dodgson condensation the historical and mathematical development of an experimental method abstract dodgson s condensation method has become a powerful tool in the automation of determinant evaluations in this expository paper i describe its 19 th century roots and the major steps on the path that began in the 20 th century when the iteration of an identity derived by dodgson first was studied including its role in the discovery of the alternating sign matrix conjecture the evaluation of an important 19 th century determinant in partition theory as well as a combinatorial proof of it i then discuss additional developments that have led the way to its use in modern experimental mathematics

minors of the moore penrose inverse abstract let q k n 1 k 1 1 k n denote the strictly increasing sequences of k elements from 1 n for q k n we denote by a the submatrix of a with rows indexed by columns by the submatrix obtained by deleting the rows and columns is denoted by a lsqb rsqb for nonsingular a r n n the jacobi identity relates the minors of the inverse a 1 to those of a det a 1 lsqb rsqb 1 k i 1 i k i 1 i det a lsqb rsqb det a for any q k n we generalize the jacobi identity to matrices a r m n r expressing the minors of the moore penrose inverse a in terms of the minors of the maximal nonsingular submatrices a ij of a in our notation det a lsqb rsqb 1 vol 2 a i j n det a ij a a ij for any q k m q k n 1 k r where vol 2 a denotes the sum of squares of determinants of r r submatrices of a we apply our results to questions concerning the nonnegativity of principal minors of the moore penrose inverse

a general orthogonalization technique with applications to time series analysis and signal processing a new orthogonalization technique is presented for computing the qr factorization of a general n x p matrix of full rank p n 2 p the method is based on the use of projections to solve increasingly larger subproblems recursively and has an o np2 operation count for general matrices the technique is readily adaptable to solving linear least squares problems if the initial matrix has a circulant structure the algorithm simplifies significantly and gives the so called lattice algorithm for solving linear prediction problems from this point of view it is seen that the lattice algorithm is really an efficient way of solving specially structured least squares problems by orthogonalization as opposed to solving the normal equations by fast toeplitz algorithms

computation of rational interpolants with prescribed poles a constructive proof for existence and unicity of the rational r belonging to gm v m 0 n 0 having prescribed n poles and interpolating m 1 hermite data is given it is based upon explicit computation of the confluent cauchy vandermonde determinant in terms of the nodes and the poles an algorithm to compute r z is presented it calculates this value from a triangular field of values of rational interpolants of lowest possible orders the algorithm is based upon a neville aitken interpolation formula and has arithmetical complexity 0 l l max m 1 n

some schur complement identities and applications to matrix extrapolation methods the schur complement has many applications in linear algebra and in numerical analysis schur complement identities have been used for implementing some scalar and vector extrapolation methods in this paper we define some matrix sequence transformations and give two new schur complement identities that will be used to derive algorithms for implementing these transformations

about several classes of bi orthogonal polynomials and discrete integrable systems by introducing some special bi orthogonal polynomials we derive the so called discrete hungry quotient difference dhqd algorithm and a system related to the qd type discrete hungry lotka volterra qd type dhlv system together with their lax pairs these two known equations can be regarded as extensions of the qd algorithm when this idea is applied to a higher analogue of the discrete time toda hadt equation and the quotient quotient difference qqd scheme proposed by spicer nijhoff and van der kamp two extended systems are constructed we call these systems the hungry forms of the higher analogue discrete time toda hhadt equation and the quotient quotient difference hqqd scheme respectively in addition the corresponding lax pairs are provided

the principal rank characteristic sequence of a real symmetric matrix abstract given a vector u r 2 n the principal minor assignment problem asks when is there an n n matrix having its 2 n principal minors given by u this paper explores the following related problem given a sequence r 0 r 1 r n of 0s and 1s does there exist an n n real symmetric matrix that has a principal submatrix of rank k if and only if r k 1 for all 0 k n certain conditions are shown to be necessary in order for this question to have an affirmative answer several families of matrices are constructed to attain certain classes of sequences the problem is solved completely for n 6 and for 7 n 10 in the case of sequences beginning with 010

ect b splines defined by generalized divided differences ect spline curves are generated from different local ect systems via connection matrices if they are nonsingular lower triangular and totally positive there is a basis of the space of ect splines consisting of functions having minimal compact supports normalized either to form a nonnegative partition of unity or to have integral one in this paper such ect b splines are defined by generalized divided differences this definition reduces to the classical one in case of a schoenberg space under suitable assumptions it leads to a recursive method for computing the ect b splines that reduces to the de boor mansion cox recursion in case of ordinary polynomial splines and to lyche s recursion in case of tchebycheff splines muhlbach and tang calculation of ect b splines and of ect spline curves recursively in preparation there is an ect spline space naturally adjoint to every ect spline space we also construct b splines via generalized divided differences for this space and study relations between the two adjoint spaces

