different complex zfs leading to different complex znn models for time varying complex generalized inverse matrices as a special class of recurrent neural network zhang neural network znn has been recently proposed since 2001 for solving various time varying problems and has shown high efficiency and excellent performance for solving the problems in the real domain in this paper to solve online the time varying complex generalized inverse in most cases the pseudoinverse problem in the complex domain a new type of complex valued znn is further proposed and investigated the design of such a complex znn is based on a complex zhang function zf which is indefinite and quite different from the usual error function specially the scalar valued energy function in the studies of conventional algorithms by introducing five different complex zfs five different complex znn models termed complex znn i znn ii znn iii znn iv and znn v models are proposed developed and investigated for the online solution of the time varying complex generalized inverse matrices theoretical results of convergence analysis are presented to show the desirable properties of complex znn models in addition we discover the link between the proposed complex znn models and the getz marsden dynamic system in the complex domain computer simulation results further demonstrate the effectiveness of complex znn models based on different complex zfs for the time varying complex generalized inverse matrices

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

asymptotic completeness of short range many body systems on annonce une demonstration de la completude asymptotique pour des systemes de mecanique quantique constitues de nombres arbitraires de particules interagissant par des forces a courte portee

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

full rank and determinantal representation of the drazin inverse abstract in this article we introduce a full rank representation of the drazin inverse a d of a given complex matrix a which is based on an arbitrary full rank decomposition of a l l k where k is the index of a using this general representation we introduce a determinantal representation of the drazin inverse more precisely we represent elements of the drazin inverse a d as a fraction of two expressions involving minors of the order rank a k k ind a taken from the matrices a and rank invariant powers a l l k also we examine conditions for the existence of the drazin inverse for matrices whose elements are taken from an integral domain finally a few correlations between the minors of the drazin inverse a d powers of the drazin inverse and the minors of the matrix a k k ind a are explicitly derived

innovation based on gaussian elimination to compute generalized inverse at s 2 abstract in this paper we execute elementary row and column operations on the partitioned matrix g a g g g 0 into i s 0 0 0 0 0 a t s 2 to compute generalized inverse a t s 2 of a given complex matrix a where g is a matrix such that r g t and n g s the total number of multiplications and divisions operations is t m n s 2 m n 2 4 m s 1 2 n s m s n s m n s and the upper bound of t m n s is less than 6 m n 2 3 2 n 3 1 2 n 2 when n m a numerical example is shown to illustrate that this method is correct

recurrent neural network for computing outer inverse two linear recurrent neural networks for generating outer inverses with prescribed range and null space are defined each of the proposed recurrent neural networks is based on the matrix valued differential equation a generalization of dynamic equations proposed earlier for the nonsingular matrix inversion the moore penrose inversion as well as the drazin inversion under the condition of zero initial state the application of the first approach is conditioned by the properties of the spectrum of a certain matrix the second approach eliminates this drawback though at the cost of increasing the number of matrix operations the cases corresponding to the most common generalized inverses are defined the conditions that ensure stability of the proposed neural network are presented illustrative examples present the results of numerical simulations

the 2 inverse with applications in statistics an inverse g of a given matrix a which satisfies the property gag g is known as a 2 inverse this paper presents a three phase inversion procedure for which the 2 inverse is a special case we present the geometry of 2 inverses and show that starting from 2 inverses various types of generalized inverses can be derived two examples of the occurrence of 2 inverses in statistics are given one concerning the constrained least squares estimator the other concerning a necessary and sufficient condition for a quadratic form of singular multivariate normal variates to follow a chi square distribution

perturbation theory for the generalized bott duffin inverse and its applications a perturbation theory for the bott duffin inverse a l of a with respect to a subspace l is developed the perturbation bound for the solution of the constrained system ax y b email protected l email protected l is established where email protected m n c a subspace email protected c n and email protected c n meanwhile this paper shows the generalized bott duffin condition number k g b d a email protected email protected email protected a l to be minimum in the inequality of error analysis

neural network approach to computing outer inverses based on the full rank representation abstract the first intention of this paper is to emphasize equivalence between two well known general representations of outer inverses with prescribed range and null space of a given matrix two dynamical state equations corresponding to particular expressions which are included in these representations are defined within the second objective in this way two gradient based neural networks initiated by introduced dynamical equations are exploited in generating the class of outer inverses the results corresponding to most common generalized inverses are obtained in particular cases numerical results are presented

