finite difference preconditioning cubic spline collocation method of elliptic equations we discuss a finite difference preconditioner for the c 1 interpolatory cubic spline collocation method for a uniformly elliptic operator a defined by au delta u a 0 u in omega the unit square with homogeneous dirichlet boundary conditions using the generalized field of values arguments we discuss the eigenvalues of the preconditioned matrix hat l n 2 1 hat a n 2 where hat a n 2 is the matrix of the collocation discretization operator a n 2 corresponding to a and hat l n 2 is the matrix of the finite difference operator l n 2 corresponding to the uniformly elliptic operator l given by l v delta v v in omega with homogeneous dirichlet boundary conditions finally we mention a bound of h 1 singular values of hat l 1 n 2 hat a n 2 for a general elliptic operator au delta u a 1 u x a 2 u y a 0 u in omega

tensor product generalized adi methods for separable elliptic problems we consider solving separable second order linear elliptic partial differential equations if an elliptic problem is separable then for certain discretizations the matrices involved in the corresponding discrete problem can be expressed in terms of tensor products of lower order matrices in the most general case the discrete problem can be written in the form a 1 otimes b 2 b 1 otimes a 2 c f we present a new tensor product generalized alternating direction implicit tpgadi iterative method for solving such discrete problems we prove convergence and establish computational efficiency the tpgadi method is applied to the hermite bicubic collocation equations we conclude that the tpgadi method is an effective tool for solving the discrete elliptic problems arising from a large class of elliptic problems

g 1 approximation of conic sections by quartic bezier curves this paper proposes a new approximation method for conic sections by quartic bezier curves with g 1 end point continuity we give an upper bound on the hausdorff distance between the conic section and the quartic bezier approximation curve and also show that the approximation order is eight moreover using the subdivision scheme at the shoulder point of the conic section the resulting approximation curve has g 2 curvature continuity with the conic section at the endpoints finally we prove that our approximation has a smaller error bound than previous quartic bezier approximations and also present some numerical examples to demonstrate the validity and effectiveness of our method

an approximation method to circular arcs abstract modern cad cam systems are not able to dispose the circles represented by the implicit equations or the parametric equations with trigonometric function parametric rational polynomials are used to approximate the circular arcs a new approximation method by means of quartic bezier curves is presented to effectively approximate the circular arcs it is shown that our method provides the approximation of order eight and yields a smaller error than previous methods by equally spaced subdivision of circular arcs the curvature continuous spline approximation algorithm of the circular arcs is given

a nonoverlapping domain decomposition method for orthogonal spline collocation problems a nonoverlapping domain decomposition approach is used on uniform and matching grids to first define and then to compute the orthogonal spline collocation solution of the dirichlet boundary value problem for poisson s equation on an l shaped region we prove existence and uniqueness of the collocation solution and derive optimal order hs norm error bounds for s 0 1 2 the collocation solution on two interfaces is computed using the preconditioned conjugate gradient method and the collocation solution on three squares is computed by a matrix decomposition method that uses fast fourier transforms the total cost of the algorithm is o n2 log n where the number of unknowns in the collocation solution is o n2

optimal parameter values for approximating conic sections by the quartic bzier curves the previous approximation curves of conic section by quartic bzier curves interpolate the conic section at the specified parameter values in this paper by solving the minimax problem we present the optimal parameter values for approximating conic sections by the quartic bzier curves the upper bound on the hausdorff distance between the conic section and the approximation curves is minimized the method of solving the minimax problem is changed to solve a quartic algebraic equation by delicate reasoning

tensorial basis spline collocation method for poisson s equation this paper aims to describe the tensorial basis spline collocation method applied to poisson s equation in the case of a localized 3d charge distribution in vacuum this direct method based on a tensorial decomposition of the differential operator is shown to be competitive with both iterative bscm and fft based methods we emphasize the o h4 and o h6 convergence of tbscm for cubic and quintic splines respectively we describe the implementation of this method on a distributed memory parallel machine performance measurements on a cray t3e are reported our code exhibits high performance and good scalability as an example a 27 gflops performance is obtained when solving poisson s equation on a 2563 non uniform 3d cartesian mesh by using 128 t3e 750 processors this represents 215 mflops per processors

preconditioning cubic spline collocation discretizations of elliptic equations this work considers the uniformly elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 and partial u over partial nu alpha u on gamma 1 and its discretization based on hermite cubic spline spaces and collocation at the gauss points using an interpolatory basis with support on the gauss points one obtains the matrix hat a n h 1 over n we discuss the h 1 condition numbers and the distribution of tilde beta n singular values of the preconditioned matrices tilde beta n 1 hat a n where tilde beta n is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator b given by b v delta v b 0 v in omega with boundary conditions v 0 on gamma 0 partial v over partial nu beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by gauss points or the space of continuous functions which are linear on the triangles of the triangulation of omega using the gauss points when a b we obtain results on the eigenvalues of tilde beta n 1 hat a n in the general case we obtain bounds and clustering results on the tilde beta n singular values of tilde beta n 1 hat a n these results are related to the results of manteuffel and parter mp parter and wong pw and wong w for finite element discretizations as well as the results of parter and rothman pr for discretizations based on legendre spectral collocation

