condition numbers for the tensor rank decomposition abstract the tensor rank decomposition problem consists of recovering the parameters of the model from an identifiable low rank tensor these parameters are analyzed and interpreted in many applications as tensors are often perturbed by measurement errors in practice one must investigate to what extent the unique parameters change in order to preserve the validity of the analysis the magnitude of this change can be bounded asymptotically by the product of the condition number and the norm of the perturbation to the tensor this paper introduces a condition number that admits a closed expression as the inverse of a particular singular value of terracini s matrix which represents the tangent space to the set of tensors of fixed rank a practical algorithm for computing this condition number is presented the latter s elementary properties such as scaling and orthogonal invariance are established rank 1 tensors are always well conditioned the class of weak 3 orthogonal tensors which includes orthogonally decomposable tensors contains both well conditioned and ill conditioned problems numerical experiments confirm that the condition number yields a good estimate of the magnitude of the change of the parameters when the tensor is perturbed

improving newton raphson method for nonlinear equations by modified adomian decomposition method in this paper we present some efficient numerical algorithms for solving nonlinear equations based on newton raphson method the modified adomian decomposition method is applied to construct the numerical algorithms some numerical illustrations are given to show the efficiency of algorithms

classroom note geometry and convergence of euler s and halley s methods we investigate the global convergence of euler s and halley s methods by using their geometric interpretation a generalization of these methods is also briefly discussed

homotopy techniques for tensor decomposition and perfect identifiability let t be a general complex tensor of format n1 nd when the fraction i ni 1 i ni 1 is an integer and a natural inequality called balancedness is satisfied it is expected that t has finitely many minimal decomposition as a sum of decomposable tensors we show how homotopy techniques allow us to find all the decompositions of t starting from a given one in particular this gives a guess true with high probability about the total number of these decompositions this guess matches exactly with all cases previously known and predicts several unknown cases some surprising experiments yielded two new cases of generic identifiability formats 3 4 5 and 2 2 2 3 which have a unique decomposition as the sum of 6 and 4 decomposable tensors respectively we conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds i e the only identifiable cases building on the computational experiments we also use basic tools from algebraic geometry to prove these two new cases are indeed generically identifiable

induction for secant varieties of segre varieties this paper studies the dimension of secant varieties to segre varieties the problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry an inductive procedure is built around the ideas of successive specializations of points and projections this reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases as applications of the technique we give a complete classification of defective p secant varieties to segre varieties for p 6 we generalize a theorem of catalisano geramita gimigliano on non defectivity of tensor powers of pn we determine the set of p for which unbalanced segre varieties have defective p secant varieties in addition we completely describe the dimensions of the secant varieties to the deficient segre varieties p1 x p1 x pn x pn and p2 x p3 x p3 in the final section we propose a series of conjectures about defective segre varieties

typical tensorial rank abstract upper bounds on the typical rank r n m l of tensors maximal border rank rank of almost all tensors of a given shape n m l are presented these improve previous results by a tkinson and lloyd for cubic shape tensors the typical rank is determined exactly r n n n n3 3n 2 n 3

an efficient hybrid method for solving systems of nonlinear equations this paper focuses on methods of solving systems of nonlinear equations that require fewer computations than those using regular methods the proposed method presents a new technique to approximate the jacobian matrix needed in the process which still retains the good features of newton s type methods but which can reduce the amount of work

on computational efficiency for multi precision zero finding methods an adapted definition related to multi precision arithmetic zero finding methods is considered and several improvements to iterative methods to compute nonlinear equation solutions which increase the local order of convergence are revisited furthermore a higher efficiency when we use a definition related to adaptive multi precision arithmetic is also given numerical results computed with a floating point system representing 200 and 1000 decimal digits support this theory

a family of three point methods of optimal order for solving nonlinear equations a family of three point iterative methods for solving nonlinear equations is constructed using a suitable parametric function and two arbitrary real parameters it is proved that these methods have the convergence order eight requiring only four function evaluations per iteration in this way it is demonstrated that the proposed class of methods supports the kung traub hypothesis 1974 3 on the upper bound 2 n of the order of multipoint methods based on n 1 function evaluations consequently this class of root solvers possesses very high computational efficiency numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations

predictor corrector halley method for nonlinear equations abstract in this paper we suggest and analyze a new two step predictor corrector type iterative methods for solving nonlinear equations of the type f x 0 by using the technique of updating the solution this method can be viewed as a predictor corrector iterative halley s method we also consider the convergence analysis of the proposed method to illustrate the efficiency of this new method we give several examples and a comparison with the method of grau and diaz barrero m grau j l diaz barrero an improvement to ostrowski root finding method appl math comput 173 2006 450 456 is given this method can be considered as a refinement and improvement of the halley method

ostrowski type methods for solving systems of nonlinear equations abstract four generalized algorithms builded up from ostrowski s method for solving systems of nonlinear equations are written and analyzed a development of an inverse first order divided difference operator for functions of several variables is presented as well as a direct computation of the local order of convergence for these variants of ostrowski s method furthermore a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced

a new iteration method for solving algebraic equations in this paper a nonlinear algebraic equation f x 0 by applying taylor s theorem is converted into a coupled iteration systeml x n 1 g x n 0andg x n f x n l x n wherel x ax 2 bx c the formula is of high convergence some examples are given

modified iterative methods with cubic convergence for solving nonlinear equations abstract in this paper we suggest and analyze two new predictor corrector methods for solving nonlinear equations f x 0 these methods can be considered as two step methods we show that these methods have cubic convergence using this one of the new methods real or complex roots for certain nonlinear equations can be obtained several numerical examples are given to illustrate the efficiency and performance of the new methods our method can be considered as an improvement of the existing methods and can be viewed as an alternative to the existing methods

the chiral anomaly from path integral method to differential geometric approach as a continuation of a previous investigation the exterior differential formulae for non abelian anomalies are derived within the reformulated path integral scheme and a number of differential geometric objectives are traced out reversely with the help of these objects a compact form for the gauged wess zumino effective action induced by finite chiral transformation is also derived in the path integral formalism

a new modified halley method without second derivatives for nonlinear equation in a recent paper noor and noor k inayat noor m aslam noor predictor corrector halley method for nonlinear equations appl math comput in press doi 10 1016 j amc 11 023 have suggested and analyzed a predictor corrector method halley method for solving nonlinear equations in this paper we modified this method by using the finite difference scheme which has a quintic convergence we have compared this modified halley method with some other iterative of fifth orders convergence methods which shows that this new method is a robust one several examples are given to illustrate the efficiency and the performance of this new method

iterative methods improving newton s method by the decomposition method in this paper we present a sequence of iterative methods improving newton s method for solving nonlinear equations the adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses the orders of convergence are derived analytically and then rederived by applying symbolic computation of maple some numerical illustrations are given

the derivation of the regularized chiral jacobian using the zeta function method using the zeta function method a general formula for the regularized chiral jacobian to theories including non hermitian dirac operators d defined in arbitrary even dimensional euclidean space is derived the agreement of this formula with the results obtained in the differential geometric approach is also clarified

improvements of the efficiency of some three step iterative like newton methods an improvement of the local order of convergence is presented to increase the efficiency of the iterative method with an appropriate number of evaluations of the function and its derivative the third and fourth order of known two step like newton methods have been improved and the efficiency has also been increased

on waringʼs problem for systems of skew symmetric forms abstract this paper is devoted to a problem of finding the smallest positive integer s m n k such that m 1 generic skew symmetric k 1 forms in n 1 variables as linear combinations of the same s m n k decomposable skew symmetric k 1 forms this problem is analogous to a well known problem called waring s problem for symmetric forms and can be very naturally translated into a classical problem in algebraic geometry in this paper we will go through some basics of algebraic geometry describe how objects in algebraic geometry can be associated to systems of skew symmetric forms and discuss algebro geometric approaches to establish the existence of triples m n k where s m n k is more than expected

modified householder iterative method free from second derivatives for nonlinear equations abstract in this paper we suggest and analyze a new two step predictor corrector type iterative method free from second derivatives for solving nonlinear equations of the type f x 0 this new method includes the two step newton method as a special case we prove that the new iterative method is of fourth order several examples are given to illustrate the efficiency of this new method and its comparison with other iterative methods this method can be considered as a significant improvement of the newton method and its variant forms

grassmann secants identifiability and linear systems of tensors abstract for any irreducible non degenerate variety x p r we give a criterion for the k s identifiability of x if k s 1 r then the k s identifiability holds for x if and only if the s identifiability holds for the segre product seg p k x moreover if the s th secant variety of x is not defective and it does not fill the ambient space then we can produce a family of pairs k s for which the k s identifiability holds for x

