total positivity and its applications preface i spline functions total positivity of the spline kernel and its applications b bojanov total positivity properties of lb splines t lyche l l schumaker total positivity and splines k morken totally positive systems and interpolation by subspaces of generalized splines m sommer h strauss ii matrix theory preservation of total nonnegativity under the hadamard product and related topics j garloff d g wagner vertex implications for totally nonnegative under the hadamard product and related topics j garloff on factorizations of totally positive matrices j garloff on factorizations of totally positive matrices m gasca j m pena iii geometric modeling total positivity and optimal bases j m carnicer j m pena total positivity and the shape of curves t n t goodman tchebycheff curves m l mazure h pottmann iv probability and mathematical biology analytical and probabilistic methods for totally positive matrices g s goodman some inequalities of total positivity in pure and applied mathematics s karlin z ziegler new directions and problems of multivariate total positivity motivated by genetic recombination probabilities s karlin v approximation theory tchebycheff spaces and total positivity j m carnicer j m pena cebysev and weak cebysev systems r a zalik vi complex analysis zero sets of entire fourier transforms of multiply positive density functions o m katkova vii statistics total nonnegativity of moment matrices and its application to e optimal designs in polynomial regression b heiligers analysis of regression dependence in 2 xn bivariate distributions and some applications in contingency tables s kasala m b rao viii real analysis total positivity and nonlinear analysis r q jia solutions of certain functional equations s karlin alternance and snakes in the nonlinear finite moment problem k tatalian ix combinatorics the applications of total positivity to combinatorics and conversely f brenti x integral equations spectral properties of totally positive kernels and matrices a pinkus index

on switching h controllers for a class of linear parameter varying systems abstract we consider switching h controllers for a class of linear parameter varying lpv systems scheduled along a measurable parameter trajectory the candidate controllers are selected from a given controller set according to the switching rules based on the scheduling variable we provide sufficient conditions to guarantee the stability of the switching lpv systems in terms of the dwell time and the average dwell time our results are illustrated with an example where switching between two robust controllers is performed for an lpv system

shifting strategies for the parallel qr algorithm consider the problem of finding all eigenvalues of a real square matrix a that has no special structure to exploit the qr algorithm has been the method of choice for solving such problems for the past 30 years however in recent years it has been suggested that the days of the qr algorithm may be numbered for the qr algorithm has proven difficult to parallelize such predictions are at best premature a simple modification of the shifting strategy promises to improve the parallel performance of the qr algorithm significantly and keep the competition at bay for years to come

on an integral representation for the genus series for 2 cell embeddings an integral representation for the genus series for maps on oriented surfaces is derived from the combinatorial axiomatisation of 2 cell embeddings in orientable surfaces it is used to derive an explicit expression for the genus series for dipoles the approach can be extended to vertex regular maps in general and in this way may shed light on the genus series for quadrangulations the integral representation is used in conjunction with an approach through the group algebra co of the symmetric group 11 to obtain a factorisation of certain gaussian integrals 1 a power series representation for the genus series a map is a 2 cell embedding of a connected unlabelled graph with loops and multiple edges allowed in a closed surface x without boundary which is assumed throughout to be oriented the deletion of separates x into regions homeomorphic to open discs called the faces of the map and the number of edges bordering a face is called its degree a map is rooted by distinguishing a mutually incident vertex edge and face the genus series for a class of maps is the formal generating series for the number of inequivalent maps with respect to genus and the numbers of vertices edges and faces it is assumed hereinafter that maps are rooted the general approach adopted here combines ideas of jackson and visentin 11 with those of t hooft 8 and bessis itzykson and zuber 3 who in the above terminology derived the genus series for diagrams akin to a class of maps by techniques from conformal field theory although 81 and 3 are important papers they have remained largely inaccessible to combinatorialists because of their uncertainty about the automorphisms of these diagrams as combinatorial structures in this paper an explicit construction is given for an integral representation for the genus series for general maps directly from the combinatorial axiomatisation for embeddings on oriented surfaces moreover we also develop methods which are extensible to vertex regular maps vertices have the same degree and thence by restriction to quadrangulations maps whose faces are bounded by four edges this is done by examining dipoles maps with two vertices in detail although the argument is an algebraic one based on the ring of formal power series to assert that particular series belong to the ring it is necessary to received by the editors july 1 1992 and in revised form august 15 1993 1991 mathematics subject classification primary 05a1 5 20c1 5 secondary 57n37 1994 american mathematical society 0002 9947 94 1 00 s 25 per page