shank s transformation revisited abstract a unified and self contained approach to the block structure of shank s table and its cross rules is presented wynn s regular and cordellier s singular cross rules are derived by the schur complement method in a unified manner without appealing to pade approximation moreover by extending the definition of shank s transformation to certain biinfinite sequences and by introducing a parameter it is possible to get more consistency with respect to mobius transformations it is well known that pade approximants in general don t have this property

extrapolation methods for vector sequences this is an expository paper that describes and compares five methods for extrapolating to the limit or anti limit of a vector sequence without explicit knowledge of the sequence generator the methods are the minimal polynomial extrapolation mpe studied by cabay and jackson mesina and skelboe the reduced rank extrapolation rre of eddy which we show to be equivalent to mesina s version of mpe the vector and scalar versions of the epsilon algorithm vea sea introduced by wynn and extended by brezinski and gekeler and the topological epsilon algorithm tea of brezinski we cover the derivation and error analysis of iterated versions of the algorithms as applied to both linear and nonlinear problems and we show why these versions tend to converge quadratically we also present samples from extensive numerical testing that has led us to the following conclusions a tea in spite of its role as a theoretical link between the polynomial type and the epsilon type methods has no practical appl

almost zeros of a set of polynomials of r s the notion of an exact zero of a sot of polynomials p of r s is extended to that of an almost zero the almost zeros are defined as minima of a norm function of p and their invariance under a certain equivalence relation defined on p is established the region of the complex plane which contains the prime almost zero global minimum is defined and necessary and sufficient conditions characterizing the set of almost zeros are given the role of almost zeros of p in the zero assignment problem of the combinants of p is investigated under the assumption of non assignability for p it is shown that the almost zeros act as nearly fixed zeros of the combinants every almost zero z is the centre of a disc d z r which contains at least one zero of all combinants of p the size of the radius r of this disc provides a measure of the ability of the almost zero to attract the zeros of the combinants of p a number of results for estimating upper bounds for the radius r associated with subfamilies of the ge

linear extrapolation by rational functions exponentials and logarithmic functions abstract in this paper linear extrapolation by rational functions with given poles is considered from an arithmetical point of view it is shown that the classical interpolation algorithms of lagrange neville aitken and newton which are well known for polynomial interpolation can be extended in a natural way to this problem yielding recursive methods of nearly the same complexity the proofs are based upon explicit representations of generalized vandermonde determinants which are calculated by the elimination method combined with analytical considerations as an application a regularity criterion for certain linear sequence transformations is given also by the same method simplified recurrence relations for linear extrapolation by exponentials and logarithmic functions at special knots are derived

some results on fuzzy supratopological spaces abstract in this note we study some results on the concepts of supracompactness and supracompactness in a fuzzy supratopological space we introduce the notions of hausdorffnessand lindelofness in a fuzzy supratopological space and characterize their properties interms of supracompactness we also introduce and study the concept of fuzzy supra regular space

piecewise extended chebyshev spaces given a number of extended chebyshev ec spaces on adjacent intervals all of the same dimension we join them via convenient connection matrices of the maximum order the global space is called a piecewise extended chebyshev pec space in such a space one can count the total number of zeroes of any non zero element exactly as in each ec section space when this number is bounded above in the global space the same way as in its section spaces we say that it is an extended chebyshev piecewise ecp space a thorough study of ecp spaces has been developed in the last two decades in relation to blossoms with a view to design in particular extending a classical procedure for ec spaces ecp spaces were recently proved to all be obtained by means of piecewise generalised derivatives this yields an interesting constructive characterisation of ecp spaces unfortunately except for low dimensions and for very few adjacent intervals this characterisation proved to be rather difficult to handle in practice to try to overcome this difficulty in the present article we show how to reinterpret the constructive characterisation as a theoretical procedure to determine whether or not a given pec space is an ecp space this procedure is then translated into a numerical test whose usefulness is illustrated by relevant examples

matrix pencil methodologies for computing the greatest common divisor of polynomials hybrid algorithms and their performance the computation of the greatest common divisor gcd of several polynomials is a problem that emerges in many fields of applications the gcd computation has a non generic nature and thus its numerical computation is a hard problem in this paper we examine the family of matrix pencil methods for gcd computation and investigate their performance as far as their complexity error analysis and their effectiveness for evaluating approximate solutions the relative merits of the various variants of such methods are examined for the different cases of sets of polynomials with varying number of elements and degree the developed algorithms combine symbolical and numerical programming and this is what we define here as hybrid computations the combination of numerical operations with symbolical programming can improve the nature of the methods and guarantees the stability of the algorithm furthermore it emphasizes the significance of hybrid computations in complex problems such as the computation of gcd all m