a new formulation of the path integral method for the chiral anomaly using a comoving representation the authors reformulate the path integral method for the chiral anomaly and show that the anomaly comes from the accumulating effect in argument change of diagonalised matrix elements for non hermitian operators the analysis of the wess zumino consistency condition reveals further that it is the contribution of zero modes which needs to be considered thus a new regularisation scheme is proposed which leads to a unified expression for the anomaly in 2n dimensions both for abelian and non abelian cases

a probabilistic algorithm for the secant defect of grassmann varieties abstract in this paper we study the higher secant varieties of grassmann varieties in relation to the functional waring s problem for alternating tensors and to the alexander hirschowitz theorem we show how to identify defective higher secant varieties of grassmannians using a probabilistic method involving terracini s lemma and we describe an algorithm which can compute by numerical methods dim ssg k n for n 14 our main result is that except for grassmannians of lines if n 14 and k n 1 2 if n 14 we have studied the case k 5 there are only the four known defective cases s2g 2 6 s2g 3 7 s3g 3 7 and s3g 2 8

a one parameter square root family of two step root finders a one parameter family of iterative methods for solving nonlinear equations is constructed all the methods of the proposed family are cubically convergent for a simple root except one particular method which attains the fourth order without the increase of computational cost these methods belong to the class of two step methods and require three function evaluations per iteration the square root structure of the family provides finding a complex zero of real functions in some cases a comparison analysis shows that the presented family generates the methods which are comparable or even superior than the existing two step iterative methods of the third order

new iterative methods for solving nonlinear equation by using homotopy perturbation method in this paper we suggest and analyze a new class of iterative methods for solving nonlinear equations by using the homotopy perturbation method convergence of their method is also considered here we also discuss the efficiency index and computational order of convergence of new methods several numerical examples are given to illustrate the efficiency and performance of these new methods these new iterative methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations

uniqueness of nonnegative tensor approximations we show that for a nonnegative tensor a best nonnegative rank r approximation is almost always unique its best rank one approximation may always be chosen to be a best nonnegative rank one approximation and the set of nonnegative tensors with nonunique best rank one approximations forms an algebraic hypersurface we show that the last part holds true more generally for real tensors and thereby determine a polynomial equation so that a real or nonnegative tensor that does not satisfy this equation is guaranteed to have a unique best rank one approximation we also establish an analogue for real or nonnegative symmetric tensors in addition we prove a singular vector variant of the perron frobenius theorem for positive tensors and apply it to show that a best nonnegative rank r approximation of a positive tensor can never be obtained by deflation as an aside we verify that the euclidean distance ed discriminants of the segre variety and the veronese variety are hypersurfaces and give defining equations of these ed discriminants

a link between quantum entanglement secant varieties and sphericity in this paper we shed light on the relations between three concepts studied in representation theory algebraic geometry and quantum information theory first spherical actions of reductive groups on projective spaces second secant varieties of homogeneous projective varieties and the related notions of rank and border rank third quantum entanglement our main result concerns the relation between the problem of the state reconstruction from its reduced one particle density matrices and the minimal number of separable summands in its decomposition more precisely we show that sphericity implies that states of a given rank cannot be approximated by states of a lower rank we call states for which such an approximation is possible exceptional states for three important from a quantum entanglement perspective cases of distinguishable fermionic and bosonic particles we also show that non sphericity implies the existence of exceptional states remarkably the exceptional states belong to non bipartite entanglement classes in particular we show that the w type states and their appropriate modifications are exceptional states stemming from the second secant variety for three cases above we point out that the existence of the exceptional states is a physical obstruction for deciding the local unitary equivalence of states by means of the one particle reduced density matrices finally for a number of systems of distinguishable particles with a known orbit structure we list all exceptional states and discuss their possible importance in entanglement theory

an improvement of the jarratt method in this paper we present a variant of jarratt method for solving non linear equations per iteration the new method adds the evaluation of the function at another point in the procedure iterated by jarratt method as a consequence the local order of convergence is improved from four for jarratt method to six for the new method the new multistep iteration scheme based on the new method is developed and numerical tests verifying the theory are also given

