inverse numerical range and determinantal representation abstract the numerical range w a of an n n matrix a is the collection of quadratic forms z a over unit sphere 1 the inverse numerical range problem aims to find a unit vector which corresponds to a given point z of the numerical range w a given a matrix a the helton vinnikov theorem produces a symmetric matrix s so that a and s have the same numerical range w a w s in this paper we investigate the inverse numerical range problem for the boundary points and points on the boundary generating curve of the numerical range in place of the construction of unit vectors satisfying z a we express the kernel vector function of the linear pencil x s y s z i n as a function on the abel jacobi variety of the associated elliptic curve of a the kernel function plays a key role for the inverse numerical problem we perform this process when s is a generic 3 3 symmetric matrix

randomness and recursive enumerability one recursively enumerable real alpha dominates another one beta if there are nondecreasing recursive sequences of rational numbers a n n in omega approximating alpha and b n n in omega approximating beta and a positive constant c such that for all n c alpha a n geq beta b n see r m solovay draft of a paper or series of papers on chaitin s work manuscript ibm thomas j watson research center yorktown heights ny 1974 p 215 and g j chaitin ibm j res develop 21 1977 pp 350 359 we show that every recursively enumerable random real dominates all other recursively enumerable reals we conclude that the recursively enumerable random reals are exactly the omega numbers g j chaitin ibm j res develop 21 1977 pp 350 359 second we show that the sets in a universal martin lof test for randomness have random measure and every recursively enumerable random number is the sum of the measures represented in a universal martin lof test

containment regions for zeros of polynomials from numerical ranges of companion matrices containment regions for the zeros of a monic polynomial are given with the aid of results for containment regions for the numerical range of certain bordered diagonal matrices which are applied to different types of companion matrices of the polynomial

construction of unitary and normal companion matrices abstract for a polynomial f z a0 a1z an 1zn 1 zn a0 an 1 c with complex unimodular zeros we present two different methods to construct by a finite number of rational operations and square roots a unitary companion matrix i e a unitary matrix whose characteristic polynomial is 1 nf z we also show that rational operations alone cannot suffice finally we discuss the problem of constructing a normal companion matrix for a polynomial without any restrictions on its zeros and arrive at a negative answer

when does randomness come from randomness a result of shen says that if f 2 n 2 n is an almost everywhere computable measure preserving transformation and y 2 n is martin lof random then there is a martin lof random x 2 n such that f x y answering a question of bienvenu and porter we show that this property holds for computable randomness but not schnorr randomness these results combined with other known results imply that the set of martin lof randoms is the largest subset of 2 n satisfying this property and also satisfying randomness conservation if f 2 n 2 n is an almost everywhere computable measure preserving map and if x 2 n is random then f x is random

a subclass of symmetric loewner matrices abstract the subclass of real symmetric loewner matrices with fixed interpolation nodes and row sums zero is studied it is shown that it forms a commutative semigroup with respect to multiplication and contains with each element its moore penrose inverse as well

iteration schemes for the divide and conquer eigenvalue solver we propose globally convergent iteration schemes for updating the eigenvalues of a symmetric matrix after a rank 1 modification such calculations are the core of the divide and conquer technique for the symmetric tridiagonal eigenvalue problem we prove the superlinear convergence right from the start of our schemes which allows us to improve the complexity bounds of 3 the effectiveness of our algorithms is confirmed by numerical results which are reported and discussed

singular points of the algebraic curves of symmetric hyperbolic forms abstract we investigate singular points of the algebraic curve defined by a hyperbolic form satisfying some symmetric properties we show that all singular points of such a curve are real nodes

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

method for finding multiple roots of polynomials conventional numerical methods for finding multiple roots of polynomials are inaccurate the accuracy is unsatisfactory because the derivatives of the polynomial in the intermediate steps of the associated root finding procedures are eliminated engineering applications require that this problem be solved this work presents an easy to implement method that theoretically completely resolves the multiple root issue the proposed method adopts the euclidean algorithm to obtain the greatest common divisor gcd of a polynomial and its fast derivative the gcd may be approximate because of computational inaccuracy the multiple roots are then deflated into simple ones and then determined by conventional root finding methods the multiplicities of the roots are accordingly calculated a detailed derivation and test examples are provided to demonstrate the efficiency of this method

structural and computational properties of possibly singular semiseparable matrices a classical result of structured numerical linear algebra states that the inverse of a nonsingular semiseparable matrix is a tridiagonal matrix such a property of a semiseparable matrix has been proved to be useful for devising linear complexity solvers for establishing recurrence relations among its columns or rows and moreover for efficiently evaluating its characteristic polynomial in this paper we provide sparse structured representations of a semiseparable matrix a which hold independently of the fact that a is singular or not these relations are found by pointing out the band structure of the inverse of the sum of a plus a certain sparse perturbation of minimal rank further they can be used to determine in a computationally efficient way both a reflexive generalized inverse of a and its characteristic polynomial

