higher order efficient class of chebyshev halley type methods construction of two point sixth order methods for simple root is an ambitious and challenging task in numerical analysis therefore the main aim of this paper is to introduce a new highly efficient two point sixth order class of chebyshev halley type methods free from second order derivative for the first time each member of the proposed class requires only four functional evaluations viz two evaluations of function f and two of first order derivative f per full iteration a variety of concrete numerical examples illustrate that our proposed methods are more efficient and perform better than existing two point three point sixth order methods available in the literature from their dynamical study it has been observed that our proposed methods have better stability and robustness as compared to the other existing methods

two new families of sixth order methods for solving non linear equations in this paper we developed two new families of sixth order methods for solving simple roots of non linear equations per iteration these methods require two evaluations of the function and two evaluations of the first order derivatives which implies that the efficiency indexes of our methods are 1 565 these methods have more advantages than newton s method and other methods with the same convergence order as shown in the illustration examples finally using the developing methodology described in this paper two new families of improvements of jarratt method with sixth order convergence are derived in a straightforward manner notice that kou s method in jisheng kou yitian li an improvement of the jarratt method appl math comput 189 2007 1816 1821 and wang s method in xiuhua wang jisheng kou yitian li a variant of jarratt method with sixth order convergence appl math comput 204 2008 14 19 are the special cases of the new improvements

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

determinantal representations of closed orbits abstract in this paper we show that the orbit of a point mass under a central force f r r 2 r 3 is realized as the hyperbolic curve f a 1 x y 0 associated with a nilpotent matrix a on the contrary we show that the orbit of motion of particles of infinitesimal mass in the gravitational field described by schwarzschild geodesic metric is transcendental in this case the transcendental orbit has no determinantal representations

numerical solution of a system of fuzzy polynomials by fuzzy neural network in this paper a new approach for solving systems of fuzzy polynomials based on fuzzy neural network fnn is presented this method can also lead to improve numerical methods in this work an architecture of fuzzy neural networks is also proposed to find a real root of a system of fuzzy polynomials if exists by introducing a learning algorithm finally we illustrate our approach by numerical examples

on matrices whose numerical ranges have circular or weak circular symmetry in 18 among other equivalent conditions it is proved that a square complex matrix a is permutationally similar to a block shift matrix if and only if for any complex matrix b with the same zero pattern as a w b the numerical range of b is a circular disk centered at the origin in this paper we add a long list of further new equivalent conditions the corresponding result for the numerical range of a square complex matrix to be invariant under a rotation about the origin through an angle of 2 m where m 2 is a given positive integer is also proved many interesting by products are obtained in particular on the numerical range of a square nonnegative matrix a the following unexpected results are established i when the undirected graph of a is connected if w a is a circular disk centered at the origin then so is w b for any complex matrix b with the same zero pattern as a ii when a is irreducible if is an eigenvalue in the peripheral spectrum of a that lies on the boundary of w a then is a sharp point of w a we also obtain results on the numerical range of an irreducible square nonnegative matrix which strengthen or clarify the work of issos 9 and nylen and tam 14 on this topic open questions are posed at the end

classification of joint numerical ranges of three hermitian matrices of size three abstract the joint numerical range w f of three hermitian 3 by 3 matrices f f 1 f 2 f 3 is a convex and compact subset in r 3 generically we find that w f is a three dimensional oval assuming dim w f 3 every one or two dimensional face of w f is a segment or a filled ellipse we prove that only ten configurations of these segments and ellipses are possible we identify a triple f for each class and illustrate w f using random matrices and dual varieties

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

the numerical range of a nonnegative matrix abstract we offer an almost self contained development of perron frobenius type results for the numerical range of an irreducible nonnegative matrix rederiving and completing the previous work of issos nylen and tam and tam and yang on this topic we solve the open problem of characterizing nonnegative matrices whose numerical ranges are regular convex polygons with center at the origin some related results are obtained and some open problems are also posed

the upper numerical range of a quaternionic matrix is not a complex numerical range abstract a recent result is that the quaternionic numerical range of a matrix with quaternion entries has a convex intersection with the upper half complex plane this intersection is now shown to be generally not achievable as the upper half plane part of the complex numerical range of any complex matrix a key step in the proof is that if a complex matrix has an elliptical arc as part of the boundary of its complex numerical range then the full ellipse defined by the arc is also in its complex numerical range

hyperbolic forms admit reversible weighted shift determinantal representation abstract lentzos and pasley recently showed that a weakly circular symmetric invariant hyperbolic ternary form admits a determinantal representation through a cyclic weighted shift matrix in this paper we improve the representing matrix by employing a cyclic weighted shift with reversible weights