secant varieties of the varieties of reducible hypersurfaces in pn given the space v p d n 1 n 1 1 of forms of degree d in n variables and given an integer l 1 and a partition of d d1 dr it is in general an open problem to obtain the dimensions of the l secant varieties l xn 1 for the subvariety xn 1 v of hypersurfaces whose defining forms have a factorization into forms of degrees d1 dr modifying a method from intersection theory we relate this problem to the study of the weak lefschetz property for a class of graded algebras based on which we give a conjectural formula for the dimension of l xn 1 for any choice of parameters n l and this conjecture gives a unifying framework subsuming all known results moreover we unconditionally prove the formula in many cases considerably extending previous results as a consequence of which we verify many special cases of previously posed conjectures for dimensions of secant varieties of segre varieties in the special case of a partition with two parts i e r 2 we also relate this problem to a conjecture by froberg on the hilbert function of an ideal generated by general forms

some sixth order zero finding variants of chebyshev halley methods a zero finding technique in which the order of convergence is improved and nonlinear equations are solved more efficiently than they are solved by traditional iterative methods is derived composing a modified chebyshev halley method with a variant of this method that just introduces one evaluation of the function the iterative methods presented are obtained by carrying out this procedure the output numerical results show that the new methods compete in both order and efficiency with the modified chebyshev halley methods

on generic identifiability of symmetric tensors of subgeneric rank we prove that the general symmetric tensor in sdcn 1 of rank r is identifiable provided that r is smaller than the generic rank that is its waring decomposition as a sum of r powers of linear forms is unique only three exceptional cases arise all of which were known in the literature our original contribution regards the case of cubics d 3 while for d 4 we rely on known results on weak defectivity by ballico ciliberto chiantini and mella

fifth order iterative methods for solving nonlinear equations abstract in this paper we suggest and analyze a new two step iterative method for solving nonlinear equation f x 0 by rewriting the given nonlinear equation as a coupled system of equations and using the taylor series it is shown that this new iterative method is of fifth order several examples are given to illustrate its performance and efficiency comparison with other methods is also given this method can be viewed as an improvement of the halley s method

a hummel seebeck family of iterative methods with improved convergence and efficiency several improvements to a family of iterative methods deduced from a hummel seebeck theorem are presented a symbolic computation that allows to find the best coefficients respect to the local order of convergence is also given the theoretical and computational order of convergence for all methods is increased furthermore the efficiency of these methods applied to the functions tested is improved adapting the strategy presented here a new iteration function with a new evaluation of the function is obtained and using adaptive multi precision arithmetic a smaller cost is got the numerical results computed carrying out this procedure with a floating point system representing 1000 decimal digits support this theory

geometry and the complexity of matrix multiplication it has been known for some time that algebraic geometry and representation theory are useful for proving lower bounds on the complexity of the matrix multiplication operator in this talk i will explain how geometry can be used to prove both upper and lower bounds after a discussion of the problem and its history i will present very recent work with m michalek on border rank algorithms faculty host eric allender

on a steffensen hermite method of order three in this paper we study a third order steffensen type method obtained by controlling the interpolation nodes in the hermite inverse interpolation polynomial of degree 2 we study the convergence of the iterative method and we provide new convergence conditions which lead to bilateral approximations for the solution it is known that the bilateral approximations have the advantage of offering a posteriori bounds of the errors the numerical examples confirm the advantage of considering these error bounds

the border rank of the multiplication of 2 2 matrices is seven one of the leading problems of algebraic complexity theory is matrix multiplication the naive multiplication of two n n matrices uses n multiplications in 1969 strassen 20 presented an explicit algorithm for multiplying 2 2 matrices using seven multiplications in the opposite direction hopcroft and kerr 12 and winograd 22 proved independently that there is no algorithm for multiplying 2 2 matrices using only six multiplications the precise number of multiplications needed to execute matrix multiplication or any given bilinear map is called the rank of the bilinear map a related problem is to determine the border rank of matrix multiplication or any given bilinear map first introduced in 6 5 roughly speaking some bilinear maps may be approximated with arbitrary precision by less complicated bilinear maps and the border rank of a bilinear map is the complexity of arbitrarily small good perturbations of the map these perturbed maps can give rise to fast exact algorithms for matrix multiplication see 7 the border rank made appearances in the literature in the 1980s and early 1990s see e g 6 5 19 8 15 2 10 11 4 3 17 16 18 1 9 21 but to our knowledge there has not been much progress on the question since then more precisely for any complex projective variety x cp pv and point p pv define the x rank of p to be the smallest number r such that p is in the linear span of r points of x define r x the r th secant variety of x to be the zariski closure of the set of points of x rank r and define the x border rank of p to be the smallest r such that p r x the terminology is motivated by the case x seg pa 1 pb 1 p c c the segre variety of rank one matrices then the x rank of a matrix is just its usual rank let a b c be vector spaces and let f a b c be a bilinear map i e an element of a b c let x seg pa pb pc p a b c denote the segre variety of decomposable tensors in a b c the border rank of a bilinear map is its x border rank while for the segre product of two projective spaces border rank coincides with rank here they can be quite different in this paper we prove the theorem stated in the title let mmult c c c denote the matrix multiplication operator for 2 2 matrices strassen 18 showed that mmult 5 p p p our method of proof is to decompose

