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

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

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

on positivity of principal minors of bivariate bezier collocation matrix it is well known that the bivariate polynomial interpolation problem at uniformly distributed domain points of a triangle is correct thus the corresponding interpolation matrix m is nonsingular schumaker stated the conjecture that all principal submatrices of m are nonsingular too furthermore all of the corresponding determinants the principal minors are conjectured to be positive this result would solve the constrained interpolation problem in this paper the conjecture on minors for polynomial degree 17 and conjecture for some particular configurations of domain points are confirmed

closed form solutions for the effective conductivity of two phase periodic composites with spherical inclusions in this paper we use approximate solutions of the nemat nasser et al to estimate the effective conductivity of two phase composites with non overlapping spherical inclusions systems with different inclusion distributions are considered cubic lattice distributions simple cubic body centered cubic and face centered cubic and random distributions the effective conductivities of the former are obtained in closed form and compared with exact solutions from the fast fourier transform based methods for systems containing randomly distributed spherical inclusions the solutions are shown to be directly related to the static structure factor and we obtain its analytical expression in the infinite volume limit

finding c 3 strong pseudoprimes let q 1 10 9 and so that 12 3186 65857 83403 11511 67461 24 digits 399165290221 798330580441 which was found by the author in an earlier paper we give reasons to support the conjecture the main idea of our method for finding those 21978 c 3 spsp 2 s is that we loop on candidates of signatures and kernels with heights bounded subject those candidates n q 1 q 2 q 3 of c 3 spsp 2 s and their prime factors q 1 q 2 q 3 to miller s tests and obtain the desired numbers at last we speed our algorithm for finding larger c 3 spsp s say up to 10 50 with a given signature to more prime bases comparisons of effectiveness with arnault s and our previous methods for finding c 3 strong pseudoprimes to the first several prime bases are given

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

extremal and optimal properties of b bases collocation matrices totally positive matrices are related with the shape preserving representations of a space of functions the normalized b basis of the space has optimal shape preserving properties bernstein polynomials b splines and rational bernstein bases are examples of normalized b bases it is proven that the minimal eigenvalue and singular value of a collocation matrix of a normalized b basis is bounded below by the minimal eigenvalue and singular value of the corresponding collocation matrix of any normalized totally positive basis of the same space the optimal conditioning for the infty norm of a collocation matrix of a normalized b basis among all the normalized totally positive bases of a space of functions is also shown numerical examples confirm the theoretical results and answer related questions

strong pseudoprimes to the first eight prime bases define m to be the smallest strong pseudoprime to the firstm prime bases the exact value of m is known for 1 m 8 z zhang have found a 19decimal digit number q11 3825 12305 65464 13051 which is a strong pseudoprime to the first 11 prime bases and he conjectured that 9 10 11 q11 we prove the conjecture by algorithms

a note on matrices with maximal growth factor for neville elimination neville elimination is a direct method for the solution of linear systems of equations with advantages for some classes of matrices and in the context of pivoting strategies for parallel implementations the growth factor is an indicator of the numerical stability of an algorithm in the literature bounds for the growth factor of neville elimination with some pivoting strategies have appeared in this work we determine all the matrices such that the minimal upper bound of the growth factor of neville elimination with those pivoting strategies is reached

blocking neville elimination algorithm for exploiting cache memories neville elimination is a method for solving a linear system of equations that introduces zeros in a matrix column by adding to each row an adequate multiple of the previous one in this paper we explore block algorithms for neville elimination which take into account the memory hierarchies of a computer these algorithms try to manage the memory movements to optimize them thus the matrix of the system is divided following three different strategies blocking by rows columns or submatrices in each case we study the performance of the algorithm according to the ratio of floating point operations to memory references q theoretical estimations show that q depends on data partitioning being submatrix blocks the best choice

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 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