relatively robust representations of symmetric tridiagonals let ldlt be the triangular factorization of an unreduced symmetric tridiagonal matrix t i small relative changes in the nontrivial entries of l and d may be represented by diagonal scaling matrices 1 and 2 ldlt 2l 1d 1lt 2 the effect of 2 on the eigenvalues i is benign in this paper we study the inner perturbations induced by 1 suitable condition numbers govern the relative changes in the eigenvalues i we show that when j is an eigenvalue then the relative condition number of m j m j is the same for all n twisted factorizations one of which is ldlt that could be used to represent t i see section 2 we prove that as j the smallest eigenvalue has relative condition number relcond 1 o j each relcond is a rational function of we identify the poles and then use orthogonal polynomial theory to develop upper bounds on the sum of the relconds of all the eigenvalues these bounds require o n operations for an n n matrix we show that the sum of all the relconds is bounded by trace l d lt and conjecture that n ldlt the quantity trace l d lt ldlt is a natural measure of element growth in the context of this paper an algorithm for computing numerically orthogonal eigenvectors without recourse to the gram schmidt process is sketched it requires that there exist values of close to each cluster of close eigenvalues such that all the relconds belonging to the cluster are modest say 10 the sensitivity of the other eigenvalues is not important for this reason we develop o n bounds on the sum of the relconds associated with a cluster none of our bounds makes reference to the nature of the distribution of the eigenvalues within a cluster which can be very complicated

positive definite toeplitz matrices the arnoldi process for isometric operators and gaussian quadrature on the unit circle abstract we show that the well known levinson algorithm for computing the inverse cholesky factorization of positive definite toeplitz matrices can be viewed as a special case of a more general process the latter process provides a very efficient implementation of the arnoldi process when the underlying operator is isometric this is analogous with the case of hermitian operators where the hessenberg matrix becomes tridiagonal and results in the hermitian lanczos process we investigate the structure of the hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle these eigenvalues are then interpreted as abscissas for analogs of gaussian quadrature now on the unit circle instead of the real line the trapezoidal rule appears as the analog of the gauss legendre formula

a periodic krylov schur algorithm for large matrix products stewart s recently introduced krylov schur algorithm is a modification of the implicitly restarted arnoldi algorithm which employs reordered schur decompositions to perform restarts and deflations in a numerically reliable manner this paper describes a variant of the krylov schur algorithm suitable for addressing eigenvalue problems associated with products of large and sparse matrices it performs restarts and deflations via reordered periodic schur decompositions and by taking the product structure into account it is capable of achieving qualitatively better approximations to eigenvalues of small magnitude

an analysis of the hr algorithm for computing the eigenvalues of a matrix abstract the hr algorithm a method of computing the eigenvalues of a matrix is presented it is based on the fact that almost every complex square matrix a can be decomposed into a product a hr of a so called pseudo hermitian matrix h and an upper triangular matrix r this algorithm is easily seen to be a generalization of the well known qr algorithm it is shown how it is related to the power method and inverse iteration and for special matrices the connection between the lr and hr algorithms is indicated

the periodic qr algorithm is a disguised qr algorithm the periodic qr algorithm is a strongly backward stable method for computing the eigenvalues of products of matrices or equivalently for computing the eigenvalues of block cyclic matrices the main purpose of this paper is to show that this algorithm is numerically equivalent to the standard qr algorithm it will be demonstrated how this connection may be used to develop a better understanding of the periodic qr algorithm

on tridiagonal matrices unitarily equivalent to normal matrices abstract in this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied it is well known that any matrix is unitarily equivalent to a tridiagonal matrix in case of a normal matrix the resulting tridiagonal inherits a strong relation between its super and subdiagonal elements the corresponding elements of the super and subdiagonal will have the same absolute value in this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied both an iterative reduction based on krylov sequences as a direct tridiagonalization procedure via householder transformations are reconsidered this equivalence transformation is then applied to the normal case and equality of the absolute value between the super and subdiagonals is proved self adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved properties when applying the reduction on symmetric skew symmetric hermitian skew hermitian and unitary matrices and their relations with e g complex symmetric and pseudo symmetric matrices are presented it is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the takagi factorization finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given

