on the additive structure of the inverses of banded matrices the additive structure of the inverses of banded matrices is investigated under certain conditions the inverse of a 2k 1 diagonal symmetric banded matrix can be expressed as a sum of k symmetric matrices belonging to the class of inverses of symmetric irreducible tridiagonal matrices in the nonsymmetric case a more complicated structure is obtained applications are mentioned for the resolution of constant coefficient banded linear systems in vlsi models

convergence analysis of a lasserre hierarchy of upper bounds for polynomial minimization on the sphere we study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems proposed by lasserre siam j optim 21 3 864 885 2011 for the special case when the feasible set is the unit hyper sphere the upper bound at level r n of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere when the probability density function is a sum of squares polynomial of degree at most 2r with respect to the surface measure we show that the rate of convergence is o 1 r2 and we give a class of polynomials of any positive degree for which this rate is tight in addition we explore the implications for the related rate of convergence for the generalized problem of moments on the sphere

a backward three term recurrence relation for vector valued continued fractions and its applications this paper presents a backward three term recurrence relation which is a solution of an open problem in graves morris numer math 42 1983 331 by means of this relation we succeed in establishing some conclusions about the two backward successive convergents of vector valued continued fractions moreover an expression for the vector valued continued fraction in series form is obtained and some theorems of the convergence of the vector valued continued fraction are proved by virtue of the recurrence relation

improved convergence analysis of lasserre s measure based upper bounds for polynomial minimization on compact sets we consider the problem of computing the minimum value fmin k of a polynomial f over a compact set k rn which can be reformulated as finding a probability measure on k minimizing kfd lasserre showed that it suffices to consider such measures of the form q where q is a sum of squares polynomial and is a given borel measure supported on k by bounding the degree of q by 2r one gets a converging hierarchy of upper bounds f r for fmin k when k is the hypercube 1 1 n equipped with the chebyshev measure the parameters f r are known to converge to fmin k at a rate in o 1 r2 we extend this error estimate to a wider class of convex bodies while also allowing for a broader class of reference measures including the lebesgue measure our analysis applies to simplices balls and convex bodies that locally look like a ball in addition we show an error estimate in o logr r when k satisfies a minor geometrical condition and in o log2r r2 when k is a convex body equipped with the lebesgue measure this improves upon the currently best known error estimates in o 1 r and o 1 r for these two respective cases

worst case examples for lasserre s measure based hierarchy for polynomial optimization on the hypercube we study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems proposed by lasserre siam j optim 21 3 2011 pp 864 885 and a related hierarchy by de klerk hess and laurent siam j optim 27 1 2017 pp 347 367 for polynomial optimization over the hypercube we show a refined convergence analysis for the first hierarchy we also show lower bounds on the convergence rate for both hierarchies on a class of examples these lower bounds match the upper bounds and thus establish the true rate of convergence on these examples interestingly these convergence rates are determined by the distribution of extremal zeroes of certain families of orthogonal polynomials

a fast algorithm for evaluating n th order tri diagonal determinants the cost of all existing algorithms for evaluating the nth order determinants numerical analysis 7th edition brooks cole publishing pacific grove ca 2001 is at most o n3 in the current article we present a new efficient computational algorithm for evaluating the nth order tri diagonal determinants with cost o n only the algorithm is suited for implementation using computer algebra systems such as maple and macsyma some examples are given to illustrate the algorithm

comparison of lasserre s measure based bounds for polynomial optimization to bounds obtained by simulated annealing we consider the problem of minimizing a continuous function f over a compact set k we compare the hierarchy of upper bounds proposed by lasserre lasserre jb 2011 a new look at nonnegativity on closed sets and polynomial optimization siam j optim 21 3 864 885 to bounds that may be obtained from simulated annealing we show that when f is a polynomial and k a convex body this comparison yields a faster rate of convergence of the lasserre hierarchy than what was previously known in the literature

