invertibility of a linear combination of two matrices and partial orderings abstract in this paper we present some results relating different matrix partial orderings and the invertibility of a linear combination of ep matrices

convergence properties of the sbar d transformation and a fast and accurate numerical evaluation of molecular integrals the present work concerns the study of the convergence properties of the nonlinear s transformation a fast and accurate numerical evaluation of highly oscillatory semi infinite integrals involved in the analytic expressions of molecular integrals over slater type orbitals is obtained using the s transformation these semi infinite integrals involve spherical and reduced bessel functions the convergence properties are analysed and they show that the approximations obtained using the s approach converge to the exact values of the semi infinite integrals without any constraint the numerical tables show that the s method gives unprecedented accuracy with speed up by a factor 2 over the h method and a factor 10 over the transformation this illustrates the superiority of this new approach

generalized and hypergeneralized projectors abstract generalizations of the class of projectors are considered within the classes of normal and ep matrices some properties are derived and results on the sum difference and product of generalized and hypergeneralized projectors are given

some results on partial ordering and reverse order law of elements of c algebras abstract in this paper we establish some results relating star left star right star minus ordering and the reverse order law under certain conditions on moore penrose invertible elements of c algebras

on a partial order defined by the weighted moore penrose inverse the weighted moore penrose inverse of a matrix can be used to define a partial order on the set of mxn complex matrices and to introduce the concept of weighted ep matrices in this paper we study the weighted star partial order on the set of weighted ep matrices in addition some properties that relate the eigenprojection at zero with the weighted star partial order are obtained

numerical integration of functions with a sharp peak at or near one boundary using mo bius transformations abstract a class of coordinate transformations depending on a single parameter is studied as quadrature tool working rules for the choice of the parameter are proposed numerical tests for the method are presented they show that these coordinate transformations when combined with gauss legendre quadrature rules are well suited for the numerical integration of functions possessing a sharp peak at or near one boundary of the interval of integration a method to combine the transformations with automatic quadrature routines is also proposed it seems to be useful for the numerical evaluation of integrals with the same kind of integrand behavior

the star partial order and the eigenprojection at 0 on ep matrices abstract the space of n n complex matrices with the star partial order is considered in the first part of this paper the class of ep matrices is analyzed and several properties related to this order are given in addition some information about predecessors and successors of a given ep matrix is obtained the second part is dedicated to the study of some properties that relate the eigenprojection at 0 with the star and sharp partial orders

the hd and hd methods for accelerating the convergence of three center nuclear attraction and four center two electron coulomb integrals over b functions and their convergence properties abstract three center nuclear attraction and four center two electron coulomb integrals over slater type orbitals are required for ab initio and density functional theory dft molecular structure calculations they occur in many millions of terms even for small molecules and require rapid and accurate evaluation the b functions are used as a basis set of atomic orbitals these functions are well adapted to the fourier transform method that allowed analytical expressions for the integrals of interest to be developed rapid and accurate evaluation of these analytical expressions is now made possible by applying the hd and hd methods for accelerating the convergence of the semi infinite oscillatory integrals the convergence properties of the new methods are analysed the numerical results section shows the high predetermined accuracy and the substantial gain in the calculation times obtained using the new methods

two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series two new classes of nonlinear transformations the d transformation to accelerate the convergence of infinite integrals and the d transformation to accelerate the convergence of infinite series are presented in the course of the development of these transformations two interesting asymptotic expansions one for infinite integrals and the other for infinite series are derived the transformations d and d can easily be applied to infinite integrals 0f t dt whose integrands f t satisfy linear differential equations of the form f t s m k 1p k t f k t and to infinite series s r 1f r w terms f r satisfy a linear difference equation of the form f r s m k 1p k r d kf r such that in both cases the p k have asymptotic expansions in inverse powers of their arguments in order to be able to apply these transformations successfully one need not know explicitly the differential equation that the integrand satisfies or the difference equation that the terms of the series satisfy mere knowledge of the existence of such a differential or difference equation and its order m is enough this broadens the areas to which these methods can be applied the connection between the d and d transformations with some known transformations in shown the use and the remarkable efficiency of the d and d transformations are demonstrated through several numerical examples the computational aspects of these transformations are described in detail

an extremely efficient approach for accurate and rapid evaluation of three centre two electron coulomb and hybrid integrals over b functions analytic expressions of three centre two electron coulomb and hybrid integrals over b functions are obtained using the fourier transform method thoroughly explored by steinborn s group these analytic expressions involve semi infinite integrals which are slowly convergent due to the presence of hypergeometric and spherical bessel functions in the integrands we have proven that these hypergeometric functions can be expressed as finite expansions and the integrands involving these series satisfy all the conditions required to apply the h approach which greatly simplifies the application of the nonlinear transformation this work presents a rapid and accurate evaluation of these integrals obtained by using a new approach which we called s this new method is based on the h and methods and some practical properties of spherical bessel reduced bessel and sine functions the s method has greatly simplified the calculations avoiding the long and difficult implementation of the successive zeros of the spherical bessel function and a method for solving linear systems which are required by h and