on fuzzy associative memory with multiple rule storage capacity kosko s fuzzy associative memory fam is the very first neural network model for implementing fuzzy systems despite its success in various applications the model suffers from very low storage capacity i e one rule per fam matrix a lot of hardware and computations are usually required to implement the model and hence it is limited to applications with small fuzzy rule base in this paper the inherent property for storing multiple rules in a fam matrix is identified a theorem for perfect recalls of all the stored rules is established and based upon which the hardware and computation requirements of the fam model can be reduced significantly furthermore we have shown that when the fam model is generalized to the one with max bounded product composition single matrix implementation is possible if the rule base is a set of semi overlapped fuzzy rules rule modification schemes are also developed and the inference performance of the established high capacity models is reported through a numerical example

reduction of the c numerical range to the classical numerical range abstract let a be an n n complex matrix and c c 1 c 2 c n a real n tuple the c numerical range of a is defined as the set w c a j 1 n c j x j ax j x 1 x 2 x n is an orthonormal basis for c n when c 1 0 0 w c a becomes the classical numerical range of a which is often defined as the set w a x ax x c n x x 1 we show that for any n n complex matrix a and real n tuple c there exists a complex matrix b of size at most n such that w c a w b constructions of the matrix b for some matrices a and real n tuple c are provided

an optimal reconstruction of chebyshev halley type methods for nonlinear equations having multiple zeros abstract establishment of a new optimal higher order iterative scheme for multiple zeros with known multiplicity m 1 of univariate function is one of the hard and demanding task in the area of computational mathematics and numerical analysis we write this paper with the aim to propose a new higher order reconstruction of chebyshev halley type iteration functions for multiple zeros in a simple way minimum order of convergence of our scheme is seven and maximum reaches to eight without using any extra functional evaluation with regard to computational cost each member of our scheme needs four functional evaluations at each step so the maximum efficiency index of our scheme is 1 6818 for 1 in addition we thoroughly study the convergence analysis which confirms the theoretical order of convergence moreover we check the efficiency and compare them to the existing ones on four real life problems namely fractional conversion chemical engineering continuous stirred tank reactor and eigen value problems and two well known academic problems finally we confirm on the basis of obtained results that our methods have better residual error errors between two consecutive iterations and stable computational order of convergence as compared to the other existing ones

construction of determinantal representation of trigonometric polynomials abstract for a pair of n n hermitian matrices h and k a real ternary homogeneous polynomial defined by f t x y det ti n xh yk is hyperbolic with respect to 1 0 0 the fiedler conjecture or lax conjecture is recently affirmed namely for any real ternary hyperbolic polynomial f t x y there exist real symmetric matrices s 1 and s 2 such that f t x y det ti n xs 1 ys 2 in this paper we give a constructive proof of the existence of symmetric matrices for the ternary forms associated with trigonometric polynomials

the perron eigenspace of nonnegative almost skew symmetric matrices and levinger s transformation let a be a nonnegative square matrix whose symmetric part has rank one tournament matrices are of this type up to a positive shift by 1 2i when the symmetric part of a is irreducible the perron value and the left and right perron vectors of l a 1 a at are studied and compared as functions of 0 1 2 in particular upper bounds are obtained for both the perron value and its derivative as functions of the parameter via the notion of the q numerical range

on computing the generalized crawford number of a matrix abstract we use methods of geometric computing combined with hermitean matrix eigenvalue eigenvector evaluations to find the generalized crawford number of a pair of hermitean matrices s 1 and s 2 quickly and to high precision the classical crawford number is defined as the minimal distance from zero in c to the field of values of the associated complex matrix a s 1 is 2 if zero lies outside the field of values of a we describe test and compare a geometry based matlab code for finding for the generalized crawford number that measures the smallest distance of zero from the boundary of the field of values of a even if zero lies inside

basins of attraction for several methods to find simple roots of nonlinear equations there are many methods for solving a nonlinear algebraic equation the methods are classified by the order informational efficiency and efficiency index here we consider other criteria namely the basin of attraction of the method and its dependence on the order we discuss several third and fourth order methods to find simple zeros the relationship between the basins of attraction and the corresponding conjugacy maps will be discussed in numerical experiments the effect of the extraneous roots on the basins is also discussed

an efficient optimal family of sixteenth order methods for nonlinear models abstract the principle aim of this manuscript is to propose a general scheme that can be applied to any optimal iteration function of order eight whose first substep employ newton s method to further develop new interesting optimal scheme of order sixteen this scheme requires four evaluations of the involved function and one evaluation of its first order derivative at each step so it is being optimally consistent with the conjecture of kung traub in addition theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence moreover the conjugacy maps and the strange fixed points of some iterative methods are discussed their basins of attractions are also given to show their dynamical behavior around the simple roots from the numerical experiments we find that our methods perform better than the existing ones when we checked the performance on a concrete variety of nonlinear equations