a contrast of direct and transformed nystro m pairs abstract explicit nystrom pairs provide an efficient way to solve the general second order initial value problem when stability requirements do not determine the stepsize the pairs can be derived by explicitly solving the order conditions for nystrom pairs or by applying a transformation to runge kutta pairs the two approaches lead to different classes of pairs we refer to the two types of pairs as direct and transformed pairs respectively even though transformed pairs are easier to derive existing nystrom pairs are direct pairs in this paper we compare the properties of classes of direct and transformed p 1 p pairs p 4 5 6 the properties are the efficiency of the pair the accuracy of the local error estimate and stepsize prediction and the size of the stability region we also investigate the existence of an interpolation scheme for the 3 4 pairs

near optimal analysis of lasserre s univariate measure based bounds for multivariate polynomial optimization we consider a recent hierarchy of upper approximations proposed by lasserre arxiv 1907 097784 2019 for the minimization of a polynomial f over a compact set k n this hierarchy relies on using the push forward measure of the lebesgue measure on k by the polynomial f and involves univariate sums of squares of polynomials with growing degrees 2r hence it is weaker but cheaper to compute than an earlier hierarchy by lasserre siam journal on optimization 21 3 864 885 2011 which uses multivariate sums of squares we show that this new hierarchy converges to the global minimum of f at a rate in o log2 r r2 whenever k satisfies a mild geometric condition which holds e g for convex bodies as an application this rate of convergence also applies to the stronger hierarchy based on multivariate sums of squares which extends earlier convergence results to a wider class of compact sets furthermore we show that our analysis is near optimal by proving a lower bound on the convergence rate in 1 r2 for a class of polynomials on k 1 1 obtained by exploiting a connection to orthogonal polynomials

the inverse of banded matrices the inverses of r banded matrices for r 1 2 3 have been thoroughly investigated as one can see from the references we provide let b r n email protected email protected n be an nxn matrix of entries a j i email protected email protected r email protected email protected r with the remaining un indexed entries all zeros in this paper generalizing a method of mallik 1999 5 we give the lu factorization and the inverse of the matrix b r n if it exists our results are valid for an arbitrary square matrix taking r n and so we will give a new approach for computing the inverse of an invertible square matrix our method is based on hessenberg submatrices associated to b r n

theta functions gaussian series and spatially periodic solutions of the korteweg de vries equation it has been shown by novikov funct anal appl 8 236 1974 dubrovin et al russian math surveys 31 59 1976 lax commun pure appl math 28 141 1975 mckean and van moerbeke inv math 30 217 1975 and others that the nonlinear evolution equations which admit solitary waves also have spatially periodic exact solutions polycnoidal waves which can be expressed in terms of multidimensional riemann theta functions here it is shown that via poisson summation the fourier series that define the theta functions can be transformed into an infinite series of gaussian functions because the lowest terms of the gaussian series generate the usual solitary waves it is possible to intimately explore the relationship between solitary waves and these spatially periodic polycnoidal waves also by using the gaussian series one can perturbatively calculate phase velocities and wave structure for the polycnoidal wave even in the strongly nonlinear regime for which the soliton or mult

nonlinear stability of periodic traveling wave solutions to the schrodinger and the modified korteweg de vries equations abstract this work is concerned with stability properties of periodic traveling waves solutions of the focusing schrodinger equation i u t u x x u 2 u 0 posed in r and the modified korteweg de vries equation u t 3 u 2 u x u x x x 0 posed in r our principal goal in this paper is the study of positive periodic wave solutions of the equation 3 0 called dnoidal waves a proof of the existence of a smooth curve of solutions with a fixed fundamental period l 2 2 l 2 h per 0 l is given it is also shown that these solutions are nonlinearly stable in the energy space h per 1 0 l and unstable by perturbations with period 2l in the case of the schrodinger equation

on the eigenvalues of some tridiagonal matrices a solution is given for a problem on eigenvalues of some symmetric tridiagonal matrices suggested by william trench the method presented can be generalizable to other problems