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

explicit effective elasticity tensors of two phase periodic composites with spherical or ellipsoidal inclusions the effective elasticity tensors of two phase composites are estimated by solving the localization problem in the wave vector domain for the case of non overlapping spherical or ellipsoidal inclusions with previous works showing that the effective properties can be computed from lattice sums we propose a method to compute the sums analytically and obtain the explicit expressions for the effective tensors in the case of different periodic cells leading to cubic or orthotropic elasticity tensors the effective elasticity tensors are obtained in closed forms that are in good agreement with the exact solutions for a large range of physical parameters in the random distribution cases the statistical connection of the effective tensor to the structure factor is shown and a closed form expression is obtained in the infinite volume limit

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

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

which spaces for design we determine the largest class of spaces of sufficient regularity which are suitable for design in the sense that they do possess blossoms it is the class of all spaces containing constants of which the spaces derived under differentiation are quasi extended chebyshev spaces i e they permit hermite interpolation taylor interpolation excepted it is also the class of all spaces which possess bernstein bases or of all spaces for which any associated spline space does possess a b spline basis note that blossoms guarantee that such bases are normalised totally positive bases they even are the optimal ones

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

on the zeros of complex van vleck polynomials polynomial solutions to the generalized lame equation the stieltjes polynomials and the associated van vleck polynomials have been studied extensively in the case of real number parameters in the complex case relatively little is known numerical investigations of the location of the zeros of the stieltjes and van vleck polynomials in special cases reveal intriguing patterns in the complex case suggestive of a deeper structure in this article we report on these investigations with the main result being a proof of a theorem confirming that the zeros of the van vleck polynomials lie on special line segments in the case of the complex generalized lame equation having three free parameters furthermore as a result of this proposition we are able to obtain in this case a strengthening of a classical result of heine on the number of possible van vleck polynomials associated with a given stieltjes polynomial

generalized oscillatory matrices we define a new class of generalized oscillatory matrices shortly go matrices over a noncommutative ring with identity and a positive part similarly to the classical case this class consists of square generalized totally nonnegative matrices gtn matrices of which some power is generalized totally positive using the previously defined ordering of invertible gtn matrices we study in particular so called basic go matrices which form in a sense atoms of this ordering

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

understanding recurrence relations for chebyshevian b splines via blossoms the purpose of this article is to show how naturally recurrence relations for most general chebyshevian b splines emerge from blossoms in particular this work gives a new insight into previous results by lyche a recurrence relation for chebyshevian b splines constr approx 1 1985 155 178 the importance of which it underlines

symmetric functions and the vandermonde matrix this work deduces the lower and the upper triangular factors of the inverse of the vandermonde matrix using symmetric functions and combinatorial identities the l and u matrices are in turn factored as bidiagonal matrices the elements of the upper triangular matrices in both the vandermonde matrix and its inverse are obtained recursively the particular value x i 1 q qi 1 in the indeterminates of the vandermonde matrix is investigated and it leads to q binomial and q stirling matrices it is also shown that q stirling matrices may be obtained from the pascal matrix

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

sign regular matrices and neville elimination abstract a pivoting strategy of o n operations for the neville elimination of n n nonsingular sign regular matrices is introduced among other nice properties it is proved that it preserves sign regularity it is also shown its relationship with scaled partial pivoting strategies for neville elimination

the effective conductivity of random suspensions of spherical particles the effective conductivity of an infinite random mono disperse hard sphere suspension is reported for particle to matrix conductivity ratios of 10 and 0 01 for sphere volume fractions c up to 0 6 the conductivities are computed with a method previously described by the authors which includes both far and near field interactions and the particle configurations are generated via a monte carlo method the results are consistent with the previous theoretical work of d j jeffrey to o c 2 and the bounds computed by s torquato and f lado it is also found that the clausius mosotti equation is reasonably accurate for conductivity ratios of 10 or less all the way up to 60 by volume the calculated conductivities compare very well with those of experiments in addition percolation like numerical experiments are performed on periodically replicated cubic lattices of n nearly touching spheres with an infinite particle to matrix conductivity ratio where the conductivity is computed as spheres are removed one by one from the lattice under suitable normalization of the conductivity and volume fraction it is found that the initial volume fraction must be extremely close to maximum packing in order to observe a percolation transition indicating that the near field effects must be very large relative to far field effects these percolation transitions occur at the accepted values for simple sc bodycentred bcc and face centred fcc cubic lattices also the vulnerability of the lattices computed here are exactly those of previous investigators due to limited data above the percolation threshold we could not correlate the conductivity with a power law near the threshold however it can be correlated with a power law for large normalized volume fractions in this case the exponents are found to be 1 70 1 75 and 1 79 for sc bcc and fcc lattices respectively