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

a note on companion matrices abstract we show that the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices n 1 of them being the identity matrix in which a 2 2 identity submatrix in two consecutive rows and columns is replaced by an appropriate 2 2 matrix the remaining being the identity matrix with the last entry replaced by possibly different entry by a certain similarity transformation we obtain a simple new companion matrix in a pentadiagonal form some generalizations are also possible

a qr method for computing the singular values via semiseparable matrices the standard procedure to compute the singular value decomposition of a dense matrix first reduces it into a bidiagonal one by means of orthogonal transformations once the bidiagonal matrix has been computed the qr method is applied to reduce the latter matrix into a diagonal one in this paper we propose a new method for computing the singular value decomposition of a real matrix in a first phase an algorithm for reducing the matrix a into an upper triangular semiseparable matrix by means of orthogonal transformations is described a remarkable feature of this phase is that depending on the distribution of the singular values after few steps of the reduction the largest singular values are already computed with a precision depending on the gaps between the singular values an implicit qr method for upper triangular semiseparable matrices is derived and applied to the latter matrix for computing its singular values the numerical tests show that the proposed method can compete with the standard method using an intermediate bidiagonal matrix for computing the singular values of a matrix

bounds and majorization relations for the critical points of polynomials abstract we apply several matrix inequalities to the derivative companion matrices of complex polynomials to establish new bounds and majorization relations for the critical points of these polynomials in terms of their zeros

pseudozeros of polynomials and pseudospectra of companion matrices it is well known that the zeros of a polynomial p are equal to the eigenvalues of the associated companion matrix a in this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding the epsilon pseudozero set z epsilon p is the set of zeros of all polynomials hat p obtained by coefficientwise perturbations of p of size leq epsilon this is a subset of the complex plane considered earlier by mosier and is bounded by a certain generalized lemniscate the epsilon pseudospectrum lambda epsilon a is another subset of bbb c defined as the set of eigenvalues of matrices hat a a e with vert e vert leq epsilon it is bounded by a level curve of the resolvent of a we find that if a is first balanced in the usual eispack sense then z epsilon vert p vert p and lambda epsilon vert a vert a are usually quite close to one another it follows that the matlab roots algorithm of balancing the companion matrix then computing its eigenvalues is a stable algorithm for polynomial zerofinding experimental comparisons with the jenkins traub imsl and madsen reid harwell fortran codes confirm that these three algorithms have roughly similar stability properties

sturm and sylvester algorithms revisited via tridiagonal determinantal representations first we show that sturm algorithm and sylvester algorithm which compute the number of real roots of a given univariate polynomial lead to two dual tridiagonal determinantal representations of the polynomial next we show that the number of real roots of a polynomial given by a tridiagonal determinantal representation is greater than the signature of this representation

companion matrix patterns abstract companion matrices especially the frobenius companion matrices are used in algorithms for finding roots of polynomials and are also used to find bounds on eigenvalues of matrices in 2003 fiedler introduced a larger class of companion matrices that includes the frobenius companion matrices one property of the combinatorial pattern of these companion matrices is that up to diagonal similarity they uniquely realize every possible spectrum of a real matrix we characterize matrix patterns that have this property and consequently introduce more companion matrix patterns we observe that each fiedler companion matrix is permutationally similar to a unit hessenberg matrix we provide digraph characterizations of the classes of patterns described and in particular all sparse companion matrices noting that there are companion matrices that are not sparse

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

on algebras of symmetric loewner matrices abstract we present a family of algebras of real symmetric loewner matrices and discuss their algebraic and structure properties

mathematical metaphysics of randomness abstract we consider various mathematical refinements of the notion of randomness of an infinite sequence and relations between them on the base of the game approach a new possible refinement of the notion of randomness is introduced unpredictability the case of finite sequences is also considered