the inverse of a tridiagonal matrix in this paper explicit formulae for the elements of the inverse of a general tridiagonal matrix are presented by first extending results on the explicit solution of a second order linear homogeneous difference equation with variable coefficients to the nonhomogeneous case and then applying these extended results to a boundary value problem a formula for the characteristic polynomial is obtained in the process we also establish a connection between the matrix inverse and orthogonal polynomials in addition the case of a cyclic tridiagonal system is discussed

on a constant diagonals matrix in this paper we consider a constant diagonals matrix the matrix was discussed in witula and slota r witula d slota on computing the determinants and inverses of some special type of tridiagonal and constant diagonals matrices appl math comput 189 1 2007 514 527 the authors gave some results on determinant and the inverse of the matrix for some special cases we give lu factorization and then compute determinant of the matrix we determine eigenvalues of the matrix further we obtain some relationships between permanent of the matrix and terms of a certain recurrence relation

connecting optimization with spectral analysis of tri diagonal matrices we show that the global minimum resp maximum of a continuous function on a compact set can be approximated from above resp from below by computing the smallest rest largest eigenvalue of a hierarchy of r times r tri diagonal matrices of increasing size equivalently it reduces to computing the smallest resp largest root of a certain univariate degree r orthonormal polynomial this provides a strong connection between the fields of optimization orthogonal polynomials numerical analysis and linear algebra via asymptotic spectral analysis of tri diagonal symmetric matrices

orbital stability of double solitons for the benjamin ono equation in this paper we prove the orbital stability of double solitons for the benjamin ono equation in the case of the kdv equation this stability has been proved in 17 parts of the proof given there rely on the fact that the euler lagrange equations for the conserved quantities of the kdv equation are ordinary differential equations since this is not the case for the benjamin ono equation new methods are required our approach consists in using a new invariant for multi solitons and certain new identities motivated by the sylvester law of inertia

a new look at nonnegativity on closed sets and polynomial optimization we first show that a continuous function f is nonnegative on a closed set k subseteq r n if and only if countably many moment matrices of some signed measure d nu fd mu with support equal to k are all positive semidefinite if k is compact mu is an arbitrary finite borel measure with support equal to k in particular we obtain a convergent explicit hierarchy of semidefinite outer approximations with it no lifting of the cone of nonnegative polynomials of degree at most d wen used in polynomial optimization on certain simple closed sets k like e g the whole space r n the positive orthant a box a simplex or the vertices of the hypercube it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable this convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations

improved convergence rates for lasserre type hierarchies of upper bounds for box constrained polynomial optimization we consider the problem of minimizing a given n variate polynomial f over the hypercube 1 1 n an idea introduced by lasserre is to find a probability distribution on 1 1 n with polynomial density function h of given degree r that minimizes the expectation int 1 1 n f x h x d mu x where d mu x is a fixed finite borel measure supported on 1 1 n it is known that for the lebesgue measure d mu x dx one may show an error bound o 1 sqrt r if h is a sum of squares density and an o 1 r error bound if h is the density of a beta distribution in this paper we show an error bound of o 1 r 2 if d mu x left prod i 1 n sqrt 1 x i 2 right 1 the well known measure in the study of orthogonal polynomials and h has a schmudgen type representation with respect to 1 1 n which is a more general condition than a sum of squares the convergence rate analysis relies on the theory of polynomial kernels and in particular on jackson kerne

inverses of banded matrices abstract we establish a correspondence between the vanishing of a certain set of minors of a matrix a and the vanishing of a related set of minors of a 1 in particular inverses of banded matrices are characterized we then use our results to find patterns for toeplitz matrices with banded inverses finally we give an interesting determinant formula for inverses of banded matrices and show that in general a banded partial matrix may be completed in a unique way to give a banded inverse of the same bandwidth