on long range scattering even if special cases are not well understood it pays sometimes to consider a problem at hand in the most general framework thus addressing the existence of wave operators for the most general potentials hormander ho1 has discovered a natural asymptotic evolution for the long range scattering it is determined by a solution s k t of a certain hamilton jacobi equation

the expression of the generalized bott duffin inverse and its perturbation theory let l be a subspace of c n and p l be the orthogonal projection of c n onto l then for the nxn matrix a the generalized bott duffin b d inverse a l is given by a l p l ap l i p l in this paper we will give an expression of a l in the rest of the part the perturbation analysis of a l and the least squares solution of the generalized b d equation ax y b for x l and y l will be established under certain small perturbation of p lap l

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

representation and approximation of the outer inverse at s 2 of a matrix a abstract in this paper we establish a basic representation and a representation theorem for the outer inverse a t s 2 of a matrix a which is the matrix x satisfying xax x r x t and n x s we develop several specific representations and iterative methods for a t s 2 we show that this representation includes many of the traditional generalized inverses and outer inverses and the relation of our results to these outer inverses will be explored

on an integral transform with kernel o_ v 1 2 mu cos theta and its applications to second order canonical problems of gtd an arbitrary function defined for almost every theta in infty infty can be represented under certain restrictions by an integral with kernel o v 1 2 mu cos theta this representation has some analytical properties similar to fourier integrals one of the basic postulates of gtd the geometrical theory of diffraction which states high frequency diffraction as a local phenomenon makes this representation effectively suitable to solve second order canonical problems of gtd related to edges on spherical surfaces as an illustrative example the diffraction of a whispering gallery mode by a spherical cap is studied

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

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

transition functions for high frequency diffraction by a curved perfectly conducting wedge i canonical solution for a curved sheet starting from the canonical hilbert solution for a perfectly conducting cylindrically curved sheet of semi infinite angular extent illuminated by a line source transition functions are derived for illumination and or observation directions grazing the sheet surface or its geometrical continuation near the edge this is done by using the langer olver asymptotic formulas for certain hankel and bessel function factors in the hilbert solution both the soft transverse magnetic and the hard transverse electric cases are considered depending on the positions of the source and or the observer the resulting functions include the fock functions the incomplete airy function and its derivative and a pair of new canonical functions the results obtained can serve as a check for heuristic solutions for a curved wedge

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

grobner basis computation of drazin inverses with multivariate rational function entries in this paper we show how to apply grobner bases to compute the drazin inverse of a matrix with multivariate rational functions as entries when the coefficients of the rational functions depend on parameters we give sufficient conditions for the drazin inverse to specialize properly in addition we extend the method to weighted drazin inverses we present an empirical analysis that shows a good timing performance of the method

saddle point problems bott duffin inverses abstract splines and oblique projections abstract in this note we present links between several different areas in numerical analysis and operator theory more precisely we show that several saddle point problems are solvable in the presence of a property called compatibility between a positive operator and a closed subspace of a hilbert space and the same happens with certain abstract spline problems these notions are related to the generalized bott duffin inverse

integral representations of two generalized core inverses abstract in this paper we present integral representations for the dmp and core ep inverse which based on the full rank decomposition of a given matrix in particular integral representations of the core and dual core inverse are given all of these integral representations do not require any restriction on the spectrum of a certain matrix

perturbation bounds for the generalized inverses at s 2 with application to constrained linear system in this paper the more general perturbation problem of the generalized inverse a t s 2 is considered we derive its perturbation bounds when the subspaces t and s also have perturbation and in special case when t and s have no perturbation we get some results on the condition numbers with respect to a t s 2 and constrained linear system ax b x t which are similar to the regular inverse a 1 and nonsingular linear system ax b

symbolic and recursive computation of different types of generalized inverses abstract we propose a method and algorithm for recursive computation of different classes of generalized inverses of a given one variable rational matrix and corresponding algorithm for polynomial matrix these methods and algorithms are generalizations of the method for computing the generalized inverses for constant matrices originated in f e udwadia r e kalaba a unified approach for the recursive determination of generalized inverses comp math appl 37 1999 125 130 and the partitioning method for computing the generalized inverses 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 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 algorithms are implemented in the symbolic computational package mathematica