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

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

bidiagonal factorizations and quasi oscillatory rectangular matrices abstract a real matrix a of size m n is called totally nonnegative totally positive if all its minors are nonnegative positive a variant of the neville elimination process is studied in relation to the existence of a totally nonnegative elementary bidiagonal factorization of a the class of quasi oscillatory rectangular matrices which in the square case contains the oscillatory matrices is introduced and a characterization of this class of matrices by incorporating bidiagonal factorization is showed

complexity of dense linear system solution on a multiprocessor ring abstract different algorithms based on gaussian elimination for the solution of dense linear systems of equations are discussed for a multiprocessor ring the number of processors is assumed not to exceed the problem size a fairly general model for data transfer is proposed and the algorithms are analyzed with respect to their requirements of arithmetic as well as communication times

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

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

on strong pseudoprimes to several bases with y k denoting the smallest strong pseudoprime to all of the first k primes taken as bases we determine the exact values for 5 q6 q7 q8 and give upper bounds for v 9 w t 1 we discuss the methods and underlying facts for obtaining these results 1 primality tests by means of strong pseudoprimes computer algebra systems as for instance axiom 2 use strong pseudoprimes for testing primality of integers the advantage of such tests is that they are very efficient the disadvantage is that they are only probabilistic tests when the integers are not restricted to certain intervals to make such tests deterministic for integers in prescribed intervals one has to know the exact number of necessary so called strong pseudoprimality tests for this purpose we introduce the numbers v1i v 2 for which we compute lower and upper bounds these numbers are defined and discussed in this section in 2 we derive some facts which are the basis for finding bounds for the numbers v k in 3 we discuss the methods which led to our results in view of fermat s little theorem we know that n is certainly not a prime when we have bn 1 i 1 mod n for an integer b with 1 0 and when n is a composite number then n is called a strong pseudoprime to base b if either

effective properties of composite materials with periodic microstructure a computational approach abstract this study reviews several problems which are specific of composites with periodic microstructure composed of linear or nonlinear constituents the theoretical background of the method is recalled first two different families of numerical methods are considered to solve the problem the first is based on the finite element method the concept of macroscopic degrees of freedom is presented the implementation of periodicity conditions is discussed a general framework permitting either a strain or stress control is proposed the second numerical method is based on fast fourier transforms it considers first the problem of a homogeneous linear reference material undergoing a nonhomogeneous periodic eigenstrain the solution of this problem is based on the explicit form of the periodic green s function of the reference medium the relative merits of the two methods are compared and several examples are discussed both methods give very comparable results on test examples and their domains of applications appear to be complementary

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

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 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

multiresolution analyses originated from nonstationary subdivision schemes we analyze the properties of a new class of totally positive refinable functions obtained from nonstationary subdivision schemes we show that the corresponding system of the integer translates is linearly independent satisfies a whitney schoenberg condition reproduces polynomials up to a certain degree and generates a multiresolution analysis finally pre wavelets and bases on the interval are constructed