montgomery s method of polynomial selection for the number field sieve the number field sieve is the most efficient known algorithm for factoring large integers that are free of small prime factors for the polynomial selection stage of the algorithm montgomery proposed a method of generating polynomials which relies on the construction of small modular geometric progressions montgomery s method is analysed in this paper and the existence of suitable geometric progressions is considered

pole assignment of strictly proper and proper linear systems by constant output feedback a new unifying approach for the study of pole assignment by constant output feedback cpap for strictly proper and proper linear systems is presented the multilinear nature of cpap is reduced to a linear problem and to the standard multilinear problem of decomposability of multivectors the solvability of cpap thus becomes a problem of finding real intersections of a linear variety with the grassmann variety of a projective space an alternative proof to the ml n m i n are the numbers of outputs inputs states necessary and sufficient condition for generic pole assignability of strictly proper systems by complex output feedback is given the above result is extended to the ml n 1 condition for the case of proper systems stronger necessary conditions for generic pole assignment by a real output feedback are obtained by using the recently introduced invariant the plucker matrix pn such conditions are that pn must have full rank and ml n strictly proper case or ml n 1 proper case it is sh

a schur complement approach to a general extrapolation algorithm this paper is devoted to a schur complement approach to the e transformation which is the most general scalar sequence transformation known so far for accelerating the convergence a new derivation of known results on schur complements is given in the first part of the paper then schur complements and their properties are used to obtain various interpretations of the e transformation the recursive rules of the e algorithm for its implementation are also recovered new results on its kernel are derived and issues on its convergence are discussed this approach can be extended to the vector case thus leading to new vector sequence transformations

hyperbolic polynomials interlacers and sums of squares hyperbolic polynomials are real polynomials whose real hypersurfaces are maximally nested ovaloids the innermost of which is convex these polynomials appear in many areas of mathematics including optimization combinatorics and differential equations here we investigate the special connection between a hyperbolic polynomial and the set of polynomials that interlace it this set of interlacers is a convex cone which we write as a linear slice of the cone of nonnegative polynomials in particular this allows us to realize any hyperbolicity cone as a slice of the cone of nonnegative polynomials using a sums of squares relaxation we then approximate a hyperbolicity cone by the projection of a spectrahedron a multiaffine example coming from the vamos matroid shows that this relaxation is not always exact using this theory we characterize the real stable multiaffine polynomials that have a definite determinantal representation and construct one when it exists

constructing totally positive piecewise chebyshevian b spline bases abstract we consider piecewise chebyshevian splines in the sense of splines with pieces taken from any different five dimensional extended chebyshev spaces and with connection matrices at the knots in this large context we establish necessary and sufficient conditions for the existence of totally positive refinable b spline bases these conditions are applied in many important special cases e g symmetric cardinal geometrically continuous quartic b spline parametrically continuous mixed l splines the great variety of illustrations provided proves the richness of this class of splines for design this richness can be exploited in various other fields as well

revisiting totally positive differential systems a tutorial and new results abstract a matrix is called totally nonnegative tn totally positive tp if all its minors are nonnegative positive multiplying a vector by a tn matrix does not increase the number of sign variations in the vector in a largely forgotten paper binyamin schwarz 1970 considered matrices whose exponentials are tn or tp he also analyzed the evolution of the number of sign changes in the vector solutions of the corresponding linear system his work however considered only linear systems in a seemingly different line of research smillie 1984 smith 1991 and others analyzed the stability of nonlinear tridiagonal cooperative systems by using the number of sign variations in the derivative vector as an integer valued lyapunov function we show that these two research topics are intimately related this allows to derive important generalizations of the results by smillie 1984 and smith 1991 while simplifying the proofs these generalizations are particularly relevant in the context of control systems also the results by smillie and smith provide sufficient conditions for analyzing stability based on the number of sign changes in the vector of derivatives and the connection to the work of schwarz allows to show in what sense these results are also necessary we describe several new and interesting research directions arising from this new connection

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

can sylvester s determinantal identity equivalently muir s law of extensible minors be generalized abstract we show that sylvester s classical determinantal identity is equivalent to its generalizations given by the muhlbach gasca lopez carmona ramirez identity and the beckermann muhlbach identity we also show that the generating extension principles associated with each of these three identities muir s law of extensible minors muhlbach s extension principle and the extension principle given in camargo 2013 3 are equivalent and we use this fact to conclude that all these six statements and also jacobi s identity on the minors of the adjugate and cayley s law of complementaries are actually equivalent these findings lead naturally to enquire whether there would exist a generalization of sylvester s identity or an extension principle which could not be derived from sylvester s identity itself

a new proof of the cross rule for the e algorithm based on schur complements abstract matrices of the form h ge 1 f are called schur complements they are closely related to elimination procedures and to determinantal identities in this paper the quotient property for schur complements is used to establish the schur complement method a useful tool for proving certain formulas as an example a new proof of the cross rule for the algorithm is presented