on continuity of the generalized inverse at s 2 this paper studies the continuity of the generalized inverse a t s 2 since common important generalized inverses such as a a m n a d a a d w a l 1 and a l etc are all generalized inverse a t s 2 in fact we give a unified continuity of such important generalized inverses

diffraction of high frequency waves by a cylindrically curved surface with different face impedances a high frequency solution is presented to the problem of diffraction of waves generated by a magnetic line source by the edges of a cylindrical curved surface with different face impedances by taking into account the locality of the high frequency diffraction phenomena the mixed boundary value problem is first reduced into a matrix boundary hilbert equation and then solved asymptotically by the hurd rawlins william s method explicit nonuniform expressions for the diffraction coefficients are obtained by evaluating the field integrals asymptotically

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

an improved method for the computation of the moore penrose inverse matrix abstract in this article we provide a fast computational method in order to calculate the moore penrose inverse of singular square matrices and of rectangular matrices the proposed method proves to be much faster and has significantly better accuracy than the already proposed methods while works for full and sparse matrices

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 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

spectral analysis of n body schrodinger operators for a large class of two body potentials we solve two of the main problems in the spectral analysis of multiparticle quantum hamiltonians explicitly we prove that the point spectrum lies in a closed countable set and describe that set in terms of the eigenvalues of hamiltonians of subsystems and that there is no singular continuous spectrum we accomplish this by extending mourre s work on three body problems to n body problems

the representation of generalized inverse at s 2 3 and its applications this paper presents the explicit expression for matrix right symmetry factor with prescribed rang and null space moreover the explicit expression for generalized inverse a t s 2 3 which is a 2 3 inverse of a having the prescribed rang t and null space s is derived as an application two numerical examples are given

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

on the p q outer generalized inverse in banach algebra abstract in this note the p q outer generalized inverse in banach algebra is considered a new representation based on the 1 5 inverse is presented also we construct an iterative method for computing the p q outer generalized inverse in banach algebra

iterative methods for computing the generalized inverses a 2 t s of a matrix a abstract in this paper the iterative methods for computing the generalized inverse a t s 2 the unique matrix x such that xax x r x t and n x s are established and the iterative methods for computing many important generalized inverses of matrices such as a a mn a d a a d w a l 1 a l and ap l are constructed

full rank representations of outer inverses based on the qr decomposition abstract an efficient algorithm for computing a t s 2 inverses of a given constant matrix a based on the qr decomposition of an appropriate matrix w is presented correlations between the derived representation of outer inverses and corresponding general representation based on arbitrary full rank factorization are derived in particular cases we derive representations of 2 4 and 2 3 inverses numerical examples on different test matrices dense or sparse are presented as well as the comparison with several well known methods for computing the moore penrose inverse and the drazin inverse

diffraction of an obliquely incident high frequency wave by a cylindrically curved sheet in the application of the geometrical theory of diffraction gtd to practical problems a main role is played by the diffraction coefficients of different kinds nevertheless the explicit expressions of the diffraction coefficients associated with edges on certain curved sheets are known only in the case of normal incidence the canonical problems from which these coefficients are derived can always be reduced to scalar diffraction problems involving two space variables in which either the electric or the magnetic field is parallel to the edge nevertheless when the incident ray is not normal to the edge at the diffraction point the related canonical problems inevitably involve three space coordinates and at least two dependent functions the complexity inherent in such problems caused a lack of knowledge about the diffraction coefficients connected with the edges on curved sheets in the case of oblique incidence the case of oblique incidence on a perfectly conducting cylindrical sheet is considered and some explicit expressions for the matrix diffraction coefficients are derived these matrices connect the diffracted far fields of different kinds to the incident electric and magnetic field components parallel to the edge at the diffraction point the expressions are rather simple and involve the coefficients related to the case of normal incidence in which k is replaced by k sin theta 0 k and theta 0 are the wavenumber and the incidence angle respectively the effect of the incidence angle in the attenuation constants associated with the creeping waves are dependent upon theta 0 only via the curvature radius of the geodesic lines

