wavelet like bases for the fast solutions of second kind integral equations a class of vector space bases is introduced for the sparse representation of discretizations of integral operators an operator with a smooth nonoscillatory kernel possessing a finite number of singularities in each row or column is represented in these bases as a sparse matrix to high precision a method is presented that employs these bases for the numerical solution of second kind integral equations in time bounded by o n log 2 n where n is the number of points in the discretization numerical results are given which demonstrate the effectiveness of the approach and several generalizations and applications of the method are discussed

computing outer inverses by scaled matrix iterations convergence analysis for a class of iterative methods for computing outer inverses with prescribed range and null space is studied furthermore several heuristics for accelerating iterative methods via scaling are proposed in fact we are motivated by the fact that scaling of iterative methods for computing generalized inverses is investigated so far only on low order methods our intention is to test the scaling on higher order iterative methods here we also propose a new higher order iteration although the introduced method is efficient in terms of computational efficiency index we test its acceleration through several experiments

root refinement for real polynomials using quadratic interval refinement we consider the problem of approximating all real roots of a square free polynomial f with real coefficients given isolating intervals for the real roots and an arbitrary positive integer l the task is to approximate each root to l bits after the binary point abbott has proposed the quadratic interval refinement method qir for short which is a variant of regula falsi combining the secant method and bisection we formulate a variant of qir denoted aqir approximate qir that considers only approximations of the polynomial coefficients and chooses a suitable working precision adaptively it returns a certified result for any given real polynomial whose roots are all simple in addition we propose several techniques to improve the asymptotic complexity of qir we prove a bound on the bit complexity of our algorithm in terms of the degree of the polynomial the size and the separation of the roots that is parameters exclusively related to the geometric location of the roots for integer coefficients our variant improves in theory and practice the variant with exact integer arithmetic combining our approach with multipoint evaluation we obtain near optimal bounds that essentially match the best known theoretical bounds on root approximation as obtained by very sophisticated algorithms

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

enhancement of stability and accuracy of the moving particle semi implicit method as a lagrangian gridless particle method the mps moving particle semi implicit method has been proven useful in a wide range of engineering applications up to now most of mps applications have been limited to problems with compressive stress states this paper investigates the performance and stability of mps method in simulation of more general hydrodynamic problems including those characterized by tensile stresses or by changes in the stress states it is shown that mps based simulations are prone to become destabilized in presence of attractive interparticle forces this instability appears to be similar to the so called tensile instability in sph method two new modifications namely a modified poisson pressure equation and a corrective matrix for pressure gradient model are proposed to resolve this shortcoming these two new modifications together with two previously proposed improvements are shown to stabilize and enhance the performance of mps method

ring isomorphisms and linear or additive maps preserving zero products on nest algebras it is shown that every ring isomorphism between norm closed standard subalgebras of nest algebras acting on infinite dimensional banach spaces is either a spatial isomorphism or a spatial conjugate isomorphism this is then used to characterize the linear or additive maps between nest algebras which preserve zero products

dft calculation of the generalized and drazin inverse of a polynomial matrix a new algorithm is presented for the determination of the a generalized inverse and b drazin inverse of a polynomial matrix the proposed algorithms are based on the discrete fourier transform and thus are computationally fast in contrast to other known algorithms the above algorithms are implemented in the mathematica programming language and illustrated via examples

effective partitioning method for computing generalized inverses and their gradients abstract we extend the algorithm for computing 1 1 3 1 4 inverses and their gradients from 11 to the set of multiple variable rational and polynomial matrices an improvement of this extension appropriate to sparse polynomial matrices with relatively small number of nonzero coefficient matrices as well as in the case when the nonzero coefficient matrices are sparse is introduced for that purpose we exploit two effective structures form 6 which make use of only nonzero addends in polynomial matrices and define their partial derivatives symbolic computational package mathematica is used in the implementation several randomly generated test matrices are tested and the cpu times required by two used effective structures are compared and discussed

computation of the generalized inverse of a polynomial matrix and applications abstract the computation of the generalized inverse of a constant matrix is utilized in finding the generalized inverse and its laurent expansion of a nonregular polynomial matrix the proposed algorithm constitutes a generalization of the algorithm proposed by fragulis et al for regular polynomial matrices and gives rise to numerous applications in multivariable system analysis

computing generalized inverse of polynomial matrices by interpolation abstract we investigated an interpolation algorithm for computing the moore penrose inverse of a given polynomial matrix based on the leverrier faddeev method also a method for estimating the degrees of polynomial matrices arising from the leverrier faddeev algorithm is given as the improvement of the interpolation algorithm algorithms are implemented in the symbolic programming language mathematica and tested on several different classes of test examples