grassmann invariants almost zeros and the determinantal zero pole assignment problems of linear multivariable systems the strucxpects of the rationxr spaces x c x fwhere x c is the columxof the transfer function matrix g s and x f is the space associated with right matrix fracxcriptions of g s are investigated for x f gcanonixqb s grassmgesentax g x c and g x f arxd and are shown xmplete basis free invariants for x c and x f respectively the almost zeros az and almost decoupling zeros adz of g s gdefinex local minima of a normgn defix g x c and g x f respectively the computation and certain aspects of the distribution in the complex plane of azs and adzs are examined the role of azs and adzs in the determinantal zero and pole assignment problems respectively is examined next two important families of systems are defined the strongly zero non assignable szna and the strongly pole non assignable spna systems for szna and spna systems minimal radius discs dem e z r em e z and dem f zb centred at an az and adz respectively are defined it is shown that dem e r em e z contains at least one zer

how to build all chebyshevian spline spaces good for geometric design in the present work we determine all chebyshevian spline spaces good for geometric design by chebyshevian spline space we mean a space of splines with sections in different extended chebyshev spaces and with connection matrices at the knots we say that such a spline space is good for design when it possesses blossoms to justify the terminology let us recall that in this general framework existence of blossoms defined on a restricted set of tuples makes it possible to develop all the classical geometric design algorithms for splines furthermore existence of blossoms is equivalent to existence of a b spline bases both in the spline space itself and in all other spline spaces derived from it by insertion of knots we show that chebyshevian spline spaces good for design can be described by linear piecewise differential operators associated with systems of piecewise weight functions with respect to which the connection matrices are identity matrices many interesting consequences can be drawn from the latter characterisation as an example all chebsyhevian spline spaces good for design can be built by means of integral recurrence relations

what are schur complements anyway abstract given a matrix m a b c d the schur complements of a in m are the matrices of the form s d cab where a is a generalized inverse of a we survey several recent characterizations of schur complements and discuss where they arose and how they are related

matrix representation of the shifting operation and numerical properties of the eres method for computing the greatest common divisor of sets of many polynomials the extended row equivalence and shifting eres method is a matrix based method developed for the computation of the greatest common divisor gcd of sets of many polynomials in this paper we present the formulation of the shifting operation as a matrix product which allows us to study the fundamental theoretical and numerical properties of the eres method by introducing its complete algebraic representation then we analyse in depth its overall numerical stability in finite precision arithmetic numerical examples and comparison with other methods are also presented

fuzzy sets and type 2 under algebraic product and algebraic sum abstract the concept of fuzzy sets of type 2 has been proposed by l a zadeh as an extension of ordinary fuzzy sets a fuzzy set of type 2 can be defined by a fuzzy membership function the grade or fuzzy grade of which is taken to be a fuzzy set in the unit interval 0 1 rather than a point in 0 1 this paper investigates the algebraic properties of fuzzy grades that is fuzzy sets of type 2 under the operations of algebraic product and algebraic sum which can be defined by using the concept of the extension principle and shows that fuzzy grades under these operations do not form such algebraic structures as a lattice and a semiring moreover the properties of fuzzy grades are also discussed in the case where algebraic product and algebraic sum are combined with the well known operations of join and meet for fuzzy grades and it is shown that normal convex fuzzy grades form a lattice ordered semigroup under join meet and algebraic product

recursive interpolation extrapolation and projection abstract the recursive projection algorithm derived in a previous paper is related to several well known methods of numerical analysis such as the conjugate gradient method rosen s method and henrici s it is connected with the general interpolation problem with extrapolation methods with orthogonal projection on a subspace and with fourier expansions several other connections and applications are presented

on the almost principal minors of a symmetric matrix abstract the almost principal rank characteristic sequence apr sequence of an n n symmetric matrix is introduced which is defined to be the string a 1 a 2 a n 1 where a k is either a s or n according as all some but not all or none of its almost principal minors of order k are nonzero in contrast to the other principal rank characteristic sequences in the literature the apr sequence of a matrix does not depend on principal minors the almost principal rank of a symmetric matrix b denoted by ap rank b is defined as the size of a largest nonsingular almost principal submatrix of b a complete characterization of the sequences not containing an a that can be realized as the apr sequence of a symmetric matrix over a field f is provided a necessary condition for a sequence to be the apr sequence of a symmetric matrix over a field f is presented it is shown that if b f n n is symmetric and non diagonal then rank b 1 ap rank b rank b with both bounds being sharp moreover it is shown that if b is symmetric non diagonal and singular and does not contain a zero row then rank b ap rank b