accurate singular values and differential qd algorithms in september 1991 j w demmel and w m kahan were awarded the second siam prize in numerical linear algebra for their paper accurate singular values of bidiagonal matrices 1 referred to as dk hereafter among several valuable results was the observation that the standard bidiagonal qr algorithm used in linpack 2 and in many other svd programs can be simplified when the shift is zero and of greater importance no subtractions occur the last feature permits very small singular values to be found with almost all the accuracy permitted by the data and at no extra cost

the symplectic eigenvalue problem the butterfly form the sr algorithm and the lanczos method abstract we discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices any 2n 2n symplectic matrix can be reduced to this condensed form which contains 8n 4 nonzero entries and is determined by 4n 1 parameters the symplectic eigenvalue problem can be solved using the sr algorithm based on this condensed form the sr algorithm preserves this form and can be modified to work only with the 4n 1 parameters instead of the 4n2 matrix elements the reduction of symplectic matrices to symplectic butterfly form has a close analogy to the reduction of arbitrary matrices to hessenberg form a lanczos like algorithm for reducing a symplectic matrix to butterfly form is also presented

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

on hamiltonian and symplectic lanczos processes abstract large sparse hamiltonian eigenvalue problems arise in a variety of contexts these problems can be attacked directly or they can first be transformed to problems having some related structure such as symplectic or skew hamiltonian in the interest of efficiency stability and accuracy such problems should be solved by methods that preserve the structure whether it be hamiltonian skew hamiltonian or symplectic the present work outlines krylov subspace methods for computing partial eigensystems of skew hamiltonian hamiltonian and symplectic matrices and records some of the relationships between them the ordinary unsymmetric lanczos process is a structure preserving method for skew hamiltonian matrices the hamiltonian and symplectic lanczos processes developed here are condensed versions of the processes that have been published previously the condensed hamiltonian lanczos process applied to h is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian h 2 but costs half as much to execute the condensed symplectic lanczos process applied to s is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian matrix s s 1 but also costs half as much to execute implicit restarts of the hamiltonian and symplectic lanczos processes can be effected by the sr algorithm because of the known relationship between the sr and hr algorithms the much simpler hr algorithm can be used to restart the condensed symplectic and hamiltonian lanczos processes each of these restart procedures is equivalent to restarting the unsymmetric lanczos process using the hr algorithm the hr algorithm is most effectively implemented as an implicit bulge chasing hz algorithm on a symmetric tridiagonal diagonal pencil t d

schur parameter pencils for the solution of the unitary eigenproblem abstract let u v be an n n pencil with unitary matrices u and v an algorithm is presented which reduces u and v simultaneously to unitary block diagonal matrices g o q h up and g e q h vp with block size at most two it is an o n 3 process using householder eliminations and it is backward stable in the special case v i the block diagonal matrices g o g h e can be normalized so that their entries are just the schur parameters of the hessenberg condensed form of u we call g o g e a schur parameter pencil it can also be derived from u v by a lanczos like process for the solution of the eigenvalue problem for g o g e a qr type algorithm can be developed based on this unitary reduction of a pencil u v to a schur parameter pencil the condensed form is preserved throughout the process each iteration step needs only o n operations this method of solving the unitary eigenvalue problem seems to be the closest possible analogy to the qr method for the hermitian eigenvalue problem

accurate computations with said ball vandermonde matrices abstract a generalization of the vandermonde matrices which arise when the power basis is replaced by the said ball basis is considered when the nodes are inside the interval 0 1 then those matrices are strictly totally positive an algorithm for computing the bidiagonal decomposition of those said ball vandermonde matrices is presented which allows us to use known algorithms for totally positive matrices represented by their bidiagonal decomposition the algorithm is shown to be fast and to guarantee high relative accuracy some numerical experiments which illustrate the good behaviour of the algorithm are included

