteria for the principal block b 0 g of its algebra of distributions to be of tame representation type these are employed in conjunction with galois coverings to determine the structure of g modulo its multiplicative center as well as the quiver and the relations of the algebra b 0 g

a modification of the dewilde van der veen method for inversion of finite structured matrices abstract we study a class of block structured matrices r r ij i j 1 n with a property that the solution of the corresponding system rx y of linear algebraic equations may be performed for o n arithmetic operations in this paper for finite invertible matrices we analyze in detail factorization and inversion algorithms these algorithms are related to those suggested by p m dewilde and a j van der veen time varying systems and computations kluwer academic publishers new york 1998 for a class of finite and infinite matrices with a small hankel rank the algorithms presented here are more transparent and are a modification of the algorithms from the above reference the approach and the proofs are essentially different from those in the above mentioned reference the paper contains also analysis of complexity and results of numerical experiments

a companion matrix resultant for bernstein polynomials abstract a closed form expression for a companion matrix m of a bernstein polynomial is obtained and this is used to derive an expression for a resultant matrix of two bernstein polynomials it is shown that m differs from its equivalent form for a power basis polynomial because an upper triangular hankel matrix does not define a similarity transformation between m and m t a measure of the numerical condition of a resultant matrix for polynomials in an arbitrary basis is reviewed and this is used to compare the stability of two resultant matrices in particular computational tests are performed and it is shown that the resultant matrix of two bernstein polynomials is numerically better conditioned than the resultant matrix that is obtained when a simple parameter substitution is used to transform the polynomials to the power basis

on cocommutative hopf algebras of finite representation type abstract let g be a finite algebraic group defined over an algebraically closed field k of characteristic p 0 such a group decomposes into a semidirect product g g 0 g red with a constant group g red and a normal infinitesimal subgroup g 0 if the principal block b 0 g of the group algebra h g has finite representation type then both constituents have the same property with at least one of them being semisimple we determine the structure of the infinitesimal constituent g 0 up to the classification of v uniserial groups

fast gaussian elimination with partial pivoting for matrices with displacement structure fast o n 2 implementation of gaussian elimination with partial pivoting is designed for matrices possessing cauchy like displacement structure we show how toeplitz like toeplitz plus hankel like and vandermonde like matrices can be transformed into cauchy like matrices by using discrete fourier cosine or sine transform matrices in particular this allows us to propose a new fast o n 2 toeplitz solver gko which incorporates partial pivoting a large set of numerical examples showed that gko demonstrated stable numerical behavior and can be recommended for solving linear systems especially with nonsymmetric indefinite and ill conditioned positive definite toeplitz matrices it is also useful for block toeplitz and mosaic toeplitz toeplitz block matrices the algorithms proposed in this paper suggest an attractive alternative to look ahead approaches where one has to jump over ill conditioned leading submatrices which in the worst case requires o n 3 operations

computations with quasiseparable polynomials and matrices in this paper we survey several recent results that highlight an interplay between a relatively new class of quasiseparable matrices and univariate polynomials quasiseparable matrices generalize two classical matrix classes jacobi tridiagonal matrices and unitary hessenberg matrices that are known to correspond to real orthogonal polynomials and szego polynomials respectively the latter two polynomial families arise in a wide variety of applications and their short recurrence relations are the basis for a number of efficient algorithms for historical reasons algorithm development is more advanced for real orthogonal polynomials recent variations of these algorithms tend to be valid only for the szego polynomials they are analogues and not generalizations of the original algorithms herein we survey several recent results for the superclass of quasiseparable matrices which includes both jacobi and unitary hessenberg matrices as special cases the interplay between quasiseparable matrices and their associated polynomial sequences which contain both real orthogonal and szego polynomials allows one to obtain true generalizations of several algorithms specifically we discuss the bjorck pereyra algorithm the traub algorithm certain new digital filter structures as well as qr and divide and conquer eigenvalue algorithms

on the structure of tame graded basic hopf algebras ii abstract in this paper firstly we give a complete list of tame local graded frobenius algebras then by the method of bosonization and this list we determine the structure of tame radically graded basic hopf algebras completely

galois actions and blocks of tame infinitesimal group schemes given an infinitesimal group that is defined over an algebraically closed field of characteristic we determine the block structure of the algebra of measures in case its principal block is tame and the height of the factor group of by its multiplicative center is at least two our results yield a complete description of the stable auslander reiten quiver of along with a criterion for the domesticity of

