existence of positive solutions for 2nth order singular sublinear boundary value problems abstract this paper investigates the existence of positive solutions for 2nth order n 1 singular sub linear boundary value problems a necessary and sufficient condition for the existence of c 2 n 2 0 1 as well as c 2 n 1 0 1 positive solutions is given by constructing lower and upper solutions and with the maximal theorem our nonlinearity f t x 1 x 2 x n may be singular at x i 0 i 1 2 n t 0 and or t 1

nonresonant singular boundary value problems with sign changing nonlinearities existence results are established for nonresonant singular dirichlet boundary value problems our nonlinearity may be singular in its dependent variable and is allowed to change sign

multiple positive solutions to singular boundary value problems for superlinear higher order odes for 1 k n 1 we study the existence of multiple positive solutions to the singular k n k conjugate boundary value problem 1 n ky n f t y 0 t 1 y i 0 0 0 i k 1 y i 0 0 0 i k 1 y j 1 0 0 j n k 1 we show that there are at least two positive solutions if f t y is is superlinear at and singular at u 0 t 0 and t 1 the arguments involve only positivity properties of green s function and a fixed point theorem in cones moreover the explicit formula of green s function can be found in this paper

error analysis of the bjo rck pereyra algorithms for solving vandermonde systems a forward error analysis is presented for the bjorck pereyra algorithms used for solving vandermonde systems of equations this analysis applies to the case where the points defining the vandermonde matrix are nonnegative and are arranged in increasing order it is shown that for a particular class of vandermonde problems the error bound obtained depends on the dimensionn and on the machine precision only being independent of the condition number of the coefficient matrix by comparing appropriate condition numbers for the vandermonde problem with the forward error bounds it is shown that the bjorck pereyra algorithms introduce no more uncertainty into the numerical solution than is caused simply by storing the right hand side vector on the computer a technique for computing running a posteriori error bounds is derived several numerical experiments are presented and it is observed that the ordering of the points can greatly affect the solution accuracy

a chebychev vandermonde solver abstract let t j n j 0 be a family of chebyshev polynomials for a finite interval a b let x k n k 0 be a set of distinct nodes on a b and define the transposed chebyshev vandermonde matrix w w kj n k j 0 w kj t j x k we describe a new fast algorithm for the solution of linear systems of equations of the form w a f and compare it with a fast solution scheme recently proposed by higham the latter scheme is a modification of the bjorck pereyra algorithm and the computed solution depends on the ordering of the nodes x k our algorithm is designed so that the computed solution is essentially independent of the ordering of the x k extensive computational experience indicates that our algorithm generally yields higher accuracy in the computed solution than higham s

factorizations of cauchy vandermonde matrices abstract this paper analyzes the factorization of the inverse of a cauchy vandermonde matrix as a product of bidiagonal matrices factorizations of cauchy vandermonde matrices are also considered the results are applied to derive a fast algorithm for solving linear systems whose coefficient matrices are cauchy vandermonde matrices

positive solutions for singular second order neumann boundary value problems via a cone fixed point theorem in this paper we consider the existence of positive solutions for the singular second order neumann boundary value problem x k 2x f t g t x 0

how bad are hankel matrices considered are hankel vandermonde and krylov basis matrices it is proved that for any real positive definite hankel matrix of order n its spectral condition number is bounded from below by 3 cdot 2 n 6 also proved is that the spectral condition number of a krylov basis matrix is bounded from below by 3 frac 1 2 cdot 2 frac n 2 3 for v v x 1 ldots x n a vandermonde matrix with arbitrary but pairwise distinct nodes x 1 ldots x n we show that mbox rm cond 2 v geq 2 n 2 n frac 1 2 if either x j leq 1 or x j geq 1 for all j then mbox rm cond 2 v geq 2 n 2