diffraction of the creeping waves generated on a perfectly conducting spherical scatterer by a ring source to analyze the diffraction of high frequency waves one turns often to the geometrical theory of diffraction gtd whose aim is to describe this phenomenon in terms of certain factors these factors involve among the others several diffraction coefficients showing the modifications to be considered when a ray is transformed into another one at a diffraction point the aim of this paper is to analyze the diffraction of creeping waves generated on a perfectly conducting spherical reflector and thereby to obtain explicit expressions for certain coefficients related to the diffractions occurring at the edges of spherically curved reflectors the analysis is performed by using an integral transform technique recently developed by one of the authors various ray contributions are isolated and fairly simple nonuniform expressions for the diffraction coefficients are obtained

the representation and computation of generalized inverse at s 2 this paper presents a novel representation for the generalized inverse a t s 2 based on this we give an algorithm to compute this generalized inverse as an application we use gauss jordan elimination to compute the weighted moore penrose inverse a m n and the drazin inverse a d

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

diffraction of a whispering gallery mode by the edge of a thin concave cylindrically curved surface when a cylindrically curved perfectly conducting concave surface is terminated abruptly an incident whispering gallery wg mode undergoes diffraction this phenomenon is studied here for the canonical problem of a thin boundary and an edge that is parallel to the cylinder axis to confine attention to diffraction by a single edge it is assumed that the boundary extends to infinity in an infinitely extended angular space that is equivalent to placing along some radial plane intersecting the boundary a perfect absorber for angularly propagating waves the problem can then be phrased as a functional equation problem of the hilbert type which is solved exactly an asymptotic approximation in the high frequency limit ka gg 1 where k is the wavelength and a the radius of curvature is obtained in explicit form and phrased so as to exhibit the diffraction coefficient for edge diffracted rays and the launching coefficients for creeping waves on the convex side as well as wg modes on the concave side these quantities are of interest for placing the results within the context of the geometrical theory of diffraction gtd and are relevant to the analysis of curved two dimensional reflectors of finite width where wg modes excited by diffraction of the incident field at one edge provide illumination of the other edge the induced surface current density function is also derived the asymptotic results including an observed shift of the shadow boundary relative to the direction of incidence of the wg mode are explained completely in terms of half plane diffraction of one of the modal ray congruences thereby confirming via this canonical problem the validity of the gtd for diffraction of modal fields for the limiting case a rightarrow infty all expressions are shown to reduce to those for the classical half plane problem

gauss jordan elimination method for computing outer inverses this paper deals with the algorithm for computing outer inverse with prescribed range and null space based on the choice of an appropriate matrix g and gauss jordan elimination of the augmented matrix g i the advantage of such algorithms is the fact that one can compute various generalized inverses using the same procedure for different input matrices in particular we derive representations of the moore penrose inverse the weighted moore penrose inverse the drazin inverse as well as 2 4 and 2 3 inverses numerical examples on different test matrices are presented as well as the comparison with well known methods for generalized inverses computation

full rank representations of 2 4 2 3 inverses and successive matrix squaring algorithm abstract we present the full rank representations of 2 4 and 2 3 inverses with given rank as well as with prescribed range and null space as particular cases of the full rank representation of outer inverses as a consequence two applications of the successive matrix squaring sms algorithm from p s stanimirovic d s cvetkovic ilic successive matrix squaring algorithm for computing outer inverses appl math comput 203 2008 19 29 are defined using the full rank representations of 2 4 and 2 3 inverses the first application is used to approximate 2 4 inverses the second application after appropriate modifications of the sms iterative procedure computes 2 3 inverses of a given matrix presented numerical examples clarify the purpose of the introduced methods

the generalized bott duffin inverse and its applications abstract the generalized bott duffin inverse a l of a with respect to a subspace l is defined and by means of it a unified treatment of a class of linear systems and extreme value problems with equality constraints is given

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

matrix left symmetry factor and its applications in generalized inverses a 2 4 t s in this paper we propose some definitions about matrix symmetry factors and discuss the construction of matrix left symmetry factor in detail the explicit expression of the generalized inverses at s 2 4 which is a 2 4 inverses of a having the prescribed rang t and null space s is derived furthermore using the expression at s 2 4 we get the corresponding results for six important generalized inverses

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

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

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