on developing a higher order family of double newton methods with a bivariate weighting function a high order family of two point methods costing two derivatives and two functions are developed by introducing a two variable weighting function in the second step of the classical double newton method their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases a variety of concrete numerical examples and relevant results are extensively treated to verify the underlying theoretical development in addition this paper investigates the dynamics of rational iterative maps associated with the proposed method and an existing method based on illustrated description of basins of attraction for various polynomials

boundary generating curves of the c numerical range abstract let a be an n n matrix and c c 1 c 2 c n be a real n tuple the c numerical range of a is the set w c a j 1 n c j x j ax j x 1 x 2 x n is an orthonormal basis of c n we obtain parametric representations of the boundary generating curve of the c numerical range of a matrix applying this result we generalize the result of anderson to the c numerical range furthermore we give a description of the boundary generating curve of the c numerical range of certain types of nilpotent toeplitz matrices a sufficient condition for the boundary generating curve to be rational is obtained finally we explicitly compute the boundary generating curves of the numerical ranges for several concrete matrices and classify the rationality of the curves

the numerical range of 3 3 matrices abstract let a be an n n complex matrix then the numerical range of a w a is defined to be x ax x c n x x 1 in this article a series of tests is given allowing one to determine the shape of w a for 3 3 matrices reconstruction of a up to unitary similarity from w a is also examined

extraneous fixed points basin boundaries and chaotic dynamics for schro der and ko nig rational iteration functions the schroder and konig iteration schemes to find the zeros of a polynomial functiong z represent generalizations of newton s method in both schemes iteration functionsf m z are constructed so that sequencesz n 1 f m z n converge locally to a rootz ofg z aso z n z m it is well known that attractive cycles other than the zerosz may exist for newton s method m 2 asm increases the iteration functions add extraneous fixed points and cycles whether attractive or repulsive they affect the julia set basin boundaries the konig functionsk m z appear to minimize such perturbations in the case of two roots e g g z z 2 1 cayley s classical result for the basins of attraction of newton s method is extended for allk m z the existence of chaotic z n sequences is also demonstrated for these iteration methods

fuzzy data processing using polynomial bidirectional hetero associative network abstract this investigation presents a novel method of fuzzy data processing using polynomial bidirectional hetero associative network pbhan this has a higher capacity for pattern pair storage than that of the conventional bidirectional associative memories bams and fuzzy memories in addition a new energy function is defined the pbhan takes advantage of fuzzy characteristics in evolution equations such that the signal noise ratio snr is significantly increased the energy of the pbhan defined by the proposed energy function decreases as the recall process proceeds thereby ensuring the stability of the system in this work we prove the stability of fuzzy data processing using pbhan the increase of snr consequently enhances the capacity of the pbhan the capacity of the fuzzy data processing using pbhan in the worst case is also estimated

some improvements of jarratt s method with sixth order convergence abstract in this paper we present a one parameter family of variants of jarratt s fourth order method for solving nonlinear equations it is shown that the order of convergence of each family member is improved from four to six even though it adds one evaluation of the function at the point iterated by jarratt s method per iteration several numerical examples are given to illustrate the performance of the presented methods

on the lengths of jordan chains for eigenvalues on the boundary of numerical range of quadratic operator polynomial abstract we describe a connection between the geometrical structure of the numerical range w l of a selfadjoint quadratic operator polynomial l and the lengths of jordan chains for eigenvalues on the boundary of the numerical range also for each pair of natural numbers m n 1 m 2 n we construct a monic quadratic polynomial l m n 2 c b with selfadjoint n n matrices c b which has an eigenvalue 0 w l m n with a jordan chain of length m

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

perron frobenius type results on the numerical range we present results connecting the shape of the numerical range to intrinsic properties of a matrix a w hena is a nonnegative matrix these results are to a large extent analogous to the perron frobenius theory especially as it pertains to irreducibility and cyclicity in the combinatorial sense special attention is given to polygonal circular and elliptic numerical ranges the main vehicles for obtaining these results are the hermitian and skew hermitian parts of a as well as levinger s transformation aa 1 a a 2002 elsevier science

on the geometry of numerical ranges in spaces with an indefinite inner product geometric properties of the numerical ranges of operators on an indefinite inner product space are investigated in particular classes of matrices are presented such that the boundary generating curves of the j numerical range are hyperbolical the curvature of the j numerical range at a boundary point is studied generalizing results of fiedler on the classical numerical range