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

nonlinear mappings on upper triangular matrices derivable at zero point abstract let t n f be the set of all n n upper triangular matrices over a field f recently by using a technique from graph theory l wang in 8 proved that a bijective mapping on t n f with f a finite field preserves zero product i e x y 0 whenever x y 0 if and only if can be decomposed into the product of an inner automorphism an automorphism of t n f induced by an automorphism of f and a regular automorphism of the zero divisor graph of t n f algebraic group theory shows that derivations of an algebra often go parallel with automorphisms of the algebra this viewpoint encourages us to determine nonlinear mappings on t n f which are derivable at zero point i e x y x y 0 whenever x y 0 it is proved that a mapping on t n f is derivable at zero point if and only if is the sum of an inner derivation an additive derivation of t n f induced by an additive derivation of f and a regular mapping on t n f

additive maps derivable or jordan derivable at zero point on nest algebras abstract let alg n be a nest algebra associated with the nest n on a real or complex banach space x assume that every n n is complemented whenever n n let alg n alg n be an additive map it is shown that the following three conditions are equivalent 1 is derivable at zero point i e ab a b a b whenever ab 0 2 is jordan derivable at zero point i e ab ba a b a b b a b a whenever ab ba 0 3 has the form a a ca for some additive derivation and some scalar c it is also shown that is generalized derivable at zero point i e ab a b a b a i b whenever ab 0 if and only if is an additive generalized derivation finer characterizations of above maps are given for the case dim x

point estimation of simultaneous methods for solving polynomial equations a survey ii the construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root finding algorithm is one of the most important problems in solving nonlinear equations smale s point estimation theory from 1981 was a great advance in this topic it treats convergence conditions and the domain of convergence in solving an equation f z 0 using only the information of f at the initial point z 0 the study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult even in the case of algebraic polynomials in the light of smale s point estimation theory an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials we state new improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations ehrlich aberth s method ehrlich aberth s method with newton s correction borsch supan s method with weierstrass correction and halley like or wang zheng method the introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods but also explicitly gives their order of convergence the stated initial conditions are of significant practical importance since they are computationally verifiable they depend only on the coefficients of a given polynomial its degree n and initial approximations to polynomial zeros

maps determined by action on identity product elements abstract let m n r be the algebra of all n n matrices over a unital commutative ring r with 6 invertible for a given element z m n r a map on m n r is called preserving z product if x y z whenever xy z a map on m n r is called derivable at the given point z if x y x y z whenever xy z using elementary matrix technique we show that if an invertible linear map on m n r preserves identity product then it is a jordan automorphism and a linear map on m n r is derivable at the identity matrix if and only if it is an inner derivation

an efficient matrix iteration for computing weighted moore penrose inverse the goal of this study is threefold in order to calculate the weighted moore penrose inverse we first derive a new matrix iteration for computing the inverse of non singular square matrices we then analytically extend the obtained results so as to compute the moore penrose generalized inverse of a non square matrix subsequently these results will again be theoretically extended to find the weighted moore penrose inverse the computational efficiency of the presented scheme is rigourously studied and compared with the existing matrix iterations to show that it is economic some applications are given as well

a biparametric family of optimally convergent sixteenth order multipoint methods with their fourth step weighting function as a sum of a rational and a generic two variable function a biparametric family of four step multipoint iterative methods of order sixteen to numerically solve nonlinear equations are developed and their convergence properties are investigated the efficiency indices of these methods are all found to be 16 1 5 1 741101 being optimally consistent with the conjecture of kung traub numerical examples as well as comparison with existing methods developed by kung traub and neta are demonstrated to confirm the developed theory in this paper

root finding by divided differences a recursive method is presented for computing a simple zero of an analytic functionf from information contained in a table of divided differences of its reciprocalh 1 f a good deal of flexibility is permitted in the choice of ordinate and derivative values and in the choice of the number of previous points upon which to base the next estimate of the required zero the method is shown to be equivalent to a process of fitting rational functions with linear numerators to data sampled fromf asymptotic and regional convergence properties of such a process have already been studied in particular asymptotically quadratic convergence is easily obtained at the expense of only one function evaluation and a moderate amount of overhead computation per step in these respects the method is comparable with the newton form of iterated polynomial inverse interpolation while its regional convergence characteristics may be superior in certain circum stances it is also shown that the method is not unduly sensitive to round off errors

