solving boundary value problems integral and integro differential equations using gegenbauer integration matrices we introduce a hybrid gegenbauer ultraspherical integration method hgim for solving boundary value problems bvps integral and integro differential equations the proposed approach recasts the original problems into their integral formulations which are then discretized into linear systems of algebraic equations using gegenbauer integration matrices gims the resulting linear systems are well conditioned and can be easily solved using standard linear system solvers a study on the error bounds of the proposed method is presented and the spectral convergence is proven for two point bvps tpbvps comparisons with other competitive methods in the recent literature are included the proposed method results in an efficient algorithm and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems the proposed method can be applied on a broad range of mathematical problems while producing highly accurate results the developed numerical scheme provides a viable alternative to other solution methods when high order approximations are required using only a relatively small number of solution nodes

the eigenvalues of second order spectral differentiation matrices the eigenvalues of the pseudospectral second derivative matrix with homogeneous dirichlet boundary conditions are important in many applications of spectral methods this paper investigates some of their properties numerical results show that a certain fraction of the eigenvalues approximate the eigenvalues of the continuous operator very accurately but the errors in the remaining ones are large it is demonstrated that the inaccurate eigenvalues correspond to those eigenfunctions of the continuous operator that cannot be resolved by polynomial interpolation in the spectral grid in particular it is proved that 7r points on average per wavelength are sufficient for successful interpolation of the eigenfunctions of the continuous operator in a chebyshev distribution of nodes and six points per wavelength for a uniform distribution these results are in agreement with the observed fractions of accurate eigenvalues by using the characteristic polynomial a bound on the spectral radius of the differentiation matrix is derived that is accurate to 2 or better the effect of filtering on the eigenvalues is studied numerically

preconditioned minimal residual methods for chebyshev spectral calculations abstract the problem of preconditioning the pseudospectral chebyshev approximation of an elliptic operator is considered the numerical sensitivity to variations of the coefficients of the operator are investigated for two classes of preconditioning matrices one arising from finite differences the other from finite elements the preconditioned system is solved by a conjugate gradient type method and by a dufort frankel method with dynamical parameters the methods are compared on some test problems with the richardson method 13 and with the minimal residual richardson method 21

a chebyshev method for the solution of boundary value problems abstract an expansion procedure using the chebyshev polynomials as base functions is proposed the method yields more accurate results than either of the galerkin or tau methods as indicated from solving the orr sommerfeld equation for both the plane poiseuille flow and the blasius velocity profile the chebyshev approximation is also applied to resolve the radial dependence of the flow field for a circular cylinder or a sphere in a uniform flow

chebyshev pseudospectral solution of second order elliptic equations with finite element preconditioning abstract a new chebyshev pseudospectral algorithm for second order elliptic equations using finite element preconditioning is proposed and tested on various problems bilinear and biquadratic lagrange elements are considered as well as bicubic hermite elements the numerical results show that bilinear elements produce spectral accuracy with the minimum computational work l shaped regions are treated by a subdomain approach

a numerical comparison of chebyshev methods for solving fourth order semilinear initial boundary value problems in solving semilinear initial boundary value problems with prescribed non periodic boundary conditions using implicit explicit and implicit time stepping schemes both the function and derivatives of the function may need to be computed accurately at each time step to determine the best chebyshev collocation method to do this the accuracy of the real space chebyshev differentiation spectral space preconditioned chebyshev tau real space chebyshev integration and spectral space chebyshev integration methods are compared in the l 2 and w 2 2 norms when solving linear fourth order boundary value problems and in the l 0 t l 2 and l 0 t w 2 2 norms when solving initial boundary value problems we find that the best chebyshev method to use for high resolution computations of solutions to initial boundary value problems is the spectral space chebyshev integration method which uses sparse matrix operations and has a computational cost comparable to fourier spectral discretization

on the eigenvalues of second order pseudospectral differentiation operators abstract recent results on the behavior of eigenvalues for the second order legendre spectral galerkin differentiation operator are extended to the eigenvalues of the corresponding legendre pseudospectral collocation operator numerical comparisons show that similar results also hold in the case of spectral chebyshev methods

the pseudospectral method for third order differential equations generalized quadrature rules are derived which assist in the selection of collocation points for the pseudospectral solution of differential equations in particular it is shown that for an nth order differential equation in one space dimension with two point derivative boundary conditions an ideal choice of interior collocation points is the set of zeros of a jacobi polynomial the pseudospectral solution of a third order initial boundary value problem is considered and accuracy is assessed by examining how well the discrete eigenproblem approximates the continuous one convergence is established for a special choice of collocation points and numerical results are included to demonstrate the viability of the approach