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

on rational b splines with prescribed poles spaces of rational splines of maximal smoothness are considered which are constructed from certain rational functions with prescribed poles for them a basis of splines having minimal compact supports was constructed in the literature these functions which are called rational b splines are obtained by solving certain linear equations with a block matrix depending on a parameter e 0 it is shown that in the limit e 0 they tend to certain polynomial b splines

a generalization of sylvester s identity on determinants and some applications abstract a generalization of sylvester s identity on determinants is proved by elimination techniques as an application some inequalities for determinants of totally positive or positive definite matrices are derived

a conjecture based on somos 4 sequence and its extension abstract in two recent papers by barry 2010 29 and 2011 30 it is conjectured that somos 4 admits a solution expressed in terms of hankel determinant with its elements satisfying a convolution recursion relation in this paper barry s conjecture on somos 4 is firstly confirmed actually we present a more generalized result the proof is mainly based on new findings on properties for so called block hankel determinants the method can also be used to prove another conjecture proposed by michael somos which has been solved by guoce xin

a direct method for evaluating some nice hankel determinants and proofs of several conjectures abstract in this paper we propose a direct method to evaluate hankel determinants for some generating functions satisfying a certain type of quadratic equations which cover generating functions of catalan numbers motzkin numbers and schroder numbers additionally four recent conjectures proposed by cigler 2011 3 are proved

on lossless feedback delay networks lossless feedback delay networks fdns are commonly used as a design prototype for artificial reverberation algorithms the lossless property is dependent on the feedback matrix which connects the output of a set of delays to their inputs and the lengths of the delays both unitary and triangular feedback matrices are known to constitute lossless fdns however the most general class of lossless feedback matrices has not been identified in this contribution it is shown that the fdn is lossless for any set of delays if all irreducible components of the feedback matrix are diagonally similar to a unitary matrix the necessity of the generalized class of feedback matrices is demonstrated by examples of fdn designs proposed in literature

generalised resultants dynamic polynomial combinants and the minimal design problem the theory of dynamic polynomial combinants is linked to the linear part of the dynamic determinantal assignment problems dap which provides the unifying description of the dynamic as well as static pole and zero dynamic assignment problems in linear systems the assignability of spectrum of polynomial combinants provides necessary conditions for solution of the original dap this paper demonstrates the origin of dynamic polynomial combinants from linear systems examines issues of their representation and the parameterisation of dynamic polynomial combinants according to the notions of order and degree and examines their spectral assignment central to this study is the link of dynamic combinants to the theory of generalised resultants which provide the matrix representation of the dynamic combinants the paper considers the case of coprime set of polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above an

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

multiply refined enumeration of alternating sign matrices four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices asms specifically these statistics are the positions of the 1 s in the first and last row and column of an asm and the numbers of generalized inversions and 1 s in an asm previously known results for the exact enumeration of asms with prescribed values of some of these statistics are reviewed in detail a quadratic relation which fully determines the generating function associated with all six statistics is then obtained the derivation of the relation involves combining the desnanot jacobi determinant identity with the izergin korepin formula for the partition function of the six vertex model with domain wall boundary conditions

an algorithm for a special case of a generalization of the richardson extrapolation process a special case of a generalization of the richardson extrapolation process is considered and its complete solution is given in closed form using this an algorithm for implementing the extrapolation is devised it is shown that this algorithm needs a very small amount of arithmetic operations and very little storage convergence and stability properties for some cases are also considered

convergence acceleration methods the past decade abstract the aim of this paper is to present a review of the most significant results obtained the past ten years on convergence acceleration methods it is divided into four sections dealing respectively with the theory of sequence transformations the algorithms for such transformations the practical problems related to their implementation and their applications to various subjects of numerical analysis

irreducibility of induced supermodules for general linear supergroups abstract in this note we determine when an induced supermodule h g 0 corresponding to a dominant integral highest weight of the general linear supergroup g g l m n is irreducible using the contravariant duality given by the supertrace we obtain a characterization of irreducibility of weyl supermodules v this extends the result of kac 12 13 who proved that for ground fields of characteristic zero v is irreducible if and only if is typical

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

computation of the gcd of polynomials using gaussian transformations and shifting abstract a new numerical method for the computation of the greatest common divisor gcd of an m set of polynomials of r s pm d of maximal degree d is presented this method is based on a recently developed theoretical algorithm karcanias 1987 that uses elementary transformations and shifting operations the present algorithm takes into account the non generic nature of gcd and thus uses steps which minimize the introduction of additional errors and defines the gcd in an approximate sense for a given set pm d with a basis matrix pm the method defines first the most orthogonal uncorrupted base pr from the rows of pm where r rank pm m by applying successively gaussian transformations and shifting on the basis matrix pr e rr d 1 we produce each time a new basis matrix pz with z rank pz r the method terminates when the rank of pz is approximately equal to 1 the coefficient vector of the gcd is then defined as a row of the unit rank matrix pz the method defines the exact degree of t

elimination techniques and interpolation a general recurrence interpolation formula due to gasca and lopez carmona j approx th 34 1982 is studied by means of elimination techniques in linear systems the repeated application of this formula is investigated in order to show its equivalence with extrapolation methods