randomness computability and density we study effectively given positive reals more specifically computably enumerable reals under a measure of relative randomness introduced by solovay manuscript ibm thomas j watson research center yorktown heights ny 1975 and studied by calude hertling khoussainov and wang theoret comput sci 255 2001 pp 125 149 calude theoret comput sci 271 2002 pp 3 14 kucera and slaman siam j comput 31 2002 pp 199 211 and downey hirschfeldt and laforte mathematical foundations of computer science 2001 springer verlag berlin 2001 pp 316 327 among others this measure is called domination or solovay reducibility and is defined by saying that alpha dominates beta if there are a constant c and a partial computable function varphi such that for all positive rationals q alpha we have varphi q downarrow beta and beta varphi q leqslant c alpha q the intuition is that an approximating sequence for alpha generates one for beta whose rate of convergence is not much slower than that of the original sequence it is not hard to show that if alpha dominates beta then the initial segment complexity of alpha is at least that of beta in this paper we are concerned with structural properties of the degree structure generated by solovay reducibility we answer a natural question in this area of investigation by proving the density of the solovay degrees we also provide a new characterization of the random computably enumerable reals in terms of splittings in the solovay degrees specifically we show that the solovay degrees of computably enumerable reals are dense that any incomplete solovay degree splits over any lesser degree and that the join of any two incomplete solovay degrees is incomplete so that the complete solovay degree does not split at all the methodology is of some technical interest since it includes a priority argument in which the injuries are themselves controlled by randomness considerations

process and truth table characterisations of randomness this paper uses quick process machines to provide characterisations of computable randomness schnorr randomness and weak randomness the quick process machine is a type of process machine first considered in work of levin and zvonkin a new technique for building process machines and quick process machines is presented this technique is similar to the kc theorem for prefix free machines using this technique a method of translating computable martingales to quick process machines is given this translation forms the basis for these new randomness characterisations quick process machines are also used to provide characterisations of computable randomness schnorr randomness and weak randomness in terms of truth table reducibility

polynomial zerofinding iterative matrix algorithms abstract newbery s method is completed to a method for the construction of a complex symmetric or nonsymmetric matrix with a given characteristic polynomial the methods of fiedler schmeisser and dorfler and schmeisser for similar constructions of symmetric matrices are reviewed polynomials found in the literature are solved iteratively by one of fiedler s methods with initial values supplied either by schmeisser s method or taken on a large circle or randomly in a region of the complex plane the determinental equations are solved by the qr algorithm fiedler s method used iteratively exhibits fast convergence to simple roots even in the presence of multiple roots if at some iteration step the values of the iterates which are converging to a multiple root are averaged according to the hull mathon procedure then fast convergence is also attained for multiple roots this combination appears to have nice features for polynomials of small to moderate degree

on kurtz randomness kurtz randomness is a notion of algorithmic randomness for real numbers in particular a real x is called kurtz random or weakly random iff it is contained in every computably enumerable set u of lebesgue measure 1 we prove a number of characterizations of this notion relating it to other notions of randomness such as the well known notions of computable randomness martin lof randomness and schnorr randomness for the first time we give machine characterizations of kurtz randomness whereas the turing degree of every martin lof random c e real is the complete degree and the degrees of schnorr random c e reals are all high we show that kurtz random c e reals occur in every non zero c e degree additionally we show that the sets that are low for kurtz randomness are all hyperimmune and include those that are low for schnorr randomness characterized previously by terwijn and email protected 2004 elsevier b v all rights reserved

fast inversion algorithms for a class of structured operator matrices we study linear complexity inversion algorithms for diagonal plus semiseparable operator matrices applications to integral and differential equations and block matrices are obtained comparison of different algorithms is performed complexity choice of parameters and results of numerical experiments are analyzed

polynomial roots the ultimate answer we show that it is always possible to transform the problem of tinding the mots of a generic polynomial to the problem of determining the eigenvalues of tridiagonal matrices having only simple eigenvalues since this problem is very efficiently solved for example with the familiar qr method it seems that the present veiy simple approach bas the potentiality to supplant the existing ones

strong reductions in effective randomness we study generalizations of demuth s theorem which states that the image of a martin lof random real under a tt reduction is either computable or turing equivalent to a martin lof random real we show that demuth s theorem holds for schnorr randomness and computable randomness answering a question of franklin but that it cannot be strengthened by replacing the turing equivalence in the statement of the theorem with wtt equivalence we also provide some additional results about the turing and tt degrees of reals that are random with respect to some computable measure

modified gershgorin disks for companion matrices all the zeros of a polynomial are contained in the union of gershgorin disks derived from its companion matrix a consequence of gershgorin s theorem however this theorem does not exploit the structure of the companion matrix we will use this structure to obtain smaller zero inclusion regions thereby providing some nonstandard results to accompany and illustrate this frequently covered topic in numerical and matrix analysis

polynomials and hankel matrices abstract compatibility of a hankel n n matrix h and a polynomial f of degree m m n is defined if m n compatibility means that hc t f c f h where c f is t companion matrix of f with a suitable generalization of c f this theorem is generalized to the case that m n