the algorithm for computing the drazin inverses of two variable polynomial matrices we propose a finite algorithm for computing drazin inverse of two variable polynomial matrices based on greville s finite algorithm for computing drazin inverse of a constant matrix also a three dimensional recursive algorithm to do that is deduced examples show that these methods are feasible and the implementation is developed in the symbolic package matlab

symbolic computation of weighted moore penrose inverse using partitioning method abstract we propose a method and algorithm for computing the weighted moore penrose inverse of one variable rational matrices continuing this idea we develop an algorithm for computing the weighted moore penrose inverse of one variable polynomial matrix these methods and algorithms are generalizations of the method for computing the weighted moore penrose inverse for constant matrices originated in wang and chen g r wang y l chen a recursive algorithm for computing the weighted moore penrose inverse a mn j comput math 4 1986 74 85 and the partitioning method for computing the moore penrose inverse of rational and polynomial matrices introduced in stanimirovic and tasic p s stanimirovic m b tasic partitioning method for rational and polynomial matrices appl math comput 155 2004 137 163 algorithms are implemented in the symbolic computational package mathematica

on the extension of householder s method for weighted moore penrose inverse this note discusses how to extend and use the well known method of householder 1964 6 for finding weighted moore penrose inverse the discussions cover both theoretical and numerical aspects an approach for accelerating the initial phase of convergence will be contributed some tests will also be employed to check the validity and superiority of the results over the schulz method

optimal and nearly optimal algorithms for approximating polynomial zeros abstract we substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p x our new algorithms save both boolean and arithmetic sequential time versus the previous best algorithms of schonhage 1 pan 2 and neff and reif 3 in parallel nc implementation we dramatically decrease the number of processors versus the parallel algorithm of neff 4 which was the only nc algorithm known for this problem so far specifically under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p x to the unit disc x x 1 our algorithms which promise to be practically effective approximate all the zeros of p x within the absolute error bound 2 b by using order of n arithmetic operations and order of b n n 2 boolean bitwise operations in both cases up to within polylogarithmic factors the algorithms allow their optimal work preserving nc parallelization so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or b n n 2 boolean processors all the cited bounds on the computational complexity are within polylogarithmic factors from the optimum in terms of n and b under both arithmetic and boolean models of computation in the boolean case under the additional realistic assumption that n o b

generalized derivations on unital algebras determined by action on zero products abstract let a be a unital algebra having a nontrivial idempotent and let m be a unitary a bimodule we consider linear maps f g a m satisfying f x y x g y 0 whenever x y a are such that x y 0 for example when a is zero product determined algebra e g algebra generated by idempotents f and g are generalized derivations f x f 1 x d x and g x x g 1 d x for all x a where d a m is a derivation if a is not generated by idempotents then there exist also nonstandard solutions for maps f and g in the case of a being a triangular algebra under some condition on bimodule m the characterization of maps f and g is given we also consider conditions on algebra a making it a zero product determined algebra

a new approach for the numerical simulation of free surface incompressible flows using a meshfree method abstract this paper proposes a novel approach to solve free surface incompressible flows using finite pointset method fpm our approach is based in the chorin uzawa s projection method and it corresponds to an extension of the work of tiwari and kuhnert 2007 to solve general elliptic equations in a meshfree framework the main features behind this approach as well as details of the computational implementation are presented finally the numerical results of the simulation of some test problems using this approach with first and second order accuracy in time projection schemes are reported and compared with published experimental and theoretical results which show that this approach is promising for simulating free surface incompressible flows

an improved newton iteration for the weighted moore penrose inverse abstract in this paper we showed that newton iteration may be used to compute the weighted moore penrose inverse of an arbitrary matrix we also take advantage of several new acceleration procedures to reduce the cost of newton iteration which extends the earlier work r schreiber computing generalized inverses and eigenvalues of symmetric matrices using systolic arrays in r glowinski j l lious eds computing methods in applied science and engineering north holland amsterdam 1984 t soderstorm g w stewant on the numerical properties of an iterative method for computing the moore penrose generalized inverse siam j numer anal 11 1974 61 74