q difference and confluent forms of the lattice boussinesq equation and the relevant convergence acceleration algorithms a q difference lattice boussinesq equation and its confluent form are proposed with the help of determinant identity their molecule solutions are constructed it is shown that the q difference lattice boussinesq equation can be used as a numerical convergence acceleration algorithm to compute the approximation limt f t numerical examples with the application of this algorithm are presented

an invitation to topological structures of first order genuine sets the aim of this note is to construct several topological structures on first order genuine sets and it contains only the first steps to the topological properties on first order genuine sets

an extended multistep shanks transformation and convergence acceleration algorithm with their convergence and stability analysis the molecule solution of an extended discrete lotka volterra equation is constructed from which a new sequence transformation is proposed a convergence acceleration algorithm for implementing this sequence transformation is found it is shown that our new sequence transformation accelerates some kinds of linearly convergent sequences and factorially convergent sequences with good numerical stability some numerical examples are also presented

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

other manifestations of the schur complement abstract many series and sequence transformations used in numerical analysis are defined as ratios of determinants they are implemented by recursive algorithms based on determinantal identities the aim of this paper is to study the connections of these transformations and algorithms with the schur complement of a matrix the schur s formula is extended to the vector case thus providing the same treatment for vector sequence transformations and the corresponding recursive algorithms thanks to these connections particular rules for avoiding division by zero or numerical instability are obtained for these algorithms some fixed point methods and continued fractions also fit in this framework

necessary and sufficient conditions for zero assignment by constant squaring down the problem of zero assignment by constant squaring down czap is studied for minimal systems described by a transfer function g s rm l s m l using tools from exterior algebra and algebraic geometry conditions for its solvability are given in terms of the numbers m l the forney dynamical order and the rank of the plucker matrix p for the cases l 1 or l m 1 it is shown that g s is completely zero assignable cza if and only if 1 for the cases l 1 m 1 it is proved that g s is cza or generically zero assignable gza under a complex squaring down if and only if l m l 1 and 1 the latter conditions also provide necessary conditions for the existence of a real solution for a generic g s sufficient conditions for solvability of czap by a real squaring down are that l m l 1 and that the number g a0 al 1 1 a0 al 1 i j ai aj is odd for some set a0 al 1 satisfying 1 l 1i 0ai l l 1 2 and 0 a0 a1 al 1 m 1 apart from the existence results the present approach also allows the computation of the solutions whenever they exist finally it is proved that the only fixed zeros of czap are the zeros of g s and a procedure for computing the solutions of czap whenever such solutions exist based on an optimization problem is given

composite sequence transformations the aim of this work is to introduce the new concept of composite sequence transformations and to show by very simple examples and theorems that it can be useful in accelerating the convergence of sequences generalizations of classical transformations and results are obtained

normal factorisation of polynomials and computational issues abstract this paper introduces the notion of normal factorisation of polynomials and then presents a new numerical algorithm for the computation of this factorisation this procedure avoids computation of roots of polynomials and it is based on the use of algorithms determining the greatest common divisor of polynomials in this paper the general aspects of the algorithm are discussed and a symbolic implementation is given the theoretical algorithm provides the basis for a numerical procedure the general aspects of which are also discussed here the advantage of such a factorisation is that it handles the determination of multiplicities and produces factors of lower degree and with distinct roots for such polynomials robust numerical techniques based on finding roots may then be used if it is desired to work out the usual irreducible factorisation a detailed description of the implementation of the algorithm is presented the problem considered here is an integral part of computations for algebraic control problems

inverse m matrix inequalities and generalized ultrametric matrices abstract we use weighted directed graphs to introduce a class of nonnegative matrices which under a simple condition are inverse m matrices we call our class the generalized ultrametic matrices since it contains the class of symmetric ultrametric matrices and some unsymmetric matrices we show that a generalized ultrametric matrix is the inverse of a row and column diagonally dominant m matrix if and only if it contains no zero row and no two of its rows are identical this theorem generalizes the known result that a symmetric strictly ultrametric matrix is the inverse of a strictly diagonally dominant m matrix we also present inequalities and conditions for equality among the entries of the inverse of a row diagonally dominant m matrix some of these inequalities and conditions for equality generalize results of stieltjes on inverses of symmetric diagonally dominant m matrices

decentralized determinantal assignment problem fixed and almost fixed modes and zeros the decentralized determinantal assignment problem ddap is defined as the unifying description for the study of pole and zero assignment problems under decentralized output state feedback dof dsf and decentralized squaring down dsd respectively ddap is reduced to a linear problem of zero assignment of polynomial combinants and a multilinear problem of restricted decomposability of multivectors the decentralization characteristic dc and the decentralized polynomial grassmann representative d s gr of ddap are defined the fixed zero polynomial of ddap is then determined as the zero polynomial of d s gr the canonical d s gr cd r s gr and the decentralized plucker matrix dpm of ddap are introduced and necessary conditions for arbitrary assignment of the non fixed zeros are given in terms of the dpm the family of strongly zero non assignable sna systems is defined and for such systems the notion of the fixed zero is extended to that of the almost fixed zero afz an