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

two kinds of strong pseudoprimes up to 10 36 let n 1 be an odd composite integer write n 1 2 s d with d odd if either b d 1 mod n or b 2r d 1 mod n for some r 0 1 s 1 then we say that n is a strong pseudoprime to base b or spsp b for short define t to be the smallest strong pseudoprime to all the first t prime bases if we know the exact value of t we will have for integers n 10 36 and to give reasons and numerical evidence of k2 and c 3 spsp s 12 where t resp t is the smallest k2 resp c 3 strong pseudoprime to all the first t prime bases for this purpose we describe procedures for computing and enumerating the two kinds of spsp s 10 36 to the first 9 prime bases the entire calculation took about 4000 hours on a pc pentium iv 1 8ghz recall that a k2 spsp is an spsp of the form n pq with p q primes and q 1 2 p 1 and that a c 3 spsp is an spsp and a carmichael number of the form n q 1 q 2 q 3 with each prime factor q i 3 mod 4

equal entries in totally positive matrices we show that the maximal number of equal entries in a totally positive resp totally nonsingular n by n matrix is n 4 3 resp n 3 2 relationships with point line incidences in the plane bruhat order of permutations and tp completability are also presented we also examine the number and positionings of equal 2 by 2 minors in a 2 by n tp matrix and give a relationship between the location of equal 2 by 2 minors and outerplanar graphs

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

total positivity of a kind of generalized toric bernstein basis abstract the normalized totally positive bases are widely used in many fields based on the generalized vandermonde determinant the normalized total positivity of a kind of generalized toric bernstein basis is proved which is defined on a set of real points by this result then the progressive iterative approximation property of the generalized toric bezier curve is obtained and some examples are provided to illustrate this property

the hadamard core of the totally nonnegative matrices abstract an m by n matrix a is called totally nonnegative if every minor of a is nonnegative the hadamard product of two matrices is simply their entry wise product this paper introduces the subclass of totally nonnegative matrices whose hadamard product with any totally nonnegative matrix is again totally nonnegative many properties concerning this class are discussed including a complete characterization for min m n a characterization of the zero nonzero patterns for which all totally nonnegative matrices lie in this class and connections to oppenheim s inequality

totally nonnegative cells and matrix poisson varieties we describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices that is the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by a postnikov in order to do this we relate the totally nonnegative cells to torus orbits of symplectic leaves of the poisson varieties of complex matrices in particular we describe the minors that vanish on a torus orbit of symplectic leaves we prove that such families of minors are exactly the admissible families and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices

the effective conductivity of a periodic array of spheres we consider the problem of determining the effective conductivity k of a composite material consisting of equal sized spheres of conductivity alpha arranged in a cubic array within a homogeneous matrix of unit conductivity we modify zuzovski brenner9s 1977 method and thereby obtain a set of infinite linear equations for the coefficients of the formal solution equivalent to that derived by mckenzie et al 1978 using a method originally devised by rayleigh 1892 on solving these equations we derive expressions for k ast to o c 9 c being the volume fraction of the spheres for simple body centred and face centred cubic arrays and also obtain numerical values for k over the whole range of alpha and c we show that these results for cubic arrays can be used to estimate ast for random arrays of identical spheres for arrays of highly conducting and nearly touching spheres batchelor o9brien 1977 showed that begin equation k ast sim begin cases k 1 ln 1 chi k 2 alpha infty chi c c max frac 1 3 rightarrow 1 2k 1 ln alpha k9 2 quad chi 1 alpha rightarrow infty end cases end equation where c max corresponds to the volume fraction when the spheres are actually touching each other and determined k 1 for the three cubic arrays our numerical results are consistent with the above asymptotic expressions except for the fact that the numerical values for the constants k 2 and k9 2 thereby obtained do not quite satisfy the relation k9 2 k 1 3 9 ln 2 k 2 given by batchelor o9brien we have been unable to find the reason for this slight discrepancy

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

shape preserving interpolants with high smoothness the aim of the present paper is to introduce a new method to construct a c3 function which interpolates a given set of data in a shape preserving way the resulting function is a parametric quintic curve whose shape is controlled by tension parameters which have a direct geometric meaning

shape preserving c1 hermite interpolants generated by a gori pitolli subdivision scheme in this paper we present a method for the construction of c 1 hermite interpolants obtained from a particular family of refinable spline functions introduced by gori and pitolli they constitute a one parameter subfamily of the hermite interpolants generated by the general merrien s subdivision scheme we compare this family to the other one parameter subfamily studied by merrien and sablonniere and lyche and merrien on the solution of two points hermite interpolation problems with arbitrary monotonicity or convexity constraints

comparing pivoting strategies for almost strictly sign regular matrices abstract in this paper some properties of two determinant pivoting for neville elimination are presented in particular we consider a zero increasing property and we show an optimal normwise growth factor comparisons with other pivoting strategies for neville elimination and with gaussian elimination with partial pivoting of almost strictly sign regular matrices are performed numerical examples are included