efficient polynomial root refiners a survey and new record efficiency estimates a typical iterative polynomial root finder begins with a relatively slow process of computing a crude but sufficiently close initial approximation to a root and then rapidly refines it the policy of using the same iterative process at both stages of computing an initial approximation and refining it however is neither necessary nor most effective the efficiency of an iteration at the former stage resists formal study and is usually decided empirically whereas formal study of the efficiency at the latter stage of refinement is not hard and is the subject of the current paper we define this local efficiency as log 1 0qd log 1 0 q 1 d q is the convergence order and d is the number of function evaluations per iteration it is inversely proportional to the number of flops involved assuming that about 2n flops are needed per evaluation of a polynomial of a degree n at a single point we extend the definition to cover the recent matrix methods for polynomial root finding as well as some methods that combine n approximations to all n roots to refine them simultaneously for the approximation of a single root of a polynomial of degree n the maximum local efficiency achieved so far is log 1 02 0 301 but we show its growth to infinity for simultaneous approximation of all n roots as n grows to infinity

a finite algorithm for the drazin inverse of a polynomial matrix based on greville s finite algorithm for drazin inverse of a constant matrix we propose a finite numerical algorithm for the drazin inverse of polynomial matrices we also present a new proof for decell s finite algorithm through greville s finite algorithm

generalized derivable mappings at zero point on some reflexive operator algebras abstract let a be a subalgebra with the unit operator i in b h we say that a linear mapping from a into b h is a generalized derivable mapping at zero point if st s t s t s i t for any s t a with st 0 in this paper we show the following main result every norm continuous generalized derivable mapping at zero point on finite csl algebras is a generalized derivation

the amended dsesc power method for polynomial root finding cardinal s matrix version of the sebastiao e silva polynomial root finder rapidly approximates the roots as the eigenvalues of the associated frobenius matrix we preserve rapid convergence to the roots but amend the algorithm to allow input polynomials with multiple roots and root clusters as in cardinal s algorithm we repeatedly square the frobenius matrix in nearly linear arithmetic time per squaring which yields dramatic speedup versus the recent effective polynomial root finder based on the application of the inverse power method to the frobenius matrix

computation of the inverse of a polynomial matrix and evaluation of its laurent expansion abstract an algorithm is given which constitutes a generalization of the algorithm for the inversion of a pencil se a due to mertzios 1984 in the case of general systems described in differential operator form recursive formulae are obtained for the calculation of both the coefficient matrices of the adjoint of the polynomial matrix as well as for the coefficients of the characteristic polynomial of the polynomial matrix a simple method is also presented that allows the evaluation of the laurent expansion for the inverse of a polynomial matrix of any order specifically an arma model is given for the computation of the coefficient matrices of the laurent expansion in terms of the coefficient matrices of the polynomial matrix without inverting the latter the laurent expansion of a polynomial matrix is used for the analysis and synthesis of the polynomial matrix descriptions that constitute a further generalization of the singular or generalized systems

an accelerated iterative method for computing weighted moore penrose inverse the goal of this paper is to present an accelerated iterative method for computing weighted moore penrose inverse analysis of convergence is included to show that the proposed scheme has sixth order convergence using a proper initial matrix a sequence of iterates will be produced which is convergent to the weighted moore penrose inverse numerical experiments are reported to show the efficiency of the new method

characterizations of lie derivations of triangular algebras abstract let t be a triangular algebra over a commutative ring r in this paper under some mild conditions on t we prove that if t t is an r linear map satisfying x y x y x y for any x y t with xy 0 resp xy p where p is the standard idempotent of t then d where d is a derivation of t and t z t where z t is the center of t is an r linear map vanishing at commutators x y with xy 0 resp xy p

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

characterizations of isomorphisms and derivations of some algebras abstract let be a zero product preserving bijective bounded linear map from a unital algebra a onto a unital algebra b such that 1 k we show that if a is a csl algebra on a hilbert space or a j lattice algebra on a banach space then there exists an isomorphism from a onto b such that k for a nest algebra a in a factor von neumann algebra we characterize the linear maps on a such that x y x y 0 for all x y a with x y 0

the representation and approximation for the generalized inverse at s 2 we present a unified representation theorem for the generalized inverses a t s 2 which are 2 or outer inverses having the prescribed range t and null space s specific expressions and computational procedures for the generalized inverses a t s 2 can be uniformly derived