diagonal pivoting for partially reconstructible cauchy like matrices with applications to toeplitz like linear equations and to boundary rational matrix interpolation problems abstract in an earlier paper we exploited the displacement structure of cauchy like matrices to derive for them a fast o n 2 implementation of gaussian elimination with partial pivoting one application is to the rapid and numerically accurate solution of linear systems with toeplitz like coefficient matrices based on the fact that the latter can be transformed into cauchy like matrices by using the fast fourier sine or cosine transform however symmetry is lost in the process and the algorithm given is not optimal for hermitian coefficient matrices in this paper we present a new fast o n 2 implementation of symmetric gaussian elimination with partial diagonal pivoting for hermitian cauchy like matrices and show how to transform hermitian toeplitz like matrices to hermitian cauchy like matrices obtaining algorithms that are twice as fast as those in the earlier work numerical experiments indicate that in order to obtain not only fast but also numerically accurate methods it is advantageous to explore the important case in which the corresponding displacement operators have nontrivial kernels this situation gives rise to what we call partially reconstructible matrices which are introduced and studied in the present paper we extend the transformation technique and the generalized schur algorithms i e fast displacement based implementations of gaussian elimination to partially reconstructible matrices we show by a variety of computed examples that the incorporation of diagonal pivoting methods leads to high accuracy we focused in this paper on the design of new numerically reliable algorithms for hermitian toeplitz like matrices however the proposed algorithms have other important applications in particular we briefly describe how they recursively solve a boundary interpolation problem for j unitary rational matrix functions

fast gaussian elimination with partial pivoting for matrices with displacement structure fast o n 2 implementation of gaussian elimination with partial pivoting is designed for matrices possessing cauchy like displacement structure we show how toeplitz like toeplitz plus hankel like and vandermonde like matrices can be transformed into cauchy like matrices by using discrete fourier cosine or sine transform matrices in particular this allows us to propose a new fast o n 2 toeplitz solver gko which incorporates partial pivoting a large set of numerical examples showed that gko demonstrated stable numerical behavior and can be recommended for solving linear systems especially with nonsymmetric indefinite and ill conditioned positive definite toeplitz matrices it is also useful for block toeplitz and mosaic toeplitz toeplitz block matrices the algorithms proposed in this paper suggest an attractive alternative to look ahead approaches where one has to jump over ill conditioned leading submatrices which in the worst case requires o n 3 operations

a lagrange matrices approach to confluent cauchy matrices we present a new approach to study inversion and factorization properties of confluent cauchy matrices we consider a class of generalized vandermonde matrices called lagrange matrices that are connected in a simple way with the cauchy matrices we apply the methods introduced in adv appl math 10 1989 348 to obtain inversion and factorization theorems for lagrange matrices and their confluent forms and then derive corresponding results for cauchy matrices a lagrange matrix v is associated with a pair of vectors x0 x1 xn and y0 y1 yn v is the matrix representation of the substitution operator that sends the vector p x0 p x1 p xn t to the vector p y0 p y1 p yn t for any polynomial p of degree at most n

a fast and stable algorithm for downdating the singular value decomposition abstract in this paper we modify a classical downdating svd algorithm and reduce its complexity significantly we use a structured low rank approximation algorithm to compute an hierarchically semiseparable hss matrix approximation to the eigenvector matrix of a diagonal matrix plus rank one modification the complexity of our downdating algorithm is analyzed we further show that the structured low rank approximation algorithm is backward stable numerous experiments have been done to show the efficiency of our algorithm for some matrices with large dimensions our algorithm can be much faster than that using plain matrix matrix multiplication routine in intel mkl in both sequential and parallel cases

factorizations of cauchy matrices the development of fast algorithms for the solution of linear systems of equations with a cauchy matrix has recently received considerable attention several of these algorithms factor a cauchy matrix or its inverse into triangular and possibly diagonal matrices the numerical properties of the factorization methods depend on the selection of pivots this note presents elementary derivations of some factorization methods and describes a new strategy for searching both rows and columns for suitable pivots

existence of positive solutions for 2nth order singular superlinear boundary value problems this paper investigates the existence of positive solutions for 2nth order n 1 singular superlinear boundary value problems a necessary and sufficient condition for the existence of c 2 n 2 0 1 as well as c 2 n 1 0 1 positive solutions is given by constructing a special cone and with the e norm