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

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

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

finding strong pseudoprimes to several bases define m to be the smallest strong pseudoprime to all the first m prime bases if we know the exact value of m we will have for integers n m a deterministic primality testing algorithm which is not only easier to implement but also faster than either the jacobi sum test or the elliptic curve test thanks to pornerance et al and jaeschke m are known for 1 m 8 upper bounds for 9 10 and 1 were given by jaeschke in this paper we tabulate all strong pseudoprimes spsp s n 10 24 to the first ten prime bases 2 3 29 which have the form n pq with p q odd primes and q 1 k p 1 k 2 3 4 there are in total 44 such numbers six of which are also spsp 31 and three numbers are spsp s to both bases 31 and 37 as a result the upper bounds for 10 and 11 are lowered from 28 and 29 decimal digit numbers to 22 decimal digit numbers and a 24 decimal digit upper bound for 12 is obtained the main tools used in our methods are the biquadratic residue characters and cubic residue characters we propose necessary conditions for n to be a strong pseudoprime to one or to several prime bases comparisons of effectiveness with both jaeschke s and arnault s methods are given

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

some examples of kinetic schemes whose diffusion limit is il in s exponential fitting this paper is concerned with diffusive approximations of some numerical schemes for several linear or weakly nonlinear kinetic models which are motivated by wide range applications including radiative transfer or neutron transport run and tumble models of chemotaxis dynamics and vlasov fokker planck plasma modeling the well balanced method applied to such kinetic equations leads to time marching schemes involving a scattering s matrix itself derived from a normal modes decomposition of the stationary solution one common feature these models share is the type of diffusive approximation their macroscopic densities solve drift diffusion systems for which a distinguished numerical scheme is il in scharfetter gummel s exponential fitting discretization we prove that these well balanced schemes relax within a parabolic rescaling towards such type of discretization by means of an appropriate decomposition of the s matrix hence are asymptotic preserving

total positivity and its applications preface i spline functions total positivity of the spline kernel and its applications b bojanov total positivity properties of lb splines t lyche l l schumaker total positivity and splines k morken totally positive systems and interpolation by subspaces of generalized splines m sommer h strauss ii matrix theory preservation of total nonnegativity under the hadamard product and related topics j garloff d g wagner vertex implications for totally nonnegative under the hadamard product and related topics j garloff on factorizations of totally positive matrices j garloff on factorizations of totally positive matrices m gasca j m pena iii geometric modeling total positivity and optimal bases j m carnicer j m pena total positivity and the shape of curves t n t goodman tchebycheff curves m l mazure h pottmann iv probability and mathematical biology analytical and probabilistic methods for totally positive matrices g s goodman some inequalities of total positivity in pure and applied mathematics s karlin z ziegler new directions and problems of multivariate total positivity motivated by genetic recombination probabilities s karlin v approximation theory tchebycheff spaces and total positivity j m carnicer j m pena cebysev and weak cebysev systems r a zalik vi complex analysis zero sets of entire fourier transforms of multiply positive density functions o m katkova vii statistics total nonnegativity of moment matrices and its application to e optimal designs in polynomial regression b heiligers analysis of regression dependence in 2 xn bivariate distributions and some applications in contingency tables s kasala m b rao viii real analysis total positivity and nonlinear analysis r q jia solutions of certain functional equations s karlin alternance and snakes in the nonlinear finite moment problem k tatalian ix combinatorics the applications of total positivity to combinatorics and conversely f brenti x integral equations spectral properties of totally positive kernels and matrices a pinkus index