polyhedral groups mckay quivers and the finite algebraic groups with tame principal blocks given an algebraically closed field k of characteristic p 3 we classify the finite algebraic k groups whose algebras of measures afford a principal block of tame representation type the structure of such a group mathcal g is largely determined by a linearly reductive subgroup scheme hat mathcal g of sl 2 with the mckay quiver of hat mathcal g relative to its standard module being the gabriel quiver of the principal block mathcal b 0 mathcal g the graphs underlying these quivers are extended dynkin diagrams of type tilde a tilde d or tilde e and the tame blocks are morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type

a new approach to fast polynomial interpolation and multipoint evaluation abstract the fastest known algorithms for the problems of polynomial evaluation and multipoint interpolation are devastatingly unstable numerically because of their recursive use of polynomial divisions we apply a completely distinct approach to compute approximate solutions to both problems equally fast but with improved numerical stability our approach relies on new techniques so far not used in this area we reduce the problems to vandermonde matrix computations and then exploit some recent methods for improving computations with structured matrices

on auslander reiten components of blocks and self injective biserial algebras we investigate the existence of auslander reiten components of euclidean type for special biserial self injective algebras and for blocks of group algebras in particular we obtain a complete description of stable auslander reiten quivers for the tame self injective algebras considered here

nearly optimal solution of rational linear systems of equations with symbolic lifting and numerical initialization abstract hensel s symbolic lifting is a highly effective method for the solution of a general or structured e g toeplitz or hankel linear system of equations with integer or rational coefficients of bounded length it can handle ill conditioned inputs for which numerical methods become costly lifting amounts to recursive multiplications by vectors of the input coefficient matrices and its precomputed inverse modulo a fixed integer s such multiplications only involve small numbers of data movements and arithmetic operations with bounded precision the known methods for precomputation of the inverse are more costly however in particular they involve more data movements as our remedy for this bottleneck stage we create an auxiliary matrix sharing its inverse modulo s with the input matrix and we readily compute this inverse by applying numerical iterative refinement which is a numerical counterpart of lifting in the case of general unstructured as well as toeplitz hankel and other popular structured inputs our hybrid algorithms involve a small number of data movements and optimal number of boolean that is bitwise operations up to a logarithmic factor we extend the algorithms to nearly optimal computation of polynomial greatest common divisors gcds least common multiples lcms and pade approximations as well as the berlekamp massey reconstruction of linear recurrences we also cover newton s lifting for matrix inversion specialize it to the case of structured input and combine it with hensel s to enhance the overall efficiency our initialization techniques work for newton s lifting as well furthermore we extend all our lifting algorithms to allow their initialization modulo powers of two thus implementing them in the binary base

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

fast algorithms for polynomial interpolation integration and differentiation for functions tabulated at chebyshev nodes on an interval spectral interpolation and indefinite integration can be performed stably and efficiently via the fast fourier transform for many other sets of nodes such as the gaussian nodes on an interval the classical interpolation and indefinite integration schemes are stable but slow requiring o n 2 arithmetic operations with n the number of nodes in the discretization of the interval in this paper a group of algorithms is presented for the efficient evaluation of lagrange polynomial interpolants at multiple points on the line and for the rapid indefinite integration and differentiation of functions tabulated at nodes other than chebyshev the interpolation scheme requires o n cdot log 1 varepsilon arithmetic operations and o n cdot log n n cdot log 1 varepsilon operations are required for the integration and differentiation schemes where varepsilon is the precision of computations and n is the number of nodes t

accelerated approximation of the complex roots and factors of a univariate polynomial the known algorithms approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and boolean time they are however quite involved and require a high precision of computing when the degree of the input polynomial is large which causes numerical stability problems we observe that these difficulties do not appear at the initial stages of the algorithms and in our present paper we extend one of these stages analyze it and avoid the cited problems still achieving the solution within a nearly optimal complexity estimates provided that some mild initial isolation of the roots of the input polynomial has been ensured the resulting algorithms promise to be of some practical value for root finding and can be extended to the problem of polynomial factorization which is of interest on its own right we conclude with outlining such an extension which enables us to cover the cases of isolated multiple roots and root clusters

complexity of multiplication with vectors for structured matrices abstract fast algorithms for computing the product with a vector are presented for a number of classes of matrices whose properties relate to the properties of toeplitz vandermonde or cauchy matrices these matrices are defined using the concept of displacement of a matrix and also for their inverses all the actions which are not dependent upon the coordinates of the input vector are singled out in a separate preprocessing stage the proposed algorithms are based on new representations of these matrices involving factor circulants