isoinertial family of operators and convergence of kdv cnoidal waves to solitons abstract in this paper we show the convergence of korteweg de vries cnoidal waves to the limit soliton it is proved that the convergence is uniform and in h 2 norm as the period of the solutions tends to infinity families of hill operators are also studied we obtain a condition under which families of operators are isoinertial this condition is satisfied for classes of hill operators that are obtained by linearization our application is to the family of linearized operators at the kdv cnoidal waves it is proved that this family is isoinertial and also the value of the inertial index is calculated

orbital stability of one parameter periodic traveling waves for dispersive equations and applications abstract this paper establishes sufficient conditions for the orbital stability of one parameter spatially periodic traveling wave solutions for one dimensional dispersive equations our method of proof combines known techniques with some new ideas as a consequence of our result we give several applications for well known dispersive equations extension of the theory to regularized equations is also established

optimal lebesgue constant for lagrange interpolation fehlberg computing 4 1969 pp 93 106 developed a family of eight stage pairs of runge kutta methods of orders 5 and 6 subsequently improved versions were derived independently by butcher the numerical analysis of ordinary differential equations john wiley new york 1987 and verner those obtained by verner are only a slight generalization of fehlberg s methods whereas those of butcher have quite a different structure and require one more stage when the high order method is propagated prince and dormand j comput appl math 7 1981 pp 67 76 have since constructed methods similar to but more general than those developed by verner this paper provides explicit formulas that relate the methods developed by fehlberg by verner and by dormand and prince in particular each of verner s methods leads to a two parameter family of dormand prince methods the formulas facilitate the study of these methods which include the most effective known methods of orders 5 and 6 a particular set

improving formulas for the eigenvalues of finite block toeplitz tridiagonal matrices abstract after a short overview improvements based on the kronecker product are proposed for the eigenvalues of n n block toeplitz tridiagonal block tt matrices with k k matrix entries common in applications some extensions of the spectral properties of the toeplitz tridiagonal matrices are pointed out the eigenvalues of diagonalizable symmetric and skew symmetric block tt matrices are studied besides if certain matrix square root is well defined it is proved that every block tt matrix with commuting matrix entries is isospectral to a related symmetric block tt one further insight about the eigenvalues of hierarchical hermitian block tt matrices of use in the solution of pdes is also achieved

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

orbital stability of periodic traveling wave solutions for the log kdv equation abstract in this paper we establish the orbital stability of periodic waves related to the logarithmic korteweg de vries equation our motivation is inspired in the recent work 3 in which the authors established the well posedness and the linear stability of gaussian solitary waves by using the approach put forward recently in 20 to construct a smooth branch of periodic waves as well as to get the spectral properties of the associated linearized operator we apply the abstract theories in 13 and 25 to deduce the orbital stability of the periodic traveling waves in the energy space

inversion of k tridiagonal matrices with toeplitz structure in this paper we consider an inverse problem with the k tridiagonal toeplitz matrices a theoretical result is obtained that under certain assumptions the explicit inverse of a k tridiagonal toeplitz matrix can be derived immediately two numerical examples are given to demonstrate the validity of our results

estimates for the inverse of tridiagonal matrices arising in boundary value problems by considering tridiagonal matrices as three term recurrence relations with dirichlet boundary conditions one can formulate their inverses in terms of green s functions this analysis is applied to three point difference schemes for 1 d problems and five point difference schemes for 2 d problems we derive either an explicit inverse of the jacobian or a sharp estimate for both uniform and nonuniform grids

embedded diagonally implicit runge kutta nystrom 4 3 pair for solving special second order ivps abstract in this paper third order 3 stage diagonally implicit runge kutta nystrom method embedded in fourth order 4 stage for solving special second order initial value problems is constructed the method has the property of minimized local truncation error as well as the last row of the coefficient matrix is equal to the vector output the stability of the method is investigated and a standard set of test problems are tested upon and comparisons on the numerical results are made when the same set of test problems are reduced to first order system and solved using existing runge kutta method the results clearly shown the advantage and the efficiency of the new method