point equation of the boundary of the numerical range of a matrix polynomial abstract the numerical range of an n n matrix polynomial p a m m a m 1 m 1 a 1 a 0 is defined by w p c x p x 0 x c n x 0 for the linear pencil p i a the range w p coincides with the numerical range of matrix a f a x ax x c n x x 1 in this paper we obtain necessary conditions for the origin to be a boundary point of f a as a consequence an algebraic curve of degree at most 2 n n 1 m which contains the boundary of w p is constructed

a family of modified ostrowski methods with accelerated sixth order convergence based on ostrowski fourth order multipoint method we derive a one parameter family of sixth order methods for solving equations each member of the family requires three evaluations of the given function and one evaluation of its derivative per iteration numerical examples are presented and performance is compared with ostrowski method

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

simply constructed family of a ostrowski s method with optimal order of convergence in this paper we propose a simple modification over chun s method for constructing iterative methods with at least cubic convergence 5 using iteration formulas of order two we now obtain several new interesting families of cubically or quartically convergent iterative methods the fourth order family of ostrowski s method is the main finding of the present work per iteration this family of ostrowski s method requires two evaluations of the function and one evaluation of its first order derivative therefore the efficiency index of this ostrowski s family is e 43 1 587 which is better than those of most third order iterative methods e 33 1 442 and newton s method e 2 1 414 the performance of ostrowski s family is compared with its closest competitors namely ostrowski s method jarratt s method and king s family in a series of numerical experiments

condition for the numerical range to contain an elliptic disc for an n by n matrix a and an elliptic disc e in the plane we show that the sum of the number of common supporting lines and the number of common intersection points to e and the numerical range w a of a should be at least 2n 1 in order to guarantee that e be contained in w a this generalizes previous results of anderson and thompson as an application our result is used to verify a special case of the poncelet property conjecture 2003 elsevier science inc all rights reserved

basin attractors for various methods abstract there are many methods for the solution of a nonlinear algebraic equation the methods are classified by the order informational efficiency and efficiency index here we consider other criteria namely the basin of attraction of the method and its dependence on the order we discuss several methods of various orders and present the basin of attraction for several examples it can be seen that not all higher order methods were created equal newton s halley s murakami s and neta johnson s methods are consistently better than the others in two of the examples neta s 16th order scheme was also as good

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

differential properties of the numerical range map of pairs of matrices abstract let a a 1 a 2 be a pair of hermitian operators in c n and a a1 ia2 we investigate certain differential properties of the numerical range map n a x a 1 x x a 2 x x with the aim of better understanding the nature of the numerical range w a of a for example the joint eigenvalues of a correspond to the stationary points of n a i e points where the derivative n a vanishes moreover points x where rank n a x 2 get mapped by n a into the interior w a of w a for n 2 it turns out that if a1 and a2 have no common invariant subspace then the image under n a of the set 1 a consisting of those points x with rank n a x 1 is precisely the boundary w a of w a and the image under n a of the rank 2 points for n a is precisely w a there are no rank 0 points for n a as a consequence for n 2 we have that a1a2 a2a1 iff 1 a n a 1 w a

a survey of canonical forms and invariants for unitary similarity abstract matrices a and b are said to be unitarily similar if u au b for some unitary matrix u this expository paper surveys results on canonical forms and invariants for unitary similarity the first half gives a detailed description of methods developed by several authors brenner littlewood mitchell mcrae radjavi serge chuk and benedetti and cragnolini using inductively defined reduction procedures to transform matrices to canonical form the matrix is partitioned and successive unitary similarities applied to reduce the submatrices to some nice form at each stage one refines the partition and restricts the set of permissible unitary similarities to those that preserve the already reduced blocks the process ends in a finite number of steps producing both the canonical form and the subgroup of the unitary group that preserves that form depending on the initial step various canonical forms may be defined the method can also be used to define canonical forms relative to certain subgroups of the unitary group and canonical forms for finite sets of matrices under simultaneous unitary similarity the remainder of the paper surveys results on unitary invariants and other topics related to unitary similarity such as the specht pearcy trace invariants the numerical range and unitary reducibility

hermitian pencils with a cubic minimal polynomial let a be an n n complex matrix and write a h ik where h and k are hermitian matrices we show that if the minimal polynomial of the pencil xh yk has degree 3 then there is a unitary matrix u such that u 1au is block diagonal with blocks of size 3 3 or smaller this is a special case of a conjecture made by kippenhahn in 1951

numerical range of matrices and levinger s theorem abstract this paper is in a sense a continuation of the author s previous paper on the numerical range of matrices in particular its connection to levinger s theorem for the case of nonnegative matrices is extended for general complex matrices an elementary proof of levinger s theorem is also included