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

the linear algebra of the pascal matrix abstract pascal s triangle can be represented as a square matrix in two basically different ways as a lower triangular matrix pn or as a full symmetric matrix qn it has been found that the pn pnt is the cholesky factorization of qn pn can be factorized by special summation matrices it can be shown that the inverses of these matrices are the operators which perform the gaussian elimination steps for calculating cholesky s factorization by applying linear algebra we produce combinatorial identities and an existence theorem for diophantine equation systems finally an explicit formula for the sum of the kth powers is given

the transmission of shifts and shift blurring in the qr algorithm abstract the qr algorithm is one of the most widely used algorithms for calculating the eigenvalues of matrices the multishift qr algorithm with multiplicity m is a version that effects m iterations of the qr algorithm at a time it is known that roundoff errors cause the multishift qr algorithm to perform poorly when m is large in this paper the mechanism by which the shifts are transmitted through the matrix in the course of a multishift qr iteration is identified numerical evidence showing that the mechanism works well when m is small and poorly when m is large is presented when the mechanism works poorly the convergence of the algorithm is degraded proportionately

numerical solution of the discrete time periodic riccati equation in this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic riccati equation this method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the euler lagrange difference equations in doing so it is possible to extract a basis for the stable deflating subspace of the extended pencil from which the riccati solution is obtained this algorithm is an extension of the standard qz algorithm and retains its attractive features such as quadratic convergence and small relative backward error a method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with

isospectral flows on a class of finite dimensional jacobi matrices abstract we present a new matrix valued isospectral ordinary differential equation that asymptotically block diagonalizes n n zero diagonal jacobi matrices employed as its initial condition this o d e features a right hand side with a nested commutator of matrices and structurally resembles the double bracket o d e studied by r w brockett in 1991 we prove that its solutions converge asymptotically that the limit is block diagonal and above all that the limit matrix is defined uniquely as follows for n even a block diagonal matrix containing 2 2 blocks such that the super diagonal entries are sorted by strictly increasing absolute value furthermore the off diagonal entries in these 2 2 blocks have the same sign as the respective entries in the matrix employed as the initial condition for n odd there is one additional 1 1 block containing a zero that is the top left entry of the limit matrix the results presented here extend some early work of kac and van moerbeke

a parallel implementation of the nonsymmetric qr algorithm for distributed memory architectures one approach to solving the nonsymmetric eigenvalue problem in parallel is to parallelize the qr algorithm not long ago this was widely considered to be a hopeless task recent efforts have led to significant advances although the methods proposed up to now have suffered from scalability problems this paper discusses an approach to parallelizing the qr algorithm that greatly improves scalability a theoretical analysis indicates that the algorithm is ultimately not scalable but the nonscalability does not become evident until the matrix dimension is enormous experiments on the intel paragon system the ibm sp2 supercomputer the sgi origin 2000 and the intel asci option red supercomputer are reported

a class of hamiltonian symplectic methods for solving the algebraic riccati equation abstract the algebraic riccati equation can be solved by finding a certain invariant subspace of a related hamiltonian matrix g ammar and v mehrmann devised a method for calculating this invariant subspace that exploits the hamiltonian structure of the matrix by using unitary symplectic similarity transformations this paper discusses a class of methods that find the invariant subspace by using symplectic similarity transformations that are not necessarily unitary the method of ammar and mehrmann is a special case

on hamiltonian and symplectic hessenberg forms abstract characterizations are given for the hamiltonian matrices that can be reduced to hamiltonian hessenberg form and the symplectic matrices that can be reduced to symplectic hessenberg form by orthogonal symplectic similarity transformations the reduction to these special hessenberg forms is the missing link in the solution of the open problem of constructing a stable structure preserving qr like method of complexity o n 3 for the computation of invariant subspaces of hamiltonian and symplectic matrices our considerations lead us to propose an approach to the computation of lagrangian invariant subspaces of a hamiltonian or symplectic matrix

a jacobi davidson type method for the product eigenvalue problem we propose a jacobi davidson type method to compute selected eigenpairs of the product eigenvalue problem a m a email protected where the matrices may be large and sparse to avoid difficulties caused by a high condition number of the product matrix we split up the action of the product matrix and work with several search spaces we generalize the jacobi davidson correction equation and the harmonic and refined extraction for the product eigenvalue problem numerical experiments indicate that the method can be used to compute eigenvalues of product matrices with extremely high condition numbers

sr and sz algorithms for the symplectic butterfly eigenproblem abstract sr and sz algorithms for the symplectic generalized eigenproblem that are based on the reduction of a symplectic matrix to symplectic butterfly form are discussed a 2 n 2 n symplectic butterfly matrix has 8 n 4 generically nonzero entries which are determined by 4 n 1 parameters while the sr algorithm operates directly on the matrix entries the sz algorithm works with the 4 n 1 parameters the algorithms are made more compact and efficient by using laurent polynomials instead of standard polynomials to drive the iterations