polynomial roots from companion matrix eigenvalues in classical linear algebra the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial an algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition we derive a first order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm our error analysis expands on work by van dooren and dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully regarding the stronger concept of a small componentwise backward error our analysis predicts a small such error on a test suite of eight random polynomials suggested by toh and trefethen however we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice we extend our results to polynomial matrices where the result is essentially the same but the geometry becomes more complicated

error estimates of finite element method about elliptic problems with singular righthand side abstract in this paper on basis of other related works see r scott optimal l estimates for the finite element method on irregular meshes math comp v 30 1976 681 697 qun lin qi ding zhu the preprocessing and postprocessing for fem shanghai science press china 1994 i babuska a miller the post processing in the finite element method parts i ii internat j numer methods engrg 20 1984 1085 1109 1111 1129 we will continue to study the theory of fem for elliptic problems with singular righthand side the main difference between this paper and other related works see scott 1976 lin and zhu 1994 babuska and miller 1984 is that it will present some local estimates for fem of elliptic problems with singular righthand side of course some numerical examples will be given to investigate those error estimates

linear construction of companion matrices this note is concerned with the following problem for a given matrix a e c ix and a vector a e c does there exist a mapping 2 assigning to each manic polynomial f of degree n a vector x f e c such that the matrix b a a x f is a companion matrix off i e the characteristic polynomial of b is t l ftf the classes of suitable matrices a and vectors a are characterized and some properties of b are described the corresponding unique mapping is determined by a system of linear equations the cases of a triangular bidiagonal or diagonal matrix a are discussed explicitly and many known companion matrices are obtained as particular cases then gershgorin s theorem is applied yielding error estimates for polynomial roots finally the extension to block companion matrices and an example of nonlinear construction are discussed

a real symmetric tridiagonal matrix with a given characteristic polynomial abstract given a polynomial u x x n a n 1 x n 1 a 0 a v r v 0 1 n 1 having only real zeros we construct a real symmetric tridiagonal matrix whose characteristics polynomial is equal to 1 n u x this is a complete solution to a problem raised and partly solved by m fiedler

some decomposition results for companion matrices let c p denote the frobenius companion matrix of the monic polynomial p with complex coefficients the singular value decomposition and the qr decomposition of c p are computed

a constrained eigenvalue problem in this paper we consider the following mathematical and computational problem given the quantities a n m by n m matrix symmetric n 0 n n m by m matrix with full rank t vector of dimension m with nt t 1 determine an x such that operatorname x t ax min subject to the constraints n t x t i x t x 1 ii variants of this problem occur in many applications 1 5 7 8 11 the problem has been studied previously when t 0 the null vector cf 4 6 when t 0 then the problem becomes more complicated

comparing notions of randomness it is an open problem in the area of effective algorithmic randomness whether kolmogorov loveland randomness coincides with martin lof randomness joe miller and andre nies suggested some variations of kolmogorov loveland randomness to approach this problem and to provide a partial solution we show that their proposed notion of injective randomness is still weaker than martin lof randomness since in this proof some of the ideas we use are clearer we also show the weaker theorem that permutation randomness is weaker than martin lof randomness

inverse power and durand kerner iterations for univariate polynomial root finding abstract univariate polynomial root finding is the oldest classical problem of mathematics and computational mathematics and is still an important research topic due to its impact on computational algebra and geometry the weierstrass durand kerner approach and its variations as well as matrix methods based on the qr algorithm are among the most popular practical choices for simultaneous approximation of all roots of a polynomial we propose an alternative application of the inverse power iteration to generalized companion matrices for polynomial root finding demonstrate its effectiveness and relate its study to unifying the derivation of the weierstrass durand kerner algorithm having quadratic convergence and its extensions having convergence rates 4 6 8 our experiments show substantial improvement versus the latter algorithm even though the inverse power iteration is most effective for the more limited tasks of approximating a single root or a few selected roots

multiplication of diagonal transforms of loewner matrices abstract a new formula for multiplication of matrices arising as appropriate diagonal transformations of loewner matrices is derived in the case of symmetric loewner matrices it is shown that special subclasses form multiplicative groups some extensions of known results about polynomials associated with hankel and loewner matrices are given also a previously found formula of the author describing the inverses of loewner matrices is shown to be a consequence of the present results

a unifying convergence analysis of second order methods for secular equations existing numerical methods of second order are considered for a so called secular equation we give a brief description of the most important of these methods and show that all of them can be interpreted as improvements of newton s method for an equivalent problem for which newton s method exhibits convergence from any point in a given interval this interpretation unifies the convergence analysis of these methods provides convergence proofs where they were lacking and furnishes ways to construct improved methods in addition we show that some of these methods are in fact equivalent a second secular equation is also briefly considered