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

on error estimates of projection methods for navier stokes equations first order schemes in this paper projection methods or fractional step methods are studied in the semi discretized form for the navier stokes equations in a two or three dimensional bounded domain error estimates for the velocity and the pressure of the classical projection scheme are established via the energy method a modified projection scheme which leads to improved error estimates is also proposed

on jordan all derivable points of b h abstract let h be a hilbert space and b h the algebra of all bounded linear operators on h in this note we show that 0 is a generalized jordan all derivable point of b h if h is infinite dimensional for any hilbert space h we also show that i is a jordan all derivable point of b h

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

modeling of two phase flows with surface tension by finite pointset method fpm a meshfree method for two phase immiscible incompressible flows including surface tension is presented the continuum surface force csf model is used to include the surface tension force the incompressible navier stokes equation is considered as the mathematical model application of implicit projection method results in linear second order partial differential equations for velocities and pressure these equations are then solved by the finite pointset method fpm which is a meshfree and lagrangian method the fluid is represented as finite number of particles and the immiscible fluids are distinguished by the color of each particle the interface is tracked automatically by advecting the color functions for each particle two test cases laplace s law and the rayleigh taylor instability in 2d have been presented the results are found to be consistent with the theoretical results

a new family of eighth order iterative methods for solving nonlinear equations a family of eighth order iterative methods with four evaluations for the solution of nonlinear equations is presented kung and traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2 n 1 the new family of eighth order methods agrees with the conjecture of kung traub for the case n 4 therefore this family of methods has efficiency index equal to 1 682 numerical comparisons are made with several other existing methods to show the performance of the presented methods

successive matrix squaring algorithm for computing outer inverses abstract in this paper we derive a successive matrix squaring sms algorithm to approximate an outer generalized inverse with prescribed range and null space of a given matrix a c r m n we generalize the results from the papers l chen e v krishnamurthy i macleod generalized matrix inversion and rank computation by successive matrix powering parallel computing 20 1994 297 311 y wei successive matrix squaring algorithm for computing drazin inverse appl math comput 108 2000 67 75 y wei h wu j wei successive matrix squaring algorithm for parallel computing the weighted generalized inverse a mn appl math comput 116 2000 289 296 and obtain an algorithm for computing various classes of outer generalized inverses of a instead of particular matrices used in these articles we use an appropriate matrix r c s n m s r numerical examples are presented

partitioning method for rational and polynomial matrices we propose an extension of the grevile s partitioning method for computing the moore penrose inverse which is applicable to the set of rational matrices also we develop an algorithm for computing the moore penrose inverse of given one variable polynomial matrix which is based on the grevile s method major problems arising in the implementation of this method are repetitive recomputations of the same values and simplification of rational and polynomial expressions which contain unknown variable these algorithms are implemented in the symbolic computational package mathematica

a nonstationary iterative second order method for solving nonlinear equations abstract this paper presents a new nonstationary iterative method of second order for solving nonlinear equations that does not require the use of any derivatives for algebraic equations our method coincides with newton s method from a certain step of iteration on due to the methodology of ostrowski 8 the efficiency index of our process equals 2 which is higher than the efficiency indices of classical iterative methods such as newton s process and the secant method to mention just a few

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

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

characterizations of derivations of banach space nest algebras all derivable points abstract let n be a nest on a complex banach space x with n n complemented in x whenever n n and let alg n be the associated nest algebra we say that an operator z alg n is an all derivable point of alg n if every linear map from alg n into itself derivable at z i e a b a b z for any a b a with ab z is a derivation in this paper it is shown that if z alg n is an injective operator or an operator with dense range or an idempotent operator with ran z n then z is an all derivable point of alg n particularly if n is a nest on a complex hilbert space then every idempotent operator with range in n every injective operator as well as every operator with dense range in alg n is an all derivable point of the nest algebra alg n

hamiltonian moving particle semi implicit hmps method for incompressible fluid flows particle methods are meshless simulation techniques in which motion of continua is approximated by discrete dynamics of a finite number of particles and thus have a great degree of flexibility for instance in dealing with the complex motion of surfaces or boundaries in this paper a particle method is developed from a direct discretization of the lagrangian for inviscid incompressible fluid flows the discretized lagrangian is transformed into a hamiltonian with holonomic constraints that stem from the incompressibility condition the rattle algorithm that is a symplectic scheme for holonomically constrained hamiltonian systems is adopted so that the resulting particle method inherits structures possessed by the partial differential equations that govern inviscid incompressible flows the algorithm of the particle method is semi implicit so we call it the hamiltonian moving particle semi implicit hmps method some numerical tests indicate the excellence of the hmps method in conservation of mechanical energy as well as linear and angular momenta