lattice quantization of phases for equal gain transmission equal gain transmission egt requires only the feedback of phase information and is known to enjoy several implementational advantages compared to maximum ratio transmission mrt this paper investigates the use of lattice quantization for quantizing phases in egt systems in practice the feedback rate is finite and the lattices need to be truncated to have finite code rate the truncation of lattices results in boundary effect hence earlier results developed for untruncated lattices e g fast algorithm and mean square error mse analysis cannot be applied directly exploiting the fact that the quantized signals are phases we show how to quantize using truncated lattices so that there is no boundary effect moreover we propose efficient methods to convert between lattice codewords and binary digits so that the lattice quantization can be used in practical systems furthermore the mean square quantization error is analyzed for several useful truncated lattices we also show how to analyze the signal to noise ratio snr loss when the proposed quantization is applied to an egt system

integration preconditioning matrix for ultraspherical pseudospectral operators in this paper we give an integration rectangular preconditioner for ultraspherical pseudospectral operators we present an explicit solution of matrix equations with the structure provided by the pseudospectral discretization of the nth order differential operator subject to various types of boundary conditions this occurs in two steps in the first step we introduce an integration rectangular preconditioner in the second step we use a new strategy to impose the boundary conditions on the condition that the coefficients matrix has a known inverse computational examples are given

an efficient spectral method for ordinary differential equations with rational function coefficients the authors present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients they employ expansions in terms of a large class of orthogonal polynomial families including all the classical orthogonal polynomials these families obey a simple 3 term recurrence relation for differentiation which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse the inverse is an integration operator for the family and it is simply the tridiagonal coefficient matrix for the recurrence since in these families convolution operators i e matrix representations of multiplication by a function are banded for polynomials the authors are able to obtain a banded representation for linear differential operators with rational coefficients this leads to a method of solution of initial or boundary value problems that besides having an operation count that scales linearly with the order of truncation n is computationally well conditioned among the applications considered is the use of rational maps for the resolution of sharp interior layers 19 refs 2 figs 3 tabs

a well conditioned collocation method using a pseudospectral integration matrix in this paper a well conditioned collocation method is constructed for solving general p th order linear differential equations with various types of boundary conditions based on a suitable birkhoff interpolation we obtain a new set of polynomial basis functions that results in a collocation scheme with two important features the condition number of the linear system is independent of the number of collocation points and the underlying boundary conditions are imposed exactly moreover the new basis leads to an exact inverse of the pseudospectral differentiation matrix of the highest derivative at interior collocation points which is therefore called the pseudospectral integration matrix psim we show that psim produces the optimal integration preconditioner and stable collocation solutions with even thousands of points

efficient spectral galerkin methods for systems of coupled second order equations and their applications we construct in this paper two efficient spectral galerkin algorithms for solving systems of n coupled second order equations the computational complexity of these algorithms is essentially n times the cost of solving one second order equation we present numerical results which illustrate the accuracy and flexibility of these algorithms as well as several interesting and challenging applications including in particular a number of high order nonlinear parabolic type equations

spectral integration and two point boundary value problems a numerical method for two point boundary value problems with constant coefficients is developed which is based on integral equations and the spectral integration matrix for chebyshev nodes the method is stable achieves superalgebraic convergence and requires o n log n operations where n is the number of nodes in the discretization although stable spectral methods have been constructed in the past they have generally been based on reformulating the recurrence relations obtained through spectral differentiation in an attempt to avoid the ill conditioning introduced by that process

an instability phenomenon in spectral methods the eigenvalues of chebyshev and legendre spectral differentiation matrices which determine the allowable time step in an explicit time integration are extraordinarily sensitive to rounding errors and other perturbations on a grid of n points per space dimension machine rounding leads to errors in the eigenvalues of size o n 2 this phenomenon may lead to inconsistency between predicted and observed time step restrictions one consequence of it is that spectral differentiation by interpolation in legendre points which has a favorable o n 1 time step restriction for the model problem u t u x in theory is subject to an o n 2 restriction in practice the same effect occurs with chebyshev points for the model problem u t xu x another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision we know of no other examples in numerical computation of this kind of precision dependent stability

a fast and well conditioned spectral method a spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients and general boundary conditions the method leads to matrices that are al