relationships between dmp relation and some partial orders let h be an infinite dimensional complex hilbert space and let b h be the algebra of all bounded linear operators on h the equivalent definition of the dmp relation on b h using operator block matrices is introduced we present a new interpretation of this relation which allows to generalize many known results for matrices to general hilbert spaces regarding the dmp relation and its relationship with the star order and minus partial order also some properties of dmp relation defined with the help of idempotent operators are investigated

further results on generalized and hypergeneralized projectors abstract the notions of generalized and hypergeneralized projectors introduced by gros and trenkler j gro s j trenkler generalized and hypergeneralized projectors linear algebra appl 264 1997 463 474 are revisited on the one hand the present paper provides several new characterizations of these sets and on the other the properties of generalized and hypergeneralized projectors related to various matrix partial orderings are considered moreover the paper demonstrates the usefulness in studying the properties of generalized and hypergeneralized projectors of the representation of complex matrices given in corollary 6 by hartwig and spindelbock r e hartwig k spindelbock matrices for which a and a commute linear and multilinear algebra 14 1984 241 256

weighted binary relations involving the drazin inverse the drazin inverse of a matrix has been used in the literature to define a pre order on the set of square complex matrices in this paper we analyze new binary relations defined on the set of rectangular complex matrices and some relationships to the w support idempotent we introduce the class of weighted drazin equal projectors and analyze the pre orders on this class moreover adjacent matrices are studied under the considered relations finally some observations on weighted partial orders are given

efficient evaluation of coulomb integrals the nonlinear d and transformations the two electron multicentre coulomb integrals constitute the rate limiting step of ab initio and density functional theory dft molecular structure calculations speed up can be achieved by limiting the number of integrals to evaluate analytically but these analytical evaluations remain rate limiting for large molecules here we apply the nonlinear d and transformations to evaluate coulomb integrals over b functions more rapidly than the alternative transformation methods to a given predetermined high accuracy

weighted g drazin inverses and a new pre order on rectangular matrices abstract this paper deals with weighted g drazin inverses which is a new class of matrices introduced to extend to the rectangular case g drazin inverses recently considered by wang and liu for square matrices first we define and characterize weighted g drazin inverses next we consider a new pre order defined on complex rectangular matrices based on weighted g drazin inverses finally we characterize this pre order and relate it to the minus partial order and to the weighted drazin pre order

the drazin and generalized drazin invertibility of linear combinations of idempotents abstract let a be a unital complex banach algebra and p q be two idempotents in a in this paper we prove that the drazin resp generalized drazin invertibility of p q p q is independent of the choice of scalars c 0 and c we also give a simple representation formula for the drazin inverse of p q as an application we study the left resp right drazin invertibility and the property of being browder of linear combinations of two idempotent operators on a banach space

a characterization of the class of partial isometries abstract let b h be the set of all bounded linear operators on a hilbert space h in this paper we show that if s is a closed range operator with r s r s then s s s s ast 2 if and only if s is a non zero real scalar of a normal partial isometry also we find some other characterizations of this space using some inequalities related to corach porta recht inequality

nonsingularity of linear combinationsof idempotent matrices abstract gros and trenkler siam j matrix anal appl 21 1999 390 pointed out that if a difference of idempotent matrices p 1 and p 2 is nonsingular then so is their sum and koliha et al linear algebra appl in press expressed explicitly a condition which combined with the nonsingularity of p 1 p 2 ensures the nonsingularity of p 1 p 2 in the present note these results are strengthened by showing that the nonsingularity of p 1 p 2 is in fact equivalent to the nonsingularity of any linear combination c 1 p 1 c 2 p 2 wherein c1 c2 0 some other nonsingularity type relationships referring to linear combinations of p 1 and p 2 are also established

addition theorems for b functions and other exponentially declining functions in this paper addition theorems are derived for a special class of exponentially declining functions the so called b functions bmn l r e filter and e o steinborn phys rev a 18 1 1978 although these b functions have a relatively complicated analytical structure they nevertheless have some mathematical properties that are particularly advantageous in connection with multicenter problems also all the commonly occurring exponentially declining functions like for instance bound state hydrogen eigenfunctions and slater type functions can be expressed by simple finite sums of b functions consequently addition theorems for these functions can also be written down immediately the various addition theorems for b functions are derived by applying suitable generating differential operators to the well known addition theorem of the special b function bm l l which is that solution of the modified helmholtz equation that is irregular at the origin and regular at infinity e j weniger and e o

