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

towards a theory of classification abstract the well known difficulties arising in a classification which is not set theoretically trivial involving what is sometimes called a non smooth quotient have been overcome in a striking way in the theory of operator algebras by the use of what might be called a classification functor the very existence of which is already a surprise here the notion of such a functor is developed abstractly and a number of examples are considered including those which have arisen for various classes of operator algebras

total nonpositivity of nonsingular matrices in this paper nonsingular totally nonpositive matrices are studied and new characterizations are provided in terms of the signs of minors with consecutive initial rows or consecutive initial columns these characterizations extend an existing characterization that uses some restrictive hypotheses

on the cd index and γ vector of s shellable cw spheres we show that the vector of the order complex of any polytope is the f vector of a balanced simplicial complex this is done by proving this statement for a subclass of stanley s s shellable cw spheres which includes all polytopes the proof shows that certain parts of the cd index when specializing c 1 and considering the resulted polynomial in d are the f polynomials of simplicial complexes that can be colored with few colors we conjecture that the cd index of a regular cw sphere is itself the flag f vector of a colored simplicial complex in a certain sense

the rook s pivoting strategy abstract based on the geometric analysis of gaussian elimination ge found in neal and poole linear algebra appl 173 1992 239 264 and poole and neal linear algebra appl 149 1991 249 272 162 164 1992 309 324 a new pivoting strategy rook s pivoting rp was introduced in neal and poole linear algebra appl 173 1992 239 264 which encourages stability in the back substitution phase of ge while controlling the growth of round off error during the sweep out in fact foster j comput appl math 86 1997 177 194 has previously shown that rp as with complete pivoting cannot have exponential growth error empirical evidence presented in neal and poole linear algebra appl 173 1992 239 264 showed that rp produces computed solutions with consistently greater accuracy than partial pivoting that is rook s pivoting is on average more accurate than partial pivoting with comparable costs moreover the overhead to implement rook s pivoting in a scalar or serial environment is only about three times the overhead to implement partial pivoting the theoretical proof establishing this fact is presented here and is empirically confirmed in this paper and supported in foster j comput appl math 86 1997 177 194

a classification of simple limits of splitting interval algebras abstract let a be a unital simple limit of finite direct sums of sub homogeneous interval algebras of a certain type cf definition 1 1 it is proved that a can be classified by the scaled ordered group k 0 a the simplex t a and the canonical pairing between them it is also shown that k 0 a might fail to have the riesz decomposition property

rounding error bounds for the clenshaw and forsythe algorithms for the evaluation of orthogonal polynomial series rounding error bounds of the forsythe and the clenshaw smith algorithm for the evaluation of finite series of orthogonal polynomials are presented the forward error is studied by means of a direct approach that permits to obtain sharp bounds the bounds are illustrated with several numerical tests

inductive limits of interval algebras the tracial state space introduction this is the first in a series of papers where we investigate the structure of the c algebras obtained as inductive limits of interval algebras c 0 1 0m the purpose here is to show that these c algebras comprise a large stock of simple c algebras and that some of them allow endomorphisms with chaotic actions on the tracial state space to be more specific we show that to any dense subgroup g of q containing z and any metrizable choquet simplex s there is a sequence

a stable test for strict sign regularity a stable test to check if a given matrix is strictly sign regular is provided among other nice properties we prove that it has an optimal growth factor the test is compared with other alternative tests appearing in the literature and its advantages are shown

the accuracy of solutions to triangular systems triangular systems play a fundamental role in matrix computations it has been prominently stated in the literature but is perhaps not widely appreciated that solutions to triangular systems are usually computed to high accuracy higher than the traditional condition numbers for linear systems suggest this phenomenon is investigated by use of condition numbers appropriate to the componentwise backward error analysis of triangular systems results of wilkinson are unified and extended among the conclusions are that the conditioning of a triangular system depends on the right hand side as well as the coefficient matrix that use of pivoting in lu qr and cholesky factorisations can greatly improve the conditioning of a resulting triangular system and that a triangular matrix may be much more or less ill conditioned than its transpose

a survey of condition number estimation for triangular matrices we survey and compare a wide variety of techniques for estimating the condition number of a triangular matrix and make recommendations concerning the use of the estimates in applications each of the methods is shown to bound the condition number the bounds can broadly be categorised as upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms for each bound we examine by how much at worst it can overestimate or underestimate the condition number numerical experiments are presented in order to illustrate and compare the practical performance of the condition estimators