kolmogorov complexity and the recursion theorem several classes of diagonally nonrecursive dnr functions are characterized in terms of kolmogorov complexity in particular a set of natural numbers a can wtt compute a dnr function iff there is a nontrivial recursive lower bound on the kolmogorov complexity of the initial segments of a furthermore a can turing compute a dnr function iff there is a nontrivial a recursive lower bound on the kolmogorov complexity of the initial segments of a a is pa complete that is a can compute a 0 1 valued dnr function iff a can compute a function f such that f n is a string of length n and maximal c complexity among the strings of length n a t k iff a can compute a function f such that f n is a string of length n and maximal h complexity among the strings of length n further characterizations for these classes are given the existence of a dnr function in a turing degree is equivalent to the failure of the recursion theorem for this degree thus the provided results characterize those turing degrees in terms of kolmogorov complexity which no longer permit the usage of the recursion theorem

bounds for the zeros of polynomials from eigenvalues and singular values of some companion matrices abstract let a monic polynomial pn x xn a1xn 1 an aj plane1d 49e j 1 2 n be given bounds for the eigenvalues and the computation of the singular values of some special companion matrices give bounds for the zeros of the polynomial pn which improve classical and recently found ones also some new bounds for the zeros are given

scaled generalized bernstein polynomials and containment regions for the zeros of polynomials let p n x email protected k 0 email protected kb k n x a b b n 0 where b k n x a b a x k b x n k k 0 1 n a email protected c a email protected a generalized scaled bernstein polynomial extending a result of winkler a resultant matrix for scaled bernstein polynomials linear algebra appl 319 2000 179 191 a companion matrix for p n is given the application of some matrix methods to this companion matrix yields regions for the zeros of p n

fast and stable qr eigenvalue algorithms for generalized companion matrices and secular equations we introduce a class of n n structured matrices which includes three well known classes of generalized companion matrices tridiagonal plus rank one matrices comrade matrices diagonal plus rank one matrices and arrowhead matrices relying on the structure properties of we show that if a then a rq where a qr is the qr decomposition of a this allows one to implement the qr iteration for computing the eigenvalues and the eigenvectors of any a with o n arithmetic operations per iteration and with o n memory storage this iteration applied to generalized companion matrices provides new o n2 flops algorithms for computing polynomial zeros and for solving the associated rational secular equations numerical experiments confirm the effectiveness and the robustness of our approach

quadratically constrained least squares and quadratic problems we consider the following problem compute a vectorx such that ax b 2 min subject to the constraint x 2 a new approach to this problem based on gauss quadrature is given the method is especially well suited when the dimensions ofa are large and the matrix is sparse it is also possible to extend this technique to a constrained quadratic form for a symmetric matrixa we consider the minimization ofx t a x 2b t x subject to the constraint x 2 some numerical examples are given

additive preconditioning eigenspaces and the inverse iteration we incorporate our recent preconditioning techniques into the classical inverse power rayleigh quotient iteration for computing matrix eigenvectors every loop of this iteration essentially amounts to solving an ill conditioned linear system of equations due to our modification we solve a well conditioned linear system instead we prove that this modification preserves local quadratic convergence show experimentally that fast global convergence is preserved as well and yield similar results for higher order inverse iteration covering the cases of multiple and clustered eigenvalues

inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices abstract matrices represented as a sum of diagonal and semiseparable ones are considered here these matrices belong to the class of structured matrices which arises in numerous applications fast o n algorithms for their inversion were developed before under additional restrictions which were a source of instability our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms in this paper the case of semiseparable matrices of order one is considered

the local superconvergence of the trilinear element for the three dimensional poisson problem abstract in this paper we shall combine the finite element theory of green s function presented in this paper the extrapolation technique and the local symmetric technique to investigate the local superconvergence of the trilinear element for the three dimensional poisson equation

a unified approach to loewner and hankel matrices abstract we introduce so called generalized loewner matrices the definition is almost the same as antoulas and anderson s our class of generalized loewner matrices contains both classical loewner and hankel matrices as special cases the properties of such matrices are studied together with the associated interpolation problem the detailed characterization of its nonsolvable cases is given by means of the s c minimal generalized solution by means of rank properties of the loewner matrix and by means of the s c l polynomial introduced by fiedler

on computing the eigenvectors of a class of structured matrices a real symmetric matrix of order n has a full set of orthogonal eigenvectors the most used approach to compute the spectrum of such matrices reduces first the dense symmetric matrix into a symmetric structured one i e tridiagonal matrices or semiseparable matrices this step is accomplished in o n 3 operations once the latter symmetric structured matrix is available its spectrum is computed in an iterative fashion by means of the qr method in o n 2 operations in principle the whole set of eigenvectors of the latter structured matrix can be computed by means of inverse iteration in o n 2 operations the blemish in this approach is that the computed eigenvectors may not be numerically orthogonal if clusters are present in the spectrum to enforce orthogonality the gram schmidt procedure is used requiring o n 3 operations in the worst case in this paper a fast and stable method to compute the eigenvectors of tridiagonal and semiseparable matrices which does not suffer from the loss of orthogonality due to the presence of clusters in the spectrum is presented the algorithm requires o n 2 floating point operations if clusters of small size are present in the spectrum