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

an improvement to ostrowski root finding method an improvement to the iterative method based on the ostrowski one to compute nonlinear equation solutions which increases the local order of convergence is suggested the adaptation of a strategy presented here gives a new iteration function with an additional evaluation of the function it also shows a smaller cost if we use adaptive multi precision arithmetic the numerical results computed using this system with a floating point system representing 200 decimal digits support this theory

a fifth order iterative method for solving nonlinear equations in this paper we present a new fifth order method for solving nonlinear equations per iteration the new method requires two function and two first derivative evaluations it is shown that the new method is fifth order convergent several numerical examples are given to illustrate the performance of the presented method

identifiability of parameters in latent structure models with many observed variables while hidden class models of various types arise in many statistical applications it is often difficult to establish the identifiability of their parameters focusing on models in which there is some structure of independence of some of the observed variables conditioned on hidden ones we demonstrate a general approach for establishing identifiability utilizing algebraic arguments a theorem of j kruskal for a simple latent class model with finite state space lies at the core of our results though we apply it to a diverse set of models these include mixtures of both finite and nonparametric product distributions hidden markov models and random graph mixture models and lead to a number of new results and improvements to old ones in the parametric setting this approach indicates that for such models the classical definition of identifiability is typically too strong instead generic identifiability holds which implies that the set of nonidentifiable parameters has measure zero so that parameter inference is still meaningful in particular this sheds light on the properties of finite mixtures of bernoulli products which have been used for decades despite being known to have nonidentifiable parameters in the nonparametric setting we again obtain identifiability only when certain restrictions are placed on the distributions that are mixed but we explicitly describe the conditions

most secant varieties of tangential varieties to veronese varieties are nondefective we prove a conjecture stated by catalisano geramita and gimigliano in 2002 which claims that the secant varieties of tangential varieties to the dth veronese embedding of the projective n space pn have the expected dimension modulo a few wellknown exceptions it is arguably the first complete result on the dimensions of secant varieties of a classically studied variety since the work of alexander and hirschowitz in 1995 as bernardi catalisano gimigliano and ida demonstrated that the proof of this conjecture may be reduced to the case of cubics i e d 3 the main contribution of this work is the resolution of this base case the proposed proof proceeds by induction on the dimension n of the projective space via a specialization argument this reduces the proof to a large number of initial cases for the induction which were settled using a computer assisted proof the individual base cases were computationally challenging problems indeed the largest base case required us to deal with the tangential variety to the third veronese embedding of p79 in p88559

modified householder iterative method for nonlinear equations abstract in this paper we suggest and analyze a new two step predictor corrector type iterative method for solving nonlinear equations of the type f x 0 this new method includes the two step newton method as a special case we show that this new two step method is a sixth order convergent method several examples are given to illustrate the efficiency of this new method and its comparison with other sixth order methods this method can be considered as a significant improvement of the newton method and its variant forms

some developments in general variational inequalities general variational inequalities provide us with a unified natural novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences in this paper we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection wiener hopf equations updating the solution auxiliary principle inertial proximal penalty function dynamical system and well posedness we also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection type algorithms our proofs of convergence are very simple as compared with other methods our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems since the general variational inequalities include quasi variational inequalities and quasi implicit complementarity problems as special cases results presented here continue to hold for these problems several open problems have been suggested for further research in these areas