gaussian elimination when is scaling beneficial abstract for the linear system ax b the ordered pair d f of nonsingular diagonal matrices determine a scaling of the system through the two equations d af y db y f 1 x when scaling is implemented along with partial pivoting pp to solve ax b by gaussian elimination ge it is well known that certain ordered pairs d f produce better computed solutions than those obtained in the absence of scaling while others produce worse solutions the two most common explanations of this fact are 1 d f modifies magnifies or reduces the classical condition number of a and 2 d f modifies the magnitudes of the elements of a in case 2 if a scaling yields entries of approximately the same magnitude it is called an equilibration here the underlying hyperplane geometry of both the sweepout phase and the back substitution phase of ge is used to achieve a new level of understanding we present what we believe to be a better explanation of how scaling or equilibration influences pp in the selection of pivot equations a process critical to both phases of ge

a geometric analysis of gaussian elimination ii abstract in part i of this work we began a discussion of the numeric consequences of hyperplane orientation in gaussian elimination we continue this discussion by introducing the concept of back substitution phase error multipliers these error multipliers help to explain many of the previously unproven or poorly understood observations concerning gaussian elimination in a finite precision environment a new pivoting strategy designed to control both sweepout phase roundoff error and back substitution phase instability is also presented this new strategy called rook s pivoting is only slightly more expensive than partial pivoting yet produces results comparable to those produced by complete pivoting

stability of the partitioned inverse method for parallel solution of sparse triangular systems several authors have recently considered a parallel method for solving sparse triangular systems with many right hand sides the method employs a partition into sparse factors of the product form of the inverse of the coefficient matrix it is shown here that while the method can be unstable stability is guaranteed if a certain scalar that depends on the matrix and the partition is small and that this scalar is small when the matrix is well conditioned moreover when the partition is chosen so that the factors have the same sparsity structure as the coefficient matrix the backward error matrix can be taken to be sparse

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

on the classification of simple inductive limit c algebras ii the isomorphism theorem in this article it is proved that the invariant consisting of the scaled ordered k group and the space of tracial states together with the natural pairing between them is a complete invariant for the class of unital simple c algebras which can be expressed as the inductive limit of a sequence a 1 to a 2 to cdots to a n to cdots with a n bigoplus i 1 t n p n i m n i c x n i p n i where xn i is a compact metrizable space and pn i is a projection in m n i c xn i for each n and i and the spaces xn i are of uniformly bounded finite dimension note that the c algebras in the present class are not assumed to be of real rank zero as they were in eg2

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

required nonzero patterns for nonsingular sign regular matrices abstract an n n real matrix is called sign regular if for each k 1 k n all its minors of order k have the same nonstrict sign the zero entries which can appear in a nonsingular sign regular matrix depend on its signature because the signature can imply that certain entries are necessarily nonzero the patterns for the required nonzero entries of nonsingular sign regular matrices are analyzed

effect of equilibration on residual size for partial pivoting it is shown that column pivoting with row equilibration satisfies the same type of error bound as does row pivoting without scaling and that row pivoting with column equilibration satisfies the same type of bound as does column pivoting without scaling an interesting consequence for row pivoting is that column equilibration is sufficient to ensure that the norm of the residual is reasonably small

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

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

componentwise error analysis for linear systems associated with sign regular matrices in this paper we present a componentwise error analysis of solving linear systems whose coefficient matrices are sign regular matrices with the same signature sequences as those of jacobi sign regular matrices associated with corner cutting algorithms in computer aided geometric design it is interesting to show that for such sign regular matrices in finite precision floating point arithmetic the computed factors by gaussian elimination with the last two row exchanges satisfy a small componentwise relative backward error consequently the computed solutions of linear systems associated with such matrices admit pleasantly small componentwise relative backward errors

the growth factor and efficiency of gaussian elimination with rook pivoting abstract gaussian elimination is among the most widely used tools in scientific computing gaussian elimination with partial pivoting requires only o n2 comparisons beyond the work required in gaussian elimination with no pivoting but can in principle have error growth that is exponential in the matrix size n gaussian elimination with complete pivoting on the other hand cannot have exponential error growth but requires o n3 comparisons beyond the work required by gaussian elimination with no pivoting numerical experiments suggest that gaussian elimination with rook pivoting is between partial pivoting and complete pivoting in terms of efficiency and accuracy in this paper we prove that rook pivoting cannot have exponential error growth we also introduce a combination of partial pivoting and rock pivoting which we call gaussian elimination with partial rook pivoting and we prove the partial rook pivoting cannot have exponential error growth we include numerical experiments showing that on a serial computer the run times for cook pivoting are almost always close to those of partial pivoting and the run times for partial rook pivoting appear to be the same as those of partial pivoting