a family of iterative methods for computing moore penrose inverse of a matrix abstract this paper improves on generalized properties of a family of iterative methods to compute the approximate inverses of square matrices originally proposed in 1 and while the methods of 1 can be used to compute the inner inverses of any matrix it has not been proved that these sequences converge in norm to a fixed inner inverse of the matrix in this paper it is proved that the sequences indeed are convergent to a fixed inner inverse of the matrix which is the moore penrose inverse of the matrix the convergence proof of these sequences is given by fundamental matrix calculus and numerical experiments show that the third order iterations are as good as the second order iterations

a finite algorithm for generalized inverses of polynomial and rational matrices there are proposed two types of finite algorithms for symbolic computation of outer inverses of a given polynomial or rational matrix based on the extension of the leverrier faddeev algorithm the implementation of the algorithm corresponding to rational matrices is developed in the symbolic package mathematica

generalization of taylor s theorem and newton s method via a new family of determinantal interpolation formulas and its applications abstract the general form of taylor s theorem for a function f k k where k is the real line or the complex plane gives the formula f pn rn where pn is the newton interpolating polynomial computed with respect to a confluent vector of nodes and rn is the remainder whenever f 0 for each m 2 n 1 we describe a determinantal interpolation formula f pm n rm n where pm n is a rational function in x and f itself these formulas play a dual role in the approximation of f or its inverse for m 2 the formula is taylor s and for m 3 is a new expansion formula and a pade approximant by applying the formulas to pn for each m 2 p m m 1 p m m n 2 is a set of n rational approximations that includes pn and may provide a better approximation to f than pn hence each taylor polynomial unfolds into an infinite spectrum of rational approximations the formulas also give an infinite spectrum of rational inverse approximations as well as a fundamental k point iteration function bm k for each k m defined as the ratio of two determinants that depend on the first m k derivatives application of our formulas have motivated several new results obtained in sequel papers i theoretical analysis of the order of b m k k 1 m proving that it ranges from m to the limiting ratio of generalized fibonacci numbers of order m ii computational results with the first few members of bm k indicating that they outperform traditional root finding methods e g newton s iii a novel polynomial rootfinding method requiring only a single input and the evaluation of the sequence of iteration functions bm 1 at that input this amounts to the evaluation of special toeplitz determinants that are also computable via a recursive formula iv a new strategy for general root finding v new formulas for approximation of e and other special numbers

effective partitioning method for computing weighted moore penrose inverse we introduce a method and an algorithm for computing the weighted moore penrose inverse of multiple variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices these methods and algorithms are generalizations of algorithms developed in m b tasic p s stanimirovic m d petkovic symbolic computation of weighted moore penrose inverse using partitioning method appl math comput 189 2007 615 640 to multiple variable rational and polynomial matrices and improvements of these algorithms on sparse matrices also these methods are generalizations of the partitioning method for computing the moore penrose inverse of rational and polynomial matrices introduced in p s stanimirovic m b tasic partitioning method for rational and polynomial matrices appl math comput 155 2004 137 163 m d petkovic p s stanimirovic symbolic computation of the moore penrose inverse using partitioning method internat j comput math 82 2005 355 367 to the case of weighted moore penrose inverse algorithms are implemented in the symbolic computational package mathematica

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

symbolic computation of at s 2 inverses using qdr factorization abstract efficient evaluation of the full rank qdr decomposition is established a method and algorithm for efficient symbolic computation of a t s 2 inverses of a given rational matrix a is defined using the full rank qdr decomposition of an appropriate rational matrix w the algorithm is implemented using mathematica s ability to deal with symbolic expressions as well as with numbers examples including polynomial and rational matrices are presented some comparisons with well known methods for symbolic evaluation of generalized inverses are obtained

a fast implicit qr eigenvalue algorithm for companion matrices abstract an implicit version of the shifted qr eigenvalue algorithm given in bini et al d a bini y eidelman i gohberg l gemignani siam j matrix anal appl 29 2 2007 566 585 is presented for computing the eigenvalues of an n n companion matrix using o n 2 flops and o n memory storage numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method