quantitative analysis of pulmonary emphysema using local binary patterns we aim at improving quantitative measures of emphysema in computed tomography ct images of the lungs current standard measures such as the relative area of emphysema ra rely on a single intensity threshold on individual pixels thus ignoring any interrelations between pixels texture analysis allows for a much richer representation that also takes the local structure around pixels into account this paper presents a texture classification based system for emphysema quantification in ct images measures of emphysema severity are obtained by fusing pixel posterior probabilities output by a classifier local binary patterns lbp are used as texture features and joint lbp and intensity histograms are used for characterizing regions of interest rois classification is then performed using a k nearest neighbor classifier with a histogram dissimilarity measure as distance a 95 2 classification accuracy was achieved on a set of 168 manually annotated rois comprising the three classes normal tissue centrilobular emphysema and paraseptal emphysema the measured emphysema severity was in good agreement with a pulmonary function test pft achieving correlation coefficients of up to r 0 79 in 39 subjects the results were compared to ra and to a gaussian filter bank and the texture based measures correlated significantly better with pft than did ra

finite element preconditioning of g ni spectral methods several old and new finite element preconditioners for nodal based spectral discretizations of delta u f in the domain omega 1 1 d d 2 or 3 with dirichlet or neumann boundary conditions are considered and compared in terms of both condition number and computational efficiency the computational domain covers the case of classical single domain spectral approximations see c canuto et al spectral methods fundamentals in single domains springer heidelberg 2006 as well as that of more general spectral element methods in which the preconditioners are expressed in terms of local upon every element algebraic solvers the primal spectral approximation is based on the galerkin approach with numerical integration g ni at the legendre gauss lobatto lgl nodes in the domain the preconditioning matrices rely on either mathbb p 1 mathbb q 1 or mathbb q 1 ni i e with numerical integration finite elements on meshes whose vertices coincide with the lgl nodes used for the spectral approximation the analysis highlights certain preconditioners which yield the solution at an overall cost proportional to n d 1 where n denotes the polynomial degree in each direction

galerkin orthogonal polynomials abstract the galerkin method offers a powerful tool in the solution of differential equations and function approximation on the real interval 1 1 by expanding the unknown function in appropriately chosen global basis functions each of which explicitly satisfies the given boundary conditions in general this scheme converges exponentially fast and almost always supplies the most terse representation of a smooth solution to date typical schemes have been defined in terms of a linear combination of two jacobi polynomials however the resulting functions do not inherit the expedient properties of the jacobi polynomials themselves and the basis set will not only be non orthogonal but may in fact be poorly conditioned using a gram schmidt procedure it is possible to construct in an incremental fashion polynomial basis sets that not only satisfy any linear homogeneous boundary conditions but are also orthogonal with respect to the general weighting function 1 x 1 x however as it stands this method is not only cumbersome but does not provide the structure for general index n of the functions and obscures their dependence on the parameters in this paper it is shown that each of these galerkin basis functions as calculated by the gram schmidt procedure may be written as a linear combination of a small number of jacobi polynomials with coefficients that can be determined moreover this terse analytic representation reveals that for large index the basis functions behave asymptotically like the single jacobi polynomial p n x this new result shows that such galerkin bases not only retain exponential convergence but expedient function fitting properties too in much the same way as the jacobi polynomials themselves this powerful methodology of constructing galerkin basis sets is illustrated by many examples and it is shown how the results extend to polar geometries in exploring more generalised definitions of orthogonality involving derivatives we discuss how a large class of differential operators may be discretised by galerkin schemes and represented in a sparse fashion by the inverse of band limited matrices

automatic spectral collocation for integral integro differential and integrally reformulated differential equations automatic chebyshev spectral collocation methods for fredholm and volterra integral and integro differential equations have been implemented as part of the chebfun software system this system enables a symbolic syntax to be applied to numerical objects in order to pose and solve problems without explicit references to discretization the same objects can be used in matrix free iterative methods in linear algebra in order to avoid very large dense matrices or allow application to problems with nonsmooth coefficients as a further application of the ability to implement operator equations a method of greengard 1 for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems in the integral form large condition numbers associated with differentiation matrices in high order problems are avoided the ability to implement the recasting process generally follows from implementation of the operator expressions in chebfun the integral method also can be extended to first order systems although chebfun syntax does not currently allow easy implementation in this case

a new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations a new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested this method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions stability and convergence results are proven for the chebyshev approximation of linear scalar hyperbolic equations the eigenvalues of this method applied to parabolic equations are shown to be real and negative

numerical approximations using chebyshev polynomial expansions el gendi s method revisited we present numerical solutions for differential equations by expanding the unknown function in terms of chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema or zeros of the chebyshev polynomial of order n el gendi s method the solutions are exact at these points apart from round off computer errors and the convergence of other numerical methods used in solving the linear system of equations applications to initial value problems in time dependent quantum field theory and second order boundary value problems in fluid dynamics are presented