fast and efficient parallel solution of sparse linear systems this paper presents a parallel algorithm for the solution of a linear system a bf x bf b with a sparse n times n symmetric positive definite matrix a associated with the graph g a that has n vertices and has an edge for each nonzero entry of a if g a has an s n separator family and a known s n separator tree then the algorithm requires only o log 3 n time and e m s n log n processors for the evaluation of the solution vector bf x a 1 bf b where e is the number of edges in g a and m n is the number of processors sufficient for multiplying two n times n rational matrices in time o log n furthermore for this computational cost the algorithm computes a recursive factorization of a such that the solution of any other linear system a bf x bf b with the same matrix a requires only o log 2 n time and e log n s n 2 processors

modular representation theory of lie algebras on etudie les representations non necessairement restreintes de l algebre universelle enveloppante u 7b g pour une algebre de lie restreinte 7b g

linearizations of matrix polynomials in bernstein bases we discuss matrix polynomials expressed in a bernstein basis and the associated polynomial eigenvalue problems using mobius transformations of matrix polynomials large new families of strong linearizations are generated matrix polynomials that are structured with respect to a bernstein basis together with their associated spectral symmetries are also investigated the results in this paper apply equally well to scalar polynomials and include the development of new companion pencils for polynomials expressed in a bernstein basis

nearly optimal computations with structured matrices we estimate the boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices we study four basic and most popular classes that is toeplitz hankel cauchy and vandermonde matrices for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation the boolean cost estimates for the latter problems have been obtained by kirrinnis in 10 except for rational interpolation we supply them now as well as the boolean complexity estimates for the important problems of multiplication of transposed vandermonde matrix and its inverse by a vector all known boolean cost estimates from 10 for such problems rely on using kronecker product this implies the d fold precision increase for the d th degree output but we avoid such an increase by relying on distinct techniques based on employing fft furthermore we simplify the analysis and make it more transparent by combining the representations of our tasks and algorithms both via structured matrices and via polynomials and rational functions this also enables further extensions of our estimates to cover trummer s important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes as well as the transposed vandermonde matrices it is known that the solution of toeplitz hankel cauchy vandermonde and transposed vandermonde linear systems of equations is generally prone to numerical stability problems and numerical problems arise even for multiplication of cauchy vandermonde and transposed vandermonde matrices by a vector thus our fft based results on the boolean complexity of these important computations could be quite interesting because our estimates are reasonable even for more general classes of structured matrices showing rather moderate growth of the complexity as the input size increases

numerically safe gaussian elimination with no pivoting abstract gaussian elimination with no pivoting and block gaussian elimination are attractive alternatives to the customary but communication intensive gaussian elimination with partial pivoting 1 provided that the computations proceed safely and numerically safely that is run into neither division by 0 nor numerical problems empirically safety and numerical safety of genp have been consistently observed in a number of papers where an input matrix was pre processed with various structured multipliers chosen ad hoc our present paper provides missing formal support for this empirical observation and explains why it was elusive so far namely we prove that genp is numerically unsafe for a specific class of input matrices in spite of its pre processing with some well known and well tested structured multipliers but we also prove that genp and bge are safe and numerically safe for the average input matrix pre processed with any nonsingular and well conditioned multiplier this should embolden search for sparse and structured multipliers and we list and test some new classes of them we also seek randomized pre processing that universally that is for all input matrices supports i safe genp and bge with probability 1 and or ii numerically safe genp and bge with a probability close to 1 we achieve goal i with a gaussian structured multiplier and goal ii with a gaussian unstructured multiplier and alternatively with gaussian structured augmentation we consistently confirm all these formal results with our tests of genp for benchmark inputs we have extended our approach to other fundamental matrix computations and keep working on further extensions

tame biserial algebras a similar result is suspected if we drop the property representationfinite in classes 1 and 2 and we will start an investigation of biserial algebras of infinite representation type the key to the result for representation finite algebras was to observe that each representation finite biserial algebra is special 13 see also the basic notations below which means that it can be presented by a quiver with nice relations special algebras play an important role in the modular representation theory of finite groups namely each representation finite block of a group algebra and some of the tame blocks they occur only in characteristic 2 are special 7 10 4 moreover in the complex representation theory of the lorentz group the so called harish chandra modules are defined over tame special algebras 6 in this paper we show that l a 62 for any special algebra a and that special algebras of infinite representation type are tame our paper consists of four parts first we prove that each special algebra is a factor of a special symmetric algebra theorem 1 5 for the latter the methods of gelfand and ponomarev used in the classification of the indecomposable harish chandra modules of the lorentz group apply and furnish a complete list of indecomposable finitely generated modules over special algebras proposition 2 3 presented in section 2 from this list we 480 0021 8693 85 3 00