rank structures preserved by the qr algorithm the singular case in an earlier paper we introduced the classes of polynomial and rank structures both of them preserved by applying a shifted qr step on a matrix a in the present paper we further investigate the case of rank structures we show that even if a is a singular matrix a new qr iterate can be constructed having the same rank structure as the matrix a itself to this end we introduce the concepts of effectively eliminating qr decompositions and sparse givens patterns both of them being of independent interest

schnorr randomness for noncomputable measures abstract this paper explores a novel definition of schnorr randomness for noncomputable measures we say x is uniformly schnorr random if t x for all lower semicomputable functions t x such that t x d x is computable we prove a number of theorems demonstrating that this is the correct definition which enjoys many of the same properties as martin lof randomness for noncomputable measures nonetheless a number of our proofs significantly differ from the martin lof case requiring new ideas from computable analysis

root finding by expansion with independent constraints elimination methods are highly effective for the solution of linear and nonlinear systems of equations but reversal of the elimination principle can be beneficial as well competent incorporation of additional independent constraints and variables or more generally immersion of the original computational problem into a larger task defined by a larger number of independent constraints and variables can improve global convergence of iterative algorithms that is their convergence from the start a well known example is the dual linear and nonlinear programming which enhances the power of optimization algorithms we believe that this is just an ad hoc application of general principle of expansion with independent constraints it should be explored systematically for devising iterative algorithms for the solution of equations and systems of equations and for optimization at the end of this paper we comment on other applications and extensions of this principle presently we show it at work for the approximation of a single zero of a univariate polynomial p of a degree n empirical global convergence of the known algorithms for this task is much weaker than that of the algorithms for all n zeros such as weierstrass durand kerner s root finder which reduces its root finding task to viete s vieta s system of n polynomial equations with n unknowns we adjust this root finder to the approximation of a single zero of p preserve its fast global convergence and decrease the number of arithmetic operations per iteration from quadratic to linear together with computing a zero of a polynomial p the algorithm deflates this polynomial as by product and then could be reapplied to the quotient to approximate the next zero of p alternatively by using m processors that exchange no data one can concurrently approximate up to m zeros of p our tests confirm the efficiency of the proposed algorithms technically our root finding boils down to computations with structured matrices polynomials and partial fraction decompositions our study of these links can be of independent interest e g as by product we express the inverse of a sylvester matrix via its last column thus extending the celebrated result of gohberg and sementsul 1972 22 from toeplitz to sylvester matrix inverses

pencils of real symmetric matrices and real algebraic curves abstract let a rational real algebraic curve of degree n in the real projective plane x y z be given by its parametric equations let not containing the point o 0 0 1 have the property that i every line through the point o intersects in n real points counting multiplicities an explicit form is given for real symmetric matrices a and b of order n such that det xa yb zl 0 is an equation for for a diagonal b is a diagonal multiple of a loewner matrix the result partly proves a previous conjecture of the author about general real algebraic curves of degree n having the property i it is also shown that in this general case the conjecture is not true

new progress in real and complex polynomial root finding matrix methods are increasingly popular for polynomial root finding the idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial the algorithms also solve secular equation qr algorithm is the most customary method for eigen solving but we explore the inverse rayleigh quotient iteration instead which turns out to be competitive with the most popular root finders because of its excellence in exploiting matrix structure to advance the iteration we preprocess the matrix and incorporate newton s linearization repeated squaring homotopy continuation techniques and some heuristics the resulting algorithms accelerate the known numerical root finders for univariate polynomial and secular equations and are particularly well suited for the acceleration by using parallel processing furthermore even on serial computers the acceleration is dramatic for numerical approximation of the real roots in the typical case where they are much less numerous than all complex roots

unitary similarity of the determinantal representation of unitary bordering matrices abstract let f x y z be a hyperbolic ternary form of degree n the helton vinnikov theorem asserts that there exists an n n complex symmetric matrix s such that f s x y z f x y z where the determinantal hyperbolic ternary form of an n n matrix b is defined by f b x y z det x b y b z i n let a be an n n unitary bordering matrix it is known that a is unitarily similar to a symmetric matrix in this paper we investigate the unitary similarity between the unitary bordering matrix a and the helton vinnikov symmetric matrix s admitted the determinantal representation of the ternary form f a x y z satisfying f s x y z f a x y z for n 3 4