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

finding strong pseudoprimes to several bases ii define m to be the smallest strong pseudoprime to all the first m prime bases if we know the exact value of m we will have for integers n m a deterministic efficient primality testing algorithm which is easy to implement thanks to pomerance et al and jaeschke the m are known for 1 m 8 upper bounds for g 10 and 11 were first given by jaeschke and those for 10 and 11 were then sharpened by the first author in his previous paper math comp 70 2001 863 872 in this paper we first follow the first author s previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes spsp s n 1024 to the first five or six prime bases which have the form n pq with p q odd primes and q 1 k p 1 k 4 3 5 2 3 2 6 then we tabulate all carmichael numbers 1020 to the first six prime bases up to 13 which have the form n q1q2q3 with each prime factor qi 3 mod 4 there are in total 36 such carmichael numbers 12 numbers of which are also spsp s to base 17 5 numbers are spsp s to bases 17 and 19 one number is an spsp to the first 11 prime bases up to 31 as a result the upper bounds for 9 10 and 11 are lowered from 20 and 22 decimal digit numbers to a 19 decimal digit number 9 10 11 q11 3825 12305 65464 13051 19 digits 149491 747451 34233211 we conjecture that 9 10 11 3825 12305 65464 13051 and give reasons to support this conjecture the main idea for finding these carmichael numbers is that we loop on the largest prime factor q3 and propose necessary conditions on n to be a strong pseudoprime to the first 5 prime bases comparisons of effectiveness with arnault s bleichenbacher s jaeschke s and pinch s methods for finding carmichael numbers with three prime factors which are strong pseudoprimes to the first several prime bases are given

brief paper necessary conditions for hadamard factorizations of hurwitz polynomials a property of hurwitz polynomials is related with the hadamard product garloff and wagner proved that hadamard products of hurwitz polynomials are hurwitz polynomials and garloff and shrinivasan shown that there are hurwitz polynomials of degree 4 which do not have a hadamard factorization into two hurwitz polynomials of the same degree 4 in this paper we give necessary conditions for an even degree hurwitz polynomial to have a hadamard factorization into two even degree hurwitz polynomials such conditions are given in terms of the coefficients of the given polynomial alone furthermore we show that if an odd degree hurwitz polynomial has a hadamard factorization then a system of nonlinear inequalities has at least one solution

characterizations of rectangular totally and strictly totally positive matrices abstract an n m real matrix a is said to be totally positive strictly totally positive if every minor is nonnegative positive in this paper we study characterizations of these classes of matrices by minors by their full rank factorization and by their thin qr factorization

accurate eigenvalues of certain sign regular matrices we present a new o n 3 algorithm for computing all eigenvalues of certain sign regular matrices to high relative accuracy in floating point arithmetic the accuracy and cost are unaffected by the conventional eigenvalue condition numbers a matrix is called sign regular when the signs of its nonzero minors depend only of the order of the minors the sign regular matrices we consider are the ones which are nonsingular and whose kth order nonzero minors are of sign 1 k k 1 2 for all k this class of matrices can also be characterized as nonsingular totally nonnegative matrices with columns in reverse order we exploit a characterization of these particular sign regular matrices as products of nonnegative bidiagonals and the reverse identity we arrange the computations in such a way that no subtractive cancellation is encountered thus guaranteeing high relative forward accuracy

a class of oscillatory matrices with exponent n 1 abstract a matrix a is called an oscillatory matrix if it is totally nonnegative and there exists a positive integer k such that a k is totally positive in this paper we describe the class of n n oscillatory matrices which have maximal exponent equal to n 1