full rank representation of generalized inverse at s 2 and its application in this paper we introduce a full rank representation of the generalized inverse a t s 2 of a given complex matrix a which is based on an arbitrary full rank decomposition of g where g is a matrix such that r g t and n g s using this representation we introduce the minor of the generalized inverse a t s 2 as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained as an application we use an example to demonstrate that this representation is correct

the approximation of a totally positive band matrix by a strictly banded totally positive one every nonsingular totally positive m banded matrix is shown to be the product of m totally positive one banded matrices and therefore the limit of strictly m banded totally positive matrices this result is then extended to bi infinite m banded totally positive matrices with linearly independent rows and columns in the process such matrices are shown to possess at least one diagonal whose principal sections are all nonzero as a consequence such matrices are seen to be approximable by strictly m banded totally positive ones

hyperdeterminantal relations among symmetric principal minors abstract the principal minors of a symmetric n n matrix form a vector of length 2 n we characterize these vectors in terms of algebraic equations derived from the 2 2 2 hyperdeterminant

piecewise polynomial spaces and geometric continuity of curves we say that a curve has geometric continuity if its curvatures and frenet frame are continuous in this paper we introduce spaces of piecewise polynomials which can be used to model space curves which have geometric continuity we show that the basic theoretical properties of ordinary spline functions also hold for these spaces these results extend and unify recent work on beta splines and nu splines which are used as a design tool in computer aided geometric design of free form curves and surfaces

invariance properties and characterization of the greatest common divisor of a set of polynomials on m sets of polynomials from r s of degree d x1d4c5 m d the notion of extended r equivalence emd under x1d4c5 m d is characterized by a complete invariant and a canonical form it is shown that the greatest common divisor g c d of x1d4c5 m d is an invariant of x1d4c5 m d for proper sets x1d4c5 m d it is also shown that the g c d is invariant under the combined action of e r e transformations and shifting operations finally a systematic procedure for the computation of the g c d of x1d4c5 mdis given this procedure is based on the use of elementary row operations on real matrices and shifting operations

nearest common root of a set of polynomials a structured singular value approach abstract the paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients in particular we seek the minimum magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root it is shown that the problem is equivalent to the solution of a structured singular value problem arising in robust control for which numerous techniques are available it is also shown that the method can be extended to the calculation of an approximate gcd of fixed degree by introducing the notion of the generalized structured singular value of a matrix the work generalizes previous results by the authors involving the calculation of the approximate gcd of two polynomials although the general case considered here is considerably harder and relies on a matrix dilation approach and several preliminary transformations

chio s and dodgson s determinantal identities abstract in this expository paper we analyze and compare two determinantal identities constructed in the mid nineteenth century chio s and dodgson s from the perspective of their origins in earlier work by j j sylvester and c g j jacobi all the known proofs of chio s and dodgson s identities are cited and we present and compare them as modern algorithms for evaluating determinants restated as a recurrence we discuss the role of dodgson s identity in the development of the alternating sign matrix conjecture in the late twentieth century

squaring down and the problems of almost zeros for continuous time systems abstract in this paper we construct precompensators which square down a system such that the resulting square system has the same unstable zeros and system gains we derive the minimal gain of the precompensator to achieve such a square system if the gain of this precompensator is too large then we derive an explicit trade off between the gain of the precompensator and the number of unstable zeros we allow the precompensator to introduce in the system

cauchy binet for pseudo determinants abstract the pseudo determinant det a of a square matrix a is defined as the product of the nonzero eigenvalues of a it is a basis independent number which is up to a sign the first nonzero entry of the characteristic polynomial of a we prove det f t g p det f p det g p for any two n m matrices f g the sum to the right runs over all k k minors of a where k is determined by f and g if f g is the incidence matrix of a graph this directly implies the kirchhoff tree theorem as l f t g is then the laplacian and det 2 f p 0 1 is equal to 1 if p is a rooted spanning tree a consequence is the following pythagorean theorem for any self adjoint matrix a of rank k one has det 2 a p det 2 a p where det a p runs over k k minors of a more generally we prove the polynomial identity det 1 x f t g p x p det f p det g p for classical determinants det which holds for any two n m matrices f g and where the sum on the right is taken over all minors p understanding the sum to be 1 if p 0 it implies the pythagorean identity det 1 f t f p det 2 f p which holds for any n m matrix f and sums again over all minors f p if applied to the incidence matrix f of a finite simple graph it produces the chebotarev shamis forest theorem telling that det 1 l is the number of rooted spanning forests in the graph with laplacian l