estimation of bounds for the zeros of a polynomial using numerical radius the estimation of zeros of a polynomial have been done by many mathematicians over the years using various approaches in this paper we estimate the upper bound for the zeros of a given polynomial using hilbert space technique involving frobenius companion matrix and numerical radius we first obtain numerical range and numerical radius for certain class of matrices and use them to estimate the bounds for zeros of a given polynomial we illustrate with examples to show that the estimations obtained here is better than the previously known estimations we also obtain a sequence of real numbers which converges exactly to the spectral radius of some special class of matrices

the restarted qr algorithm for eigenvalue computation of structured matrices the qr algorithm is a popular numerical method for the computation of eigenvalues of matrices all eigenvalues of a general n n upper hessenberg matrix typically can be computed in o n3 arithmetic floating point operations using o n2 storage locations when the upper hessenberg matrix is hermitian or unitary then it can be represented by o n parameters and there are variants of the qr algorithm that reduce the operation count for computing all eigenvalues to o n2 arithmetic floating point operations and the storage requirement to o n locations however for many structured matrices that can be represented with o n storage locations available implementations of the qr algorithm require o n3 arithmetic floating point operations and o n2 storage locations to determine all eigenvalues this note shows that for some of the latter matrices the operation count can be reduced to o n2 arithmetic floating point operations and the memory requirement to o n storage locations by periodically restarting the qr algorithm

locating multiple zeros interactively abstract we present a globally convergent algorithm which has been implemented interactively for the total or partial factorisation of a polynomial initially a region is found containing all the zeros of a polynomial a parallel search algorithm is then used to locate successively smaller regions containing zeros this continues until it is possible to estimate the multiplicities of the zeros reliably when a rapidly converging iteration function if takes over at this point the user using an interactive terminal can elect to investigate further all of the zeros or only those in certain regions the if is selected from a class of such functions which exhibit certain desirable mathematical and computational properties a synopsis of computational results is given

the cayley hamilton theorem and inverse problems for multiparameter systems abstract we review some of the current research in multiparameter spectral theory we prove a version of the cayley hamilton theorem for multiparameter systems and list a few inverse problems for such systems some consequences of results on determinantal representations proved by dixon dickson and vinnikov for the inverse problems are discussed

confluent cauchy and cauchy vandermonde matrices abstract the s c confluent cauchy and confluent cauchy vandermonde matrices are introduced corresponding to an interpolation problem with multiple interpolation nodes inversion formulas are derived and a factorization of confluent loewner matrices by means of confluent cauchy vandermonde matrices is described

hankel and loewner matrices abstract mutual relations between the hankel toeplitz bezout and loewner matrices as well as further connections to rational interpolation and projective geometry are investigated

high effective finite element algorithm for elliptic partial differential equation in this paper we will discuss elliptic partial differential equation whose coefficients are constants on basis of a new computing technique for green function we will propose a new effective finite element algorithm for that problem finally we will give some numerical examples to investigate that finite element algorithm

numerical solution of a secular equation a method is proposed for the solution of a secular equation arising in modified symmetric eigenvalue problems and in several other areas this equation has singularities which make the application of standard root finding methods difficult in order to solve the equation a class of transformations of variables is considered which transform the equation into one for which newton s method converges from any point in a certain given interval in addition the form of the transformed equation suggests a convergence accelerating modification of newton s method the same ideas are applied to the secant method and numerical results are presented

on the refinement of cauchy s theorem and pellet s theorem using the relationship of a polynomial and its associated polynomial we derived a necessary and sufficient condition for determining all roots of a given polynomial on the circumference of a circle defined by its associated polynomial by employing the technology of analytic inequality and the theory of distribution of zeros of meromorphic function we refine two classical results of cauchy and pellet about bounds of modules of polynomial zeros sufficient conditions are obtained for the polynomial whose cauchy s bound and pellet s bounds are strict bounds the characteristics is given for the polynomial whose cauchy s bound or pellet s bounds can be achieved by the modules of zeros of the polynomial

spectral radius inequalities for sums of operators with an application to the problem of bounding the zeros of polynomials abstract we prove spectral radius inequalities for sums of operators our results are employed to establish a new bound for the zeros of polynomials

fast and stable eigendecomposition of symmetric banded plus semi separable matrices abstract a new fast and stable algorithm to reduce a symmetric banded plus semi separable matrix to tridiagonal form via orthogonal similarity transformations is presented