ranks of tensors secant varieties of segre varieties and fat points abstract a classical unsolved problem of projective geometry is that of finding the dimensions of all the higher secant varieties of the segre embeddings of an arbitrary product of projective spaces an important subsidiary problem is that of finding the smallest integer t for which the secant variety of projective t spaces fills the ambient projective space in this paper we give a new approach to these problems the crux of our method is the translation of a well known lemma of terracini into a question concerning the hilbert function of fat points in a multiprojective space our approach gives much new information on the classical problem even in the case of three factors a case also studied in the area of algebraic complexity theory

ranks of tensors and a generalization of secant varieties abstract we introduce subspace rank as a tool for studying ranks of tensors and x rank more generally we derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric tensors in c 2 c b c b we review the literature from a geometric perspective

some modification of newton s method by the method of undetermined coefficients in this paper we construct some modifications of newton s method for solving nonlinear equations which is based on the method of undetermined coefficients it is shown by way of illustration that the method of undetermined coefficients is a promising tool for developing new methods and reveals its wide applicability by obtaining some existing methods as special cases two new sixth order methods are developed numerical examples are given to support that the methods thus obtained can compete with other iterative methods

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

an iterative method with cubic convergence for nonlinear equations in this paper we suggest and analyze a new three step iterative method for solving nonlinear equations we show that this new iterative method has third order convergence several numerical examples are given to illustrate the efficiency and performance of this new method new method can be viewed as an improvement of the previously known iterative methods

tensor rank is not multiplicative under the tensor product the tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors let s be a k tensor and let t be an l tensor the tensor product of s and t is a k l tensor tensor rank is sub multiplicative under the tensor product we revisit the connection between restrictions and degenerations a result of our study is that tensor rank is not in general multiplicative under the tensor product this answers a question of draisma and saptharishi specifically if a tensor t has border rank strictly smaller than its rank then the tensor rank of t is not multiplicative under taking a sufficiently hight tensor product power the tensor kronecker product from algebraic complexity theory is related to our tensor product but different namely it multiplies two k tensors to get a k tensor nonmultiplicativity of the tensor kronecker product has been known since the work of strassen it remains an open question whether border rank and asymptotic rank are multiplicative under the tensor product interestingly lower bounds on border rank obtained from generalized flattenings including young flattenings multiply under the tensor product

some higher order modifications of newton s method for solving nonlinear equations in this paper we consider constructing some higher order modifications of newton s method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units this construction can be applied to any iteration formula and per iteration the resulting methods add only one additional function evaluation to increase the order some illustrative examples are provided and several numerical results are given to show the performance of the presented methods

topology of tensor ranks abstract we study path connectedness and homotopy groups of sets of tensors defined by tensor rank border rank multilinear rank as well as their symmetric counterparts for symmetric tensors we show that over c the set of rank r tensors and the set of symmetric rank r symmetric tensors are both path connected if r is not more than the complex generic rank these results also extend to border rank and symmetric border rank over c over r the set of rank r tensors is path connected if it has the expected dimension but the corresponding result for symmetric rank r symmetric d tensors depends on the order d connected when d is odd but not when d is even border rank and symmetric border rank over r have essentially the same path connectedness properties as rank and symmetric rank over r when r is greater than the complex generic rank we are unable to discern any general pattern for example we show that border rank three tensors in r 2 r 2 r 2 fall into four connected components for multilinear rank the manifold of d tensors of multilinear rank r 1 r d in c n 1 c n d is always path connected and the same is true in r n 1 r n d unless n i r i j i r j for some i 1 d beyond path connectedness we determine over both r and c the fundamental and higher homotopy groups of the set of tensors of a fixed small rank and taking advantage of bott periodicity those of the manifold of tensors of a fixed multilinear rank we also obtain analogues of these results for symmetric tensors of a fixed symmetric rank or a fixed symmetric multilinear rank

a note on fifth order iterative methods for solving nonlinear equations abstract in noor and noor 2006 m a noor k i noor fifth order iterative methods for solving nonlinear equations appl math comput in press doi 10 1016 j amc 2006 10 007 a new two step iterative method with fifth order for solving nonlinear equations has been proposed in this note we show that the iterative method given by the authors has third order convergence not the fifth order one as claimed in their work