the qr algorithm for unitary hessenberg matrices abstract let h be an n n unitary right hessenberg matrix with positive subdiagonal elements using what we call the schur parameterization of h we show how one step of the shifted qr algorithm for h can be carried out in o n arithmetic operations coupled with the shift strategy of eberlein and huang 3 this will permit computation of the spectrum of h to machine precision in o n 2 operations one potential application is the computation of gauss szego quadrature formulas 12 given the associated schur parameters 7 the weights can also be computed by direct analogy with 6

eigenvalues of ax λbx for real symmetric matrices a and b computed by reduction to a pseudosymmetric form and the hr process abstract the paper presents a method for solving the eigenvalue problem ax bx where a and b are real symmetric but not necessarily positive definite matrices and b is nonsingular the method reduces the general case into a form cz z where c is a pseudosymmetric matrix a further reduction of c produces a tridiagonal pseudosymmetric form to which the iterative hr process is applied the tridiagonal pseudosymmetric form is invariant under the hr transformations the amount of computation is significantly less than in treating the problem by a general method

the rayleigh ritz method refinement and arnoldi process for periodic matrix pairs we extend the rayleigh ritz method to the eigen problem of periodic matrix pairs assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero we show that there exist periodic ritz values that converge to the desired periodic eigenvalues unconditionally yet the periodic ritz vectors may fail to converge to overcome this potential problem we minimize residuals formed with periodic ritz values to produce the refined periodic ritz vectors which converge under the same assumption these results generalize the corresponding well known ones for rayleigh ritz approximations and their refinement for non periodic eigen problems in addition we consider a periodic arnoldi process which is particularly efficient when coupled with the rayleigh ritz method with refinement the numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors

two connections between the sr and hr eigenvalue algorithms abstract the sr and hr algorithms are members of the family of gr algorithms for calculating eigenvalues and invariant subspaces of matrices this paper makes two connections between the sr and hr algorithms 1 an iteration of the sr algorithm on a 2 n 2 n symplectic butterfly matrix using shifts i 1 i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts i 1 i i 1 k 2 an iteration of the sr algorithm on a 2 n 2 n j tridagonal hamiltonian matrix using shifts i i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts 2 i i 1 k

extending classical multirate signal processing theory to graphs part i fundamentals signal processing on graphs finds applications in many areas in recent years renewed interest on this topic was kindled by two groups of researchers narang and ortega constructed two channel filter banks on bipartitie graphs described by laplacians sandryhaila and moura developed the theory of linear systems filtering and frequency responses for the case of graphs with arbitrary adjacency matrices and showed applications in signal compression prediction etc inspired by these contributions this paper extends classical multirate signal processing ideas to graphs the graphs are assumed to be general with a possibly nonsymmetric and complex adjacency matrix the paper revisits ideas such as noble identities aliasing and polyphase decompositions in graph multirate systems drawing such a parallel to classical systems allows one to design filter banks with polynomial filters with lower complexity than arbitrary graph filters it is shown that the extension of classical multirate theory to graphs is nontrivial and requires certain mathematical restrictions on the graph thus classical noble identities cannot be taken for granted similarly one cannot claim that the so called delay chain system is a perfect reconstruction system as in classical filter banks it will also be shown that m partite extensions of the bipartite filter bank results will not work for m channel filter banks but a more restrictive condition called m block cyclic property should be imposed such graphs are studied in detail a detailed theory for m channel filter banks is developed in a companion paper

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

convergence of algorithms of decomposition type for the eigenvalue problem we develop the theory of convergence of a generic gr algorithm for the matrix eigenvalue problem that includes the qr lr sr and other algorithms as special cases our formulation allows for shifts of origin and multiple gr steps the convergence theory is based on the idea that the gr algorithm performs nested subspace iteration with a change of coordinate system at each step thus the convergence of the gr algorithm depends on the convergence of certain sequences of subspaces it also depends on the quality of the coordinate transformation matrices as measured by their condition numbers we show that with a certain obvious shifting strategy the gr algorithm typically has a quadratic asymptotic convergence rate for matrices possessing certain special types of structure cubic convergence can be achieved