the cal d q r algorithm basic theory convergence and conditional stability we describe a fast matrix eigenvalue algorithm that uses a matrix factorization and reverse order multiply technique involving three factors and that is based on the symmetric matrix factorization as well as on d orthogonal reduction techniques where d mbox diag pm1 is computed from the given matrix a it operates on a similarity reduction of a real matrix a nn to general tridiagonal form t and computes all of a s eigenvalues in frac 4 3 n 3 o n 2 operations where the o n 2 part of the operations is possibly performed over bbb c instead of the 7 8 n 3 real flops required by the qr eigenvalue algorithm potential breakdo wn of the dqr algorithm can occur in the reduction to tridiagonal form and in the d orthogonal dqr reductions both however can be monitored during the computations the former occurs rather rarely for dimensions n leq 300 and can essentially be bypassed while the latter is extremely rare and can be bypassed as well in our conditionally stable implementation of the dqr steps we prove an implicit dq theorem which allows implicit shifts give a convergence proof for the dqr algorithm and show that dqr is conditionally stable for general balanced tridiagonal matrices t rm b

polynomials compatible with a symmetric loewner matrix abstract several mutually equivalent conditions are proved for a polynomial of degree n and a symmetric n n matrix which form a basis for so called compatibility of these two objects one of the conditions yields a new parametric expression for a symmetric loewner matrix

sensitivity analysis of all eigenvalues of a symmetric matrix given a x a ij x a n by n symmetric matrix depending smoothly on a parameter x we study the first order sensitivity of all the eigenvalues lambda m x of a x 1 le m le n under some smoothness assumption like the a ij be c 1 we prove that the directional derivatives d mapsto lambda prime m x d lim t to o lambda m x td lambda m x t do exist and give an explicit expression of them in terms of the data of the parametrized matrix the key idea to circumvent the difficulties inherent to the study of each lambda m taken separately is to consider the functions f m x 1 le m le n defined as the sums of the m largest eigenvalues of a x based on ky fan s variational formulation of f m and some chain rule from nonsmooth analysis we derive an explicit formula for the generalized gradient of f m and a computationally useful formula for the directional derivative of f m using these formulas and the relation lambda m f m f m 1 we then derive the directional derivative of lambda m some properties of this directional derivative as well as an illustrative example are presented

a new superconvergence and extrapolation for second order triangular element in this paper we find a new superconvergence and extrapolation for second order triangular element based on w m he j z cui q d zhu an improved method on superconvergence and extrapolation math numer sinica 24 2002 3 in chinese of course some numerical examples are given to investigate those estimates

local ultraconvergence of linear and bilinear finite element method for second order elliptic problems abstract in this article local ultraconvergence in gradient of the bilinear and linear finite element solutions to two dimensional second order elliptic problems has been investigated special interpolation postprocessing algorithms of numerical solutions by richardson extrapolation have been developed the local symmetry theory and estimates of discrete green functions are used in the analysis numerical experiments are provided to confirm the theoretical findings

on the enestrom kakeya theorem abstract a classical result of enestrom and kakeya if a n a n 1 a 0 0 then for z 1 n k 0 a k z k 0 is extended to polynomials whose coefficients satisfy the condition t n a n t n 1 a n 1 t 1 a 1 t a t 1 a 1 ta 1 a 0 for some t 0 and 0 n

determinantal representations of hyperbolic forms via weighted shift matrices we characterize hyperbolic ternary forms of degrees 3 4 and 5 that admit determinantal representations via cyclic weighted shift matrices

a composite polynomial zerofinding matrix algorithm abstract a globally convergent matrix algorithm is presented for finding the real and complex zeros of a complex polynomial p x a combination of schmeisser s and fiedler s matrices is used in the algorithm first the greatest common divisor g x of p x and its derivative p x is obtained and used to reduce p x to a polynomial q x p x q x with simple zeros second the zeros of q x are computed either by finding the eigenvalues of the first block of schmeisser s matrix or by applying fiedler s algorithm recursively in either case the eigenvalues are obtained by the qr algorithm third the multiplicity of each zero of p x is calculated by means of lagouanelle s modified limiting formula sample numerical results are given to demonstrate the effectiveness of the algorithm

eigenvalue inequalities and equalities abstract we consider cases of equality in three basic inequalities for eigenvalues of hermitian matrices cauchy s interlacing inequalities for principal submatrices weyl s inequalities for sums and the residual theorem several applications generalize and sharpen known results for eigenvalues of irreducible tridiagonal hermitian matrices

expressing a polynomial as the characteristic polynomial of a symmetric matrix abstract we show that given a polynomial one can without knowing the roots construct a symmetric matrix whose characteristics polynomial is the given polynomial appropriately normed for a real polynomial the matrix can have purely imaginary off diagonal entries for a polynomial with only real distinct roots the matrix can be chosen real