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

an algorithm for a generalization of the richardson extrapolation process in this paper we present a recursive method designated the w m algorithm for implementing a generalization of the richardson extrapolation process that has been ntroduced in 8 compared to the direct solution of the linear systems of equations defining the extrapolation procedure this method requires a small number of arithmetic operations and very little storage the technique is also applied to solve recursively the coefficient problem associated with the rational approximations obtained by applying the d transformation of 6 13 to power series in the course of development a new recursive algorithm for implementing a very general extrapolation procedure is introduced which is similar to that given in 2 4 for solving the same problem a fortran program for the w m algorithm is also appended

the properties of sine spherical bessel and reduced bessel functions for improving convergence of semi infinite very oscillatory integrals the evaluation of three centre nuclear attraction integrals over b functions this paper focuses on the use of useful properties of sine spherical bessel and reduced bessel functions to simplify the application of the nonlinear d and transformations for accelerating the convergence of semi infinite very oscillatory integrals and to reduce the calculation times keeping a high predetermined accuracy three centre nuclear attraction integrals which are one of the most difficult type involved in density functional theory methods when using a basis set of b functions are evaluated using the new approach the numerical results show the efficiency of the new method compared with other alternatives

the fourier transform method and the sd approach for the analytical and numerical treatment of multicenter overlap like quantum similarity integrals molecular overlap like quantum similarity measurements imply the evaluation of overlap integrals of two molecular electronic densities related by dirac delta function when the electronic densities are expanded over atomic orbitals using the usual lcao mo approach linear combination of atomic orbitals overlap like quantum similarity integrals could be expressed in terms of four center overlap integrals it is shown that by introducing the fourier transform of delta dirac function in the integrals and using the fourier transform approach combined with the so called b functions one can obtain analytic expressions of the integrals under consideration these analytic expressions involve highly oscillatory semi infinite spherical bessel functions which are the principal source of severe numerical and computational difficulties in this work we present a highly efficient algorithm for a fast and accurate numerical evaluation of these multicenter overlap like quantum similarity integrals over slater type functions this algorithm is based on the s d approach due to safouhi recurrence formulae are used for a better control of the degree of accuracy and for a better stability of the algorithm the numerical result section shows the efficiency of our algorithm compared with the alternatives using the one center two range expansion method which led to very complicated analytic expressions the epsilon algorithm and the nonlinear d transformation

three centre two electron coulomb and hybrid integrals evaluated using nonlinear d and bard transformations of the two electron integrals occuring in the molecular context the three centre coulomb and hybrid integrals are numerous and difficult to evaluate to high accuracy the analytical and numerical difficulties arise mainly from the presence of the spherical bessel function and hypergeometric series in these integrals the present work accelerates the convergence of these integrals by first manipulating the indices of the hypergeometric function and exploiting relationships to express this function as a finite expansion and exploiting the properties of bessel functions which satisfy second order linear differential equations this is a suitable form of the integrand to apply the nonlinear d due to d levin and a sidi and due to a sidi transformations the extensive numerical results section illustrates the accuracy and unprecedented efficiency of evaluation of these integrals

on group inverses and the sharp order abstract in this work the group inverse of a matrix is used to define the order on square matrices of index 1 the order is similar to the order of drazin 2 and the minus order of hartwig 6 10 and nambooripad 17 the order and the order are compared and contrasted many conditions are given which assure the equivalence of the various partial orders studied

the difference and sum of projectors abstract the aim of this paper is to present new results on the invertibility of the sum of projectors and new relations between the nonsingularity of the difference and the sum of projectors we also give simple proofs without referring to rank theory of some results on the nonsingularity of the difference of projectors obtained by gros and trenkler siam j matrix anal appl 21 1999 390

some properties of matrix partial orderings abstract the matrix partial orderings considered are 1 the star ordering and 2 the minus ordering or rank subtractivity both in the set of m n complex matrices and 3 the lowner ordering in the set of m m matrices the problems discussed are 1 inheriting certain properties under a given ordering 2 preserving an ordering under some matrix multiplications 3 relationships between an ordering among direct or kronecker and hadamard products and the corresponding orderings between the factors involved 4 orderings between generalized inverses of a given matrix and 5 preserving or reversing a given ordering under generalized inversions several generalizations of results known in the literature and a number of new results are derived

efficient and rapid numerical evaluation of the two electron four center coulomb integrals using nonlinear transformations and useful properties of sine and bessel functions two electron four center coulomb integrals are undoubtedly the most difficult type involved in ab initio and density functional theory molecular structure calculations millions of such integrals are required for molecules of interest therefore rapidity is the primordial criterion when the precision has been reached this work presents an extremely efficient approach for improving convergence of semi infinite very oscillatory integrals based on the nonlinear d transformation and some useful properties of spherical bessel reduced bessel and sine functions the new method is now shown to be applicable to evaluating the two electron four center coulomb integrals over b functions the section with numerical results illustrates the unprecedented efficiency of the new approach in evaluating the integrals of interest

left star and right star partial orderings abstract two partial orderings in the set of complex matrices are introduced by combining each of the conditions a a a b and aa ba which define the star partial ordering with one of the conditions m a m b and m a m b which define the space preordering several properties of these orderings are examined with main emphasis on comparing the new orderings with the star ordering the minus ordering and other related partial orderings moreover some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived with the main emphasis on characterizations involving reflexive generalized inverses