some nystro m pairs for the general second order initial value problem abstract existing 5 6 nystrom pairs for the general second order initial value problem either control the local error in y only or use more than eight stages we develop a class of eight stage 5 6 nystrom pairs that control the local error in y and y we first give the derivation of the class then we discuss the selection of values for the free parameters in the class finally we give numerical results to illustrate the possible gains in efficiency compared with existing eight stage 5 6 nystrom and runge kutta pairs

high order embedded runge kutta formulae abstract the criteria to be satisfied by embedded runge kutta pairs of formulae are reviewed two new formulae of orders 6 and 8 are presented together with tests on their efficiency relative to other high order formulae in current use

factorizations and representations of the backward second order linear recurrences we show the relationships between the determinants and permanents of certain tridiagonal matrices and the negatively subscripted terms of second order linear recurrences also considering how to the negatively subscripted terms of second order linear recurrences can be connected to chebyshev polynomials by determinants of these matrices we give factorizations and representations of these numbers

an analytical approach explicit inverses of periodic tridiagonal matrices abstract we derive an explicit formula for the inverse of a general periodic tridiagonal matrix our approach is to derive its l u factorization using backward continued fractions bcf which are an essential tool in number theory we then use these formulae to construct an algorithm for inverting a general periodic tridiagonal matrix which we implement in maple 1 finally we present the results of testing the efficiency of our new algorithm against another published implementation and against the library procedures available within maple to invert a general matrix and to compute its determinant

orbital stability of periodic standing waves for the logarithmic klein gordon equation abstract the main goal of this paper is to present orbital stability results of periodic standing waves for the one dimensional logarithmic klein gordon equation to do so we first use compactness arguments and a non standard analysis to obtain the existence and uniqueness of weak solutions for the associated cauchy problem in the energy space second we prove the orbital stability of standing waves using a stablity analysis of conservative systems

a new recursive algorithm for inverting general tridiagonal and anti tridiagonal matrices in the current article the authors present a new recursive symbolic computational algorithm for inverting general tridiagonal and anti tridiagonal matrices an illustrative example is given

convergence analysis for lasserre s measure based hierarchy of upper bounds for polynomial optimization we consider the problem of minimizing a continuous function f over a compact set mathbf k k we analyze a hierarchy of upper bounds proposed by lasserre siam j optim 21 3 864 885 2011 obtained by searching for an optimal probability density function h on mathbf k k which is a sum of squares of polynomials so that the expectation int mathbf k f x h x dx kf x h x dx is minimized we show that the rate of convergence is no worse than o 1 sqrt r o 1 r where 2r is the degree bound on the density function this analysis applies to the case when f is lipschitz continuous and mathbf k k is a full dimensional compact set satisfying some boundary condition which is satisfied e g for convex bodies the rth upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d and if all moments of order up to 2r d 2r d of the lebesgue measure on mathbf k k are known which holds for example if mathbf k k is a simplex hypercube or a euclidean ball

stability properties of periodic waves for the klein gordon equation with quintic nonlinearity in this paper we prove results of existence and orbital instability of periodic waves for the klein gordon equation with quintic nonlinearity the main tool used here is a recent computational approach in order to decide about the nonpositive spectrum of the linearized operator combined with the grillakis shatah and strauss stability theory

explicit formula for the inverse of a tridiagonal matrix by backward continued fractions in this paper we consider a general tridiagonal matrix and give the explicit formula for the elements of its inverse for this purpose considering usual continued fraction we define backward continued fraction for a real number and give some basic results on backward continued fraction we give the relationships between the usual and backward continued fractions then we reobtain the lu factorization and determinant of a tridiagonal matrix furthermore we give an efficient and fast computing method to obtain the elements of the inverse of a tridiagonal matrix by backward continued fractions comparing the earlier result and our result on the elements of the inverse of a tridiagonal matrix it is seen that our method is more convenient efficient and fast