regularity properties in the classification program for separable amenable c algebras we report on recent progress in the program to classify separable amenable c algebras our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank strict comparison of positive elements and z stability and on the importance of the cuntz semigroup we include a brief history of the program s successes since 1989 a more detailed look at the villadsen type algebras which have so dramatically changed the landscape and a collection of announcements on the structure and properties of the cuntz semigroup

the classification of separable simple c algebras which are inductive limits of continuous trace c algebras with spectrum homeomorphic to the closed interval 0 1 abstract a classification is given of certain separable nuclear c algebras not necessarily of real rank zero namely the class of separable simple c algebras which are inductive limits of continuous trace c algebras whose building blocks have spectrum homeomorphic to the closed interval 0 1 or to a disjoint union of copies of this space also the range of the invariant is calculated

variation diminishing transformations a direct approach to total positivity and its statistical applications abstract karlin has shown that the variation diminishing property possessed by totally positive distributions allows a unified and straightforward presentation of many basic properties of hypothesis tests his work in fact reveals that total positivity is equivalent to a suitably formulated variation diminishing property this largely expository article begins with the latter concept and consequently gives a more direct account of the theory this approach avoids the extensive mathematical preliminaries previously required and isolates the more important statistical property

the lack of influence of the right hand side on the accuracy of linear system solution it is commonly believed that a fortunate right hand side b can significantly reduce the sensitivity of a system of linear equations ax b we show both theoretically and experimentally that this is not true when the system is solved in floating point arithmetic with gaussian elimination or the qr factorization the error bounds essentially do not depend on b and the error itself seems to depend only weakly on b our error bounds are exact rather than first order they are tight and they are stronger than the bound of chan and foulser we also present computable lower and upper bounds for the relative error the lower bound gives rise to a stopping criterion for iterative methods that is better than the relative residual this is because the relative residual can be much larger and it may be impossible to reduce it to a desired tolerance

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

backward stability with almost strictly sign regular matrices a sign regular matrix is almost strictly sign regular if all its nontrivial minors of the same order have the same strict sign these matrices form a subclass of sign regular matrices matrices whose minors of the same order have the same sign in this paper the backward stability for almost strictly sign regular matrices is studied when neville elimination with two determinant pivoting strategy is applied in addition several numerical experiments are also presented

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

classification of simple c algebras inductive limits of matrix algebras over one dimensional spaces abstract in this article we will give a complete classification of simple c algebras which can be written as inductive limits of algebras of the form a n i 1 k n m n i c x n i where x n i are arbitrary variable one dimensional compact metrizable spaces the results unify and generalize the previous results for the case x n i s 1 and for the case of x n i being trees we obtain our classification results by reducing the case of general one dimensional spaces to the case of circles the techniques in this paper play important roles in the study of the case of higher dimensional spaces

nonsingular almost strictly sign regular matrices abstract the class of nonsingular almost strictly totally positive matrices has been characterized m gasca j m pena characterizations and decompositions of almost strictly positive matrices siam j matrix anal appl 28 2006 1 8 in this paper we discuss the class of almost strictly sign regular matrices that includes almost strictly totally positive matrices a characterization is provided for these matrices in terms of their nontrivial minors using consecutive rows and consecutive columns in particular we present a characterization of certain almost strictly sign regular matrices in terms of a very reduced number of boundary almost trivial minors

simulating low precision floating point arithmetic the half precision fp16 floating point format defined in the 2008 revision of the ieee standard for floating point arithmetic and a more recently proposed half precision format bfloat16 are in

monotonicity of the cd index for polytopes we prove that the cd index of a convex polytope satisfies a strong monotonicity property with respect to the cd indices of any face and its link as a consequence we prove for d dimensional polytopes a conjecture of stanley that the cd index is minimized on the d dimensional simplex moreover we prove the upper bound theorem for the cd index namely that the cd index of any d dimensional polytope with n vertices is at most that of c n d the d dimensional cyclic polytope with n vertices

reliable solution of tridiagonal systems of linear equations in this paper we present new formulas for characterizing the sensitivity of tridiagonal systems that are independent of the condition number of the underlying matrix we also introduce efficient algorithms for solving tridiagonal systems of linear equations which are stable and reliable namely stable in the backward sense and little sensitive to perturbations in the coefficients

pivoting strategies leading to diagonal dominance by rows pivoting strategies for gaussian elimination leading to upper triangular matrices which are diagonally dominant by rows are studied forward error analysis of triangular systems whose coefficient matrices are diagonally dominant by rows is performed we also obtain small bounds of the backward errors for the pivoting strategies mentioned above our examples of matrices include h matrices and some generalizations of diagonally dominant matrices and scaled partial pivoting for the 1 norm is an example of these pivoting strategies in the case of an n times n m matrix a pivoting strategy of computational complexity o n 2 is proposed which satisfies all the results of the paper