matrix extrapolation algorithms abstract extrapolation algorithms are used for accelerating scalar or vector sequences they are also used for solving systems of linear and nonlinear equations these algorithms are expressed in terms of a ratio of two determinants as the e algorithm and the general recursive projection algorithm grpa in this paper we define the matrix extrapolation problem and we use the schur complement and the sylvester identity for solving this problem we give two transformations equivalent to the e algorithm and the grpa for the matrix case we give also a characterization of their kernels

a general determinantal identity of sylvester type and some applications abstract a general determinantal identity of sylvester type over arbitrary commutative fields is derived while its proof is rather short and conceptually simpler than earlier attempts the result contains sylvester s classical determinantal identity and more recent extensions as well as some old and some new determinantal identities an application to the e algorithm is added

algebraic linearizations of matrix polynomials abstract we show how to construct linearizations of matrix polynomials z a z d 0 c 0 a z b z a z b z when deg b z deg a z and z a z d 0 b z c 0 from linearizations of the component parts a z and b z this allows the extension to matrix polynomials of a new companion matrix construction

on extending determinantal identities abstract a new principle for extending determinantal identities is established which generalizes muir s classical law of extensible minors the proof makes use of general elimination strategies and of generalized schur complements this principle allows either the list of columns or the list of rows extending the corresponding list in the given identity to depend on the latter as applications of this technique karlin s and a generalization of sylvester s identity are derived

two composition methods for solving certain systems of linear equations this note is concerned with the following problem given a systema x b of linear equations and knowing that certains of its subsystemsa 1 x 1 b 1 a m x m b m can be solved uniquely what can be said about the regularity ofa and how to find the solutionx fromx 1 x m this question is of particular interest for establishing methods computing certain linear or quasilinear sequence transformations recursively 7 13 15 in dieser arbeit wird die folgende fragestellung behandelt wenn man wei da gewisse teilsystemea 1 x 1 b 1 a m x m b m eines linearen gleichungssystemsa x b eindeutig losbar sind was kann man uber die losbarkeit des systemsa x b sagen und wie kann man gegebenenfallsx ausx 1 x m berechnen die losung dieses problems erlaubt die rekursive berechnung gewisser linearer oder quasilinearer folgentransformationen 7 13 15

the hankel pencil conjecture abstract the toeplitz pencil conjecture stated in w schmale p k sharma problem 30 3 singularity of a toeplitz matrix image 30 2003 w schmale p k sharma cyclizable matrix pairs over c x and a conjecture on toeplitz pencils linear algebra appl 389 2004 33 42 is equivalent to a conjecture for n n hankel pencils of the form h n x c i j n 1 where c 0 x is an indeterminate c l 0 for l 0 and c l c c 0 for l 1 in this paper it is shown to be implied by another conjecture which we call the root conjecture the root conjecture asserts a strong relationship between the roots of certain submaximal minors of h n x specialized to have c 1 c 2 1 we give explicit formulae in the c i for these minors and prove the root conjecture for minors m nn m n 1 n of degree 6 this implies the hankel pencil conjecture for matrices up to size 8 8 the main tools involved are a partial parametrization of the set of solutions of systems of polynomial equations that are both homogeneous and index sum homogeneous and use of the sylvester identity for matrices

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

principal minors and rhombus tilings the algebraic relations between the principal minors of a generic n n matrix are somewhat mysterious see e g lin and sturmfels 2009 j algebra 322 4121 31 we show however that by adding in certain almost principal minors the ideal of relations is generated by translations of a single relation the so called hexahedron relation which is a composition of six cluster mutations we give in particular a laurent polynomial parameterization of the space of n n matrices whose parameters consist of certain principal and almost principal minors the parameters naturally live on vertices and faces of the tiles in a rhombus tiling of a convex gon a matrix is associated to an equivalence class of tilings all related to each other by yang baxter like transformations by specializing the initial data we can similarly parameterize the space of hermitian symmetric matrices over or the quaternions moreover by further specialization we can parametrize the space of positive definite matrices over these rings this article is part of a special issue of journal of physics a mathematical and theoretical devoted to cluster algebras mathematical physics

new proofs of two representations and minor of generalized inverse a 2 t s abstract in this paper we prove the limit representation for the generalized inverse a t s 2 of a given complex matrix a in a simpler way then using this representation two known results the generalized inverse a t s 2 as an analogs of convex combination with the minor of adja and the representation of the minor of the generalized inverse a t s 2 were derived anew as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained

the generalized sine theorem and inequalities for simplices abstract for an n dimensional simplex in en we establish a sine theorem and some inequalities for k order vertex angles which are defined in this paper to prove these results we give a formula about the inverse matrix of the metric matrix of an n parallelotope

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