meshfree numerical approach based on the finite pointset method for static linear elasticity problems abstract in this work a promising meshfree numerical approach is extended and implemented for the first time in the field of linear static elasticity a real meshfree method the so called finite pointset method fpm is applied and implemented in order to solve the strong classical form of the governing partial differential equations pdes the well known navier cauchy equations fpm numerical performance is demonstrated for several classical and well known studied benchmark problems and comparison with other approaches is also reported numerical results suggest that fpm method is stable with good convergence behavior for 2d linear static elasticity problems

about the generalized lm inverse and the weighted moore penrose inverse the recursive method for computing the generalized lm inverse of a constant rectangular matrix augmented by a column vector is proposed in udwadia and phohomsiri 2007 16 17 the corresponding algorithm for the sequential determination of the generalized lm inverse is established in the present paper we prove that the introduced algorithm for computing the generalized lm inverse and the algorithm for the computation of the weighted moore penrose inverse developed by wang and chen 1986 in 23 are equivalent algorithms both of the algorithms are implemented in the present paper using the package mathematica several rational test matrices and randomly generated constant matrices are tested and the cpu time is compared and discussed

implicit double shift qr algorithm for companion matrices in this paper an implicit double shifted qr method for computing the eigenvalues of companion and fellow matrices will be presented companion and fellow matrices are hessenberg matrices that can be decomposed into the sum of a unitary and a rank 1 matrix the hessenberg the unitary as well as the rank 1 structures are preserved under a step of the qr method this makes these matrices suitable for the design of a fast qr method several techniques already exist for performing a qr step the implementation of these methods is highly dependent on the representation used unfortunately for most of the methods compression is needed since one is not able to maintain all three unitary hessenberg and rank 1 structures in this manuscript an implicit algorithm will be designed for performing a step of the qr method on the companion or fellow matrix based on a new representation consisting of givens transformations moreover no compression is needed as the specific representation of the involved matrices is maintained finally also a double shift version of the implicit method is presented

a plausible extension of standard penalty streamline upwind and immersed boundary techniques to the improved element free galerkin based solution of incompressible navier stokes equations abstract the present work has been conducted in order to propose the extension of standard penalty and stabilization techniques to the improved element free galerkin iefg method for the numerical solution of incompressible fluid flow problems in principle the numerical procedures to be implemented in this communication have been conceived for finite element method fem based solutions and these include the reduced integration penalty method ripm the streamline upwind petrov galerkin supg scheme and a penalty based immersed boundary method pbibm for the imposition of essential boundary conditions along internal fluid solid interfaces the linear momentum balance and mass conservation equations have been coupled via the ripm in order to obtain a global weak formulation where the iefg model is entirely developed in terms of improved moving least squares imls approximations of the velocity field a detailed explanation concerning the appropriate extension of both the ripm and supg procedures to the context of iefg formulations has also been provided the resulting formulation has been applied to the solution of two well known benchmark problems i lid driven square cavity flow and ii flow past a fixed cylinder regarding the flow past a fixed cylinder benchmark problem the fluid solid interaction has been imposed as an internal immersed boundary condition via the pbibm the feasibility and reliability of implementing the ripm supg and pbibm procedures in the iefg formulation have been proven by comparison with experimental and mesh based numerical results reported in the literature the results obtained in this study have revealed that a proper extension of the aforementioned penalty and stabilization techniques to the iefg formulation allows the achievement of accurate and stable numerical results during the solution of incompressible fluid dynamics problems

differentiation of generalized inverses for rational and polynomial matrices our basic motivation is a direct method for computing the gradient of the pseudo inverse of well conditioned system with respect to a scalar proposed in 13 by layton in the present paper we combine the layton s method together with the representation of the moore penrose inverse of one variable polynomial matrix from 24 and developed an algorithm for computing the gradient of the moore penrose inverse for one variable polynomial matrix moreover using the representation of various types of pseudo inverses from 26 based on the grevile s partitioning method we derive more general algorithms for computing 1 1 3 and 1 4 inverses of one variable rational and polynomial matrices introduced algorithms are implemented in the programming language mathematica illustrative examples on analytical matrices are presented

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

additive jordan derivations of reflexive algebras additive jordan derivations of certain reflexive algebras are investigated in particular additive jordan derivations of nest algebras on banach spaces are shown to be additive derivations

a class of numerical algorithms for computing outer inverses the aim of this study is to present a class of numerical algorithms for finding outer inverses with prescribed range and null space general convergence theorems are proven some particular cases of the general algorithm are introduced and demonstrated theoretical convergence speed and computational efficiency will further be supported by employing some numerical tests