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

isogemetric analysis and symmetric galerkin bem isogeometric approach applied to boundary element methods is an emerging research area see e g simpson et al 2012 33 in this context the aim of the present contribution is that of investigating from a numerical point of view the symmetric galerkin boundary element method sgbem devoted to the solution of 2d boundary value problems for the laplace equation where the boundary and the unknowns on it are both represented by b splines de boor 2001 9 we mainly compare this approach which we call iga sgbem with a curvilinear sgbem aimi et al 1999 2 which operates on any boundary given by explicit parametric representation and where the approximate solution is obtained using lagrangian basis both techniques are further compared with a standard conventional sgbem approach aimi et al 1997 1 where the boundary of the assigned problem is approximated by linear elements and the numerical solution is expressed in terms of lagrangian basis several examples will be presented and discussed underlying benefits and drawbacks of all the above mentioned approaches

the continuing search for wieferich primes a prime p satisfying the congruence 2 p 1 1 mod p 2 is called a wieferich prime although the number of wieferich primes is believed to be infinite the only ones that have been discovered so far are 1093 and 3511 this paper describes a search for further solutions the search was conducted via a large scale internet based computation the result that there are no new wieferich primes less than 1 25 10 15 is reported

generalized totally nonnegative matrices we define a new class of generalized totally nonnegative matrices shortly gtn matrices over a noncommutative ring with identity and a positive part the class of gtn matrices is closed under multiplication every submatrix of a gtn matrix is a gtn matrix as well we also define a standard form of a gtn matrix we then study the invertible gtn matrices of a fixed order and classify them the classes form a partially ordered set can be noncommutatively multiplied and multiplication preserves ordering

shape preserving properties of ω q bernstein polynomials abstract in this paper we discuss shape preserving properties of the q bernstein polynomials b n q f x introduced by lewanowicz and wozny in s lewanowicz p wo ny generalized bernstein polynomials bit 44 1 2004 63 78 for q 0 1 when 0 we recover the q bernstein polynomials introduced by phillips g m phillips bernstein polynomials based on the q integers ann numer math 4 1997 511 518 when q 1 we recover the classical bernstein polynomials for q 0 1 we show that the basic q bernstein polynomial basis is a normalized totally positive basis on 0 1 and that the q bernstein operators b n q on c 0 1 are variation diminishing monotonicity preserving and convexity preserving we also show that the q bernstein polynomials of a convex function f in the case q 0 1 are monotonic in the parameters and q for fixed n

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

q bernstein polynomials and bezier curves we define q bernstein polynomials which generalize the classical bernstein polynomials and show that the difference of two consecutive q bernstein polynomials of a function f can be expressed in terms of second order divided differences of f it is also shown that the approximation to a convex function by its q bernstein polynomials is one sided a parametric curve is represented using a generalized bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve we study the nature of degree elevation and degree reduction for this basis and show that degree elevation is variation diminishing as for the classical bernstein basis

weighted lupas q bezier curves this paper is concerned with a new generalization of rational bernstein bezier curves involving q integers as shape parameters a one parameter family of rational bernstein bezier curves weighted lupas q bezier curves is constructed based on a set of lupas q analogue of bernstein functions which is proved to be a normalized totally positive basis the generalized rational bezier curve is investigated from a geometric point of view the investigation provides the geometric meaning of the weights and the representation for conic sections we also obtain degree evaluation and de casteljau algorithms by means of homogeneous coordinates numerical examples show that weighted lupas q bezier curves have more modeling flexibility than classical rational bernstein bezier curves and lupas q bezier curves and meanwhile they provide better approximations to the control polygon than rational phillips q bezier curves

a remark on oscillatory matrices in this note we revisit a classical criterion obtained by gantmacher and krein for determining when a totally nonnegative matrix is actually oscillatory a new proof of this criterion is presented by incorporating bidiagonal factorizations of invertible totally nonnegative matrices and utilizing certain associated weighted planar diagrams

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

one sided hermite interpolation by piecewise different generalized polynomials piecewise generalized polynomials of different kinds of order n ect splines of order n are constructed from different ect systems of order n via connection matrices which are nonsingular and totally positive a well known zero count for polynomial splines is extended to ect splines it is used to construct ect b splines and to show under which conditions ect splines will solve modified hermite type interpolation problems also conditions are specified such that piecewise generalized polynomials form rect systems and the interpolation problems associated with may be solved recursively