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

joint source estimation and localization the estimation of directions of arrival is formulated as the decomposition of a 3 way array into a sum of rank one terms which is possible when the receive array enjoys some geometrical structure the main advantage is that this decomposition is essentially unique under mild assumptions if computed exactly the drawback is that a low rank approximation does not always exist therefore a coherence constraint is introduced that ensures the existence of the latter best approximate which allows to localize and estimate closely located or highly correlated sources then cramer rao bounds are derived for localization parameters and source signals assuming the others are nuisance parameters some inaccuracies found in the literature are pointed out performances are eventually compared with unconstrained reference algorithms such as esprit in the presence of additive complex gaussian noise with possibly noncircular distribution

rank and optimal computation of generic tensors abstract the typical rank maximal border rank of tensors of a given size and the set of optimal bilinear computations of typical tensors of a given rank are investigated for the size n n 3 with n odd the complement of the set of tensors of maximal border rank is a hypersurface its equation is given

sixteenth order method for nonlinear equations modification of newton s method with higher order convergence is presented the modification of newton s method is based on king s fourth order method the new method requires three step per iteration analysis of convergence demonstrates that the order of convergence is 16 some numerical examples illustrate that the algorithm is more efficient and performs better than classical newton s method and other methods

three step iterative methods with eighth order convergence for solving nonlinear equations a family of eighth order iterative methods for the solution of nonlinear equations is presented the new family of eighth order methods is based on king s fourth order methods and the family of sixth order iteration methods developed by chun et al per iteration the new methods require three evaluations of the function and one evaluation of its first derivative therefore this family of methods has the efficiency index which equals 1 682 kung and traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2 n 1 thus we provide a new example which agrees with the conjecture of kung traub for n 4 numerical comparisons are made to show the performance of the presented methods

regularized non abelian chiral jacobian factor at finite temperature the regularized non abelian chiral jacobian factor in the path integral formulation of fermions interacting with background vector and axial vector fields is investigated at finite temperature in arbitrary even dimensions after separating out the terms with normal parity it is shown that the self consistent non abelian chiral jacobian is the same at t 0 and t not 0

on the generic and typical ranks of 3 tensors abstract we study the generic and typical ranks of 3 tensors of dimension l m n using results from matrices and algebraic geometry we state a conjecture about the exact values of the generic rank of 3 tensors over the complex numbers which is verified numerically for l m n 14 we also discuss the typical ranks over the real numbers and give an example of an infinite family of 3 tensors of the form l m n m 1 2 1 m 3 4 which have at least two typical ranks

on iterative methods for nonlinear equations several iterative methods have been suggested and analyzed for solving nonlinear equation f x 0 all these methods can be classified as one step and two step methods in this paper we have suggested and analyzed several two step methods for solving nonlinear equations we have also shown that these two step methods are quadratically convergent several numerical examples are given to illustrate this comparison our results can be viewed as important refinement and improvement of the previously known results

some quadrature based three step iterative methods for non linear equations in this paper we present three step quadrature based iterative methods for solving non linear equations the convergence analysis of the methods is discussed it is established that the new methods have convergence order six seven and eight respectively numerical tests show that the new methods are comparable with the well known existing methods and in many cases give better results our results can be considered as an improvement and refinement of the previously known results in the literature

three step iterative methods with optimal eighth order convergence in this paper based on ostrowski s method a new family of eighth order methods for solving nonlinear equations is derived in terms of computational cost each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative so that their efficiency indices are 1 682 which is optimal according to kung and traub s conjecture numerical comparisons are made to show the performance of the new family

some variants of ostrowski s method with seventh order convergence in this paper we present a class of new variants of ostrowski s method with order of convergence seven per iteration the new methods require three evaluations of the function and one evaluation of its first derivative and therefore this class of methods has the efficiency index equal to 1 627 numerical tests verifying the theory are given and multistep iterations based on the present methods are developed

an infinite family of one step iterators for solving nonlinear equations to increase the order of convergence and a new algorithm of global convergence in this paper we present an infinite family of one step iterative formulas for solving nonlinear equations present method one from now on pmi that can be expressed as x n 1 f m x n with email protected m 1 we will prove that the corresponding iteration formula of the family x n 1 f m 0 x n has an order of convergence m 0 1 the increment of the velocity of convergence of the sequence of the iterator family x n 1 f m 1 x n with respect to the previous one x n 1 f m x n is attained at the expense of one derivative evaluation more besides we introduce a new algorithm present method two from now on pmii that plays the role of seeker for an initial value to guarantee the local convergence of the pmi both of them can be combined as an algorithm of global convergence included the case of singular roots that does not depend on the chosen initial value and that allows to find all the roots in a feasible interval in a general and complete way these are in my opinion the main results of this work