integration preconditioning of pseudospectral operators i basic linear operators this paper develops a family of preconditioners for pseudospectral approximations of pth order linear differential operators subject to various types of boundary conditions the approximations are based on ultraspherical polynomials with special attention being paid to legendre and chebyshev polynomial methods based on gauss lobatto quadrature points the eigenvalue spectrum of the preconditioned operators are obtained in closed analytic form and the weakly enforced boundary conditions are shown to result in a rank 2p perturbation of the identity operator i e the majority of the preconditioned eigenvalues are unity the spectrum of the preconditioned advective operator is shown to be bounded independent of the order of the approximation n however the preconditioned diffusive operator is in general indefinite with four real eigenvalues for dirichlet boundary conditions the spectral radius grows as sqrt n while it scales as n for the case of neumann boundary conditions these results are shown to be asymptotically optimal within the present framework generalizations to higher order differential operators general boundary conditions and arbitrary polynomial basis and quadrature nodes are discussed

a new dual petrov galerkin method for third and higher odd order differential equations application to the kdv equation a new dual petrov galerkin method is proposed analyzed and implemented for third and higher odd order equations using a spectral discretization the key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the dual boundary conditions the method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study of complex dynamics of higher odd order equations

superconvergence points of polynomial spectral interpolation in this work we study superconvergence properties for some high order orthogonal polynomial interpolations at the superconvergent points the error is smaller by a factor of n alpha with alpha 0 where n is the polynomial degree the results are two fold when interpolating function values we identify those points where the first and second derivatives of the interpolant converge faster when interpolating the first derivative we locate those points where the function value of the interpolant superconverges for the earlier case we use various chebyshev polynomials and for the later case we also include the counterpart legendre polynomials

preconditioning chebyshev spectral collocation method for elliptic partial differential equations in this paper we analyze a preconditioning technique for the solution of chebyshev spectral collocation equations with dirichlet boundary conditions we obtain bounds on the eigenvalues for the helmholtz equation these eigenvalue bounds are obtained as a consequence of estimates on the field of values tilde a n 2 u u l 2 tilde q n 2 u u l 2 where tilde a n 2 is the weighted collocation matrix and tilde q n 2 is the preconditioner the preconditioner tilde q n 2 is robust in the sense that it provides bounds on the h 1 0 w condition number of tilde q n 2 1 tilde l n 2 when tilde l n 2 is the weighted collocation matrix associated with the general elliptic operator lu delta u a 1u x a 2u y a 0u

a fast algorithm for gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations after studying gaussian type quadrature formulae with mixed boundary conditions we suggest a fast algorithm for computing their nodes and weights it is shown that the latter are computed in the same manner as in the theory of the classical gauss quadrature formulae in fact all nodes and weights are again computed as eigenvalues and eigenvectors of a real symmetric tridiagonal matrix hence we can adapt existing procedures for generating such quadrature formulae comparative results with various methods now in use are given in the second part of this paper new algorithms for spectral approximations for second order elliptic problems are derived the key to the efficiency of our algorithms is to find an appropriate spectral approximation by using the most accurate quadrature formula which takes the boundary conditions into account in such a way that the resulting discrete system has a diagonal mass matrix hence our algorithms can be used to introduce explicit resolutions for the time dependent problems this is the so called lumped mass method the performance of the approach is illustrated with several numerical examples in one and two space dimensions

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

preconditioning chebyshev spectral collocation by finite difference operators in 1979 orszag proposed a finite difference preconditioning of the chebyshev collocation discretization of the poisson equation in 1984 haldenwang labrosse abboudi and deville gave analytic formulae for the eigenvalues of this preconditioned operator in the one dimensional case experimental results over many years have shown the effectiveness of this procedure and appropriate bounds on the eigenvalues in two dimensions however there have been no mathematical proofs describing the behavior of the eigenvalues in two or more dimensions in this work we consider the generalized field of values u hat a nu u l nu where hat a n is the matrix of the chebyshev collocation scheme and l n is the matrix of the finite difference operator for the case of the chebyshev collocation of the helmoltz operator au delta u au a ge 0 preconditioned by the finite difference operator associated with the helmholtz operator bu delta u bu b ge 0 we prove that there are two constants 0 0 and u hat a nu u l nu le lambda 1 these results extend to higher dimensions and to bounds on l n 1 hat a n 1 w and hat a n 1 l n 1 w in the general case where au delta u a 1u x a 2u y au