upper and lower solutions method and a superlinear singular boundary value problem abstract the singular boundary value problem u g t u u 0 for t 0 1 u 0 u 1 0 is studied in this paper the singularity may appear at u 0 t 0 and t 1 and the function g may be superlinear at u and change sign the existence of solutions is obtained via an upper and lower solutions method

a bjorck pereyra type algorithm for szego vandermonde matrices based on properties of unitary hessenberg matrices abstract in this paper we carry over the bjorck pereyra algorithm for solving vandermonde linear systems to what we suggest to call szego vandermonde systems v x i e polynomial vandermonde systems where the corresponding polynomial system is the szego polynomials the properties of the corresponding unitary hessenberg matrix allow us to derive a fast o n 2 computational procedure we present numerical experiments that indicate that for ill conditioned matrices the new algorithm yields better forward accuracy than gaussian elimination

fast inversion of chebyshev vandermonde matrices this paper contains two fast algorithms for inversion of chebyshev vandermonde matrices of the first and second kind they are based on special representations of the bezoutians of chebyshev polynomials of both kinds the paper also contains the results of numerical experiments which show that the algorithms proposed here are not only much faster but also more stable than other algorithms available it is also efficient to use the above two algorithms for solving chebyshev vandermode systems of equations with preprocessing

singular boundary value problems for the one dimension p laplacian abstract the singular boundary value problem u g t u u 0 for t 0 1 u 0 u 1 0 where s s p 2s p 1 is studied in this paper the singularity may appear at u 0 t 0 and t 1 and the function g may change sign the existence of solutions is obtained via an upper and lower solution method

nonresonant nonlinear singular problems in the limit circle case existence results are presented for the resonant singular boundary value problem tfrac 1 p py prime prime lambda m qy f t y a e on 0 1 with limt 0 py y 1 0 here we donot assume int 0 1 tfrac ds p s infty but only that int 0 1 tfrac 1 p s left int 0 s p x q x dx right tfrac 1 2 ds infty

complexity of multiplication with vectors for structured matrices abstract fast algorithms for computing the product with a vector are presented for a number of classes of matrices whose properties relate to the properties of toeplitz vandermonde or cauchy matrices these matrices are defined using the concept of displacement of a matrix and also for their inverses all the actions which are not dependent upon the coordinates of the input vector are singled out in a separate preprocessing stage the proposed algorithms are based on new representations of these matrices involving factor circulants

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

nonuniform nonresonant singular dirichlet boundary value problems for the one dimensional p laplacian with sign changing nonlinearity abstract existence results are presented for the nonuniform nonresonant singular boundary value problem u f t u u 0 for t 0 1 u 0 u 1 0 where s s p 2 s p 1 the singularity may appear at u 0 t 0 or t 1 and the function f may change sign the existence of solutions is obtained via an upper and lower solutions method

pivoting and backward stability of fast algorithms for solving cauchy linear equations abstract three fast o n2 algorithms for solving cauchy linear systems of equations are proposed a rounding error analysis indicates that the backward stability of these new cauchy solvers is similar to that of gaussian elimination thus suggesting to employ various pivoting techniques to achieve a favorable backward stability it is shown that cauchy structure allows one to achieve in o n2 operations partial pivoting ordering of the rows and several other judicious orderings in advance without actually performing the elimination the analysis also shows that for the important class of totally positive cauchy matrices it is advantageous to avoid pivoting which yields a remarkable backward stability of the suggested algorithms it is shown that vandermonde and chebyshev vandermonde matrices can be efficiently transformed into cauchy matrices using discrete fourier cosine or sine transforms this allows us to use the proposed algorithms for cauchy matrices for rapid and accurate solution of vandermonde and chebyshev vandermonde linear systems the analytical results are illustrated by computed examples

a class of 2nth order singular boundary value problems abstract this paper investigates the existence of positive solutions for 2 n th order n 1 singular sub linear boundary value problems first of all we establish the maximal principle and some important lemmas then we define a partial ordering in c 2 n 2 0 1 c 2 n 0 1 and construct lower and upper solutions to give a necessary and sufficient condition for the existence of c 2 n 2 0 1 as well as c 2 n 1 0 1 positive solutions our nonlinearity f t x 1 x 2 x n may be singular at x i 0 i 1 2 n t 0 and or t 1