on k type pseudo null darboux helices in minkowski 3 space abstract in this paper we introduce k type pseudo null darboux helices in minkowski 3 space for k 1 2 3 we obtain the relationship between 1 type and 2 type pseudo null darboux helices and show that all 3 type pseudo null darboux helices are also 1 type and 2 type helices but the converse does not hold finally we give some examples

alternating direction collocation for separable elliptic partial differential equations a complete theory is given for the convergence of alternating direction collocation adc applied to a large class of linear separable elliptic partial differential equations on a rectangular domain it is shown that adc using bicubic hermite splines converges in this setting and converges to the full collocation solution moreover it is shown that a complex embedding of the problem is essential to the theory and that the associated eigensplines and generalized eigensplines can be complex with complex eigenvalues whose geometric and algebraic multiplicities need not be equal nonetheless only real computations are required

spline collocation differentiation matrices spline collocation differentiation matrices arising when solving pdes with periodic boundary conditions have block circulant structure in this paper we use this fact to present an eigenvalue analysis for a large class of such spline collocation differentiation matrices we use this to discuss the stability of collocation algorithms for parabolic and hyperbolic problems and fast collocation algorithms for elliptic pdes

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

on spline basis selection for solving differential equations the suitability of b splines as a basis for piecewise polynomial solution representation for solving differential equations is challenged two alternative local solution representations are considered in the context of collocating ordinary differential equations hermite type and lmonomial both are much easier and shorter to implement and somewhat more efficient than b splines a new condition number estimate for the b splines and hermite type representations is presented one choice of the hermite type representation is experimentally determined to produce roundoff errors at most as large as those for b splines the monomial representation is shown to have a much smaller condition number than the other ones and correspondingly produces smaller roundoff errors especially for extremely nonuniform meshes the operation counts for the two local representations considered are about the same the hermite type representation being slightly cheaper it is concluded that both representations are preferable

a fast direct solver for the biharmonic problem in a rectangular grid we present a fast direct solver methodology for the dirichlet biharmonic problem in a rectangle the solver is applicable in the case of the second order stephenson scheme j w stephenson j comput phys 55 1984 pp 65 80 as well as in the case of a new fourth order scheme which is discussed in this paper and is based on the capacitance matrix method b l buzbee and f w dorr siam j numer anal 11 1974 pp 1136 1150 p bjorstad siam j numer anal 20 1983 pp 59 71 the discrete biharmonic operator is decomposed into two components the first is a diagonal operator in the eigenfunction basis of the laplacian to which the fft algorithm is applied the second is a low rank perturbation operator given by the capacitance matrix which is due to the deviation of the discrete operators from diagonal form the sherman morrison formula g h golub and c f van loan matrix computations 3rd ed john hopkins university press baltimore md 1996 is applied to obtain a fast solution of the resulting linear system of equations

a fast solver for the orthogonal spline collocation solution of the biharmonic dirichlet problem on rectangles a fast schur complement algorithm is presented for computing the piecewise hermite bicubic orthogonal spline collocation solution of the biharmonic dirichlet problem on a rectangular region on an n n uniform partition the algorithm which involves the preconditioned conjugate gradient method and fast fourier transforms requires o n2 log2 n arithmetic operations

a general framework for the optimal approximation of circular arcs by parametric polynomial curves abstract we propose a general framework for a geometric approximation of circular arcs by parametric polynomial curves the approach is based on a constrained uniform approximation of an error function by scalar polynomials the system of nonlinear equations for the unknown control points of the approximating polynomial given in the bezier form is derived and a detailed analysis provided for some low degree cases which were not studied yet at least for these cases the solutions can be in principal written in a closed form and provide the best known approximants according to the simplified radial distance a general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given

fast direct solvers for piecewise hermite bicubic orthogonal spline collocation equations fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise hermite bicubics is employed for the approximate solution of poisson s

some new g1 quartic parametric approximants of circular arcs abstract in this paper a method for approximation of circular arc by quartic bezier curve is presented interpolation by geometrically continuous g 1 parametric polynomials is considered and necessary and sufficient conditions for the existence of the solution are established optimal asymptotic approximation order eight is also confirmed for the obtained approximant application of our new solution to approximation of conic sections is presented and some numerical examples provided

matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions abstract bialecki b and g fairweather matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions journal of computational and applied mathematics 46 1993 369 386 in recent years several matrix decomposition algorithms have been developed for the efficient solution of the linear algebraic systems arising when finite difference finite element galerkin and spectral methods are applied to separable elliptic boundary value problems in a rectangle the success of these methods depends on knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two point boundary value problems in one space dimension the primary purpose of this paper is to provide an overview of matrix decomposition algorithms and show how they can be expressed in terms of a unifying framework particular emphasis is placed on algorithms formulated recently by the authors for solving the linear systems arising in orthogonal spline collocation that is spline collocation at gauss points all of the methods discussed in this paper are modular and possess a great deal of natural parallelism

an alternating direction implicit method for orthogonal spline collocation linear systems an alternating direction implicit method is analyzed for the solution of linear systems arising in high order tensor product orthogonal spline collocation applied to some separable second order linear elliptic partial differential equations in rectangles on annxn partition with jordan s selection of the acceleration parameters the method requireso n 2 ln 2 n arithmetic operations to produce an approximation whose accuracy in theh 1 norm is that of the collocation solution

analysis on eigenvalues for preconditioning cubic spline collocation method of elliptic equations abstract in the work of solving a uniformly elliptic differential equations au u a 1 u x a 2 u y a 0 u f in the unit square with boundary conditions by the c 1 cubic spline collocation method one may need to investigate efficient preconditioning techniques for this purpose using the generalized field of values argument we show the uniform bounds of the eigenvalues of the preconditioned matrix when a full finite element preconditioning is considered

on polynomial approximation of circular arcs and helices we present a simple method for polynomial approximation of circular arcs and helices by expressing the trigonometric functions using the two point taylor expansion we obtain the degree 2n 1 polynomial for the approximation problem in an efficient way which is very convenient to increase the degree of polynomial by adding new terms an upper bound on the approximation error is available so that we can obtain the lowest degree polynomial curve that can approximate a circular arc or helix segment within any user prescribed error tolerance

finite difference preconditioning for solving orthogonal collocation equations for boundary value problems a technique to construct a low order finite difference preconditioner for solving orthogonal collocation equations for boundary value problems is presented it is shown numerically and theoretically that the spectral condition numbers of the preconditioned collocation matrices are bounded by constants independent of the number of mesh nodes when certain exact low order finite difference preconditionings are used preconditioners based on incomplete lu factorization are also discussed numerical experiments show the efficiency and robustness of the preconditioning

the best g 1 cubic and g 2 quartic bezier approximations of circular arcs we obtain cubic and quartic bezier approximations of circular arcs that respectively satisfy g 1 and g 2 end point interpolation conditions we identify the necessary and sufficient conditions for such approximations to be the best in the sense that they have the minimum hausdorff distance to the circular arc we then establish the existence and uniqueness of these best approximations and present practical methods to calculate them which are verified by examples

fractal weyl law behavior in an open hamiltonian system we numerically show fractal weyl law behavior in an open hamiltonian system that is described by a smooth potential and which supports numerous above barrier resonances this behavior holds even relatively far away from the classical limit the complex resonance wave functions are found to be localized on the fractal classical repeller

a note on quintic polynomial approximation of generalized cornu spiral segments in two recent papers cripps et al 2010 3 and cross and cripps 2012 2 quintic polynomial approximations of the generalized cornu spirals have been studied by considering g 2 continuity and g 3 continuity at the end points respectively the quintic curve is constructed so that the maximum curvature error is within the specified tolerance in this paper we provide corrections to the typing errors in cross and cripps 2012 2 and propose a simpler and more efficient method for the g 2 constrained quintic polynomial approximation where the g 2 conditions are always satisfied by four free variables also we introduce a new error measure of the maximum curvature error which can greatly reduce the computational time when looking for the solution numerical experiments demonstrate the effectiveness of the new measure

multilevel additive and multiplicative methods for orthogonal spline collocation problems multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline collocation with piecewise hermite bicubics is applied to the dirichlet boundary value problem for a self adjoint elliptic partial differential equation on a rectangle additive and multiplicative preconditioners are defined respectively as sums and products of independent operators on a sequence of nested subspaces of the fine partition approximation space a general theory of additive and multiplicative schwarz methods is used to prove that the preconditioners are spectrally equivalent to the collocation discretization of the laplacian with the spectral constants independent of the fine partition stepsize and the number of levels the preconditioned conjugate gradient and preconditioned orthomin methods are considered for the solution of collocation problems an implementation of the methods is discussed and the results of numerical experiments are presented

lyapunov stability for a generalized henon heiles system in a rotating reference frame in this paper we focus on a generalized henon heiles system in a rotating reference frame in such a way that lagrangian like equilibrium points appear our goal is to study their nonlinear stability properties to better understand the dynamics around these points we show the conditions on the free parameters to have stability and we prove the superstable character of the origin for the classical case it is a stable equilibrium point regardless of the frequency value of the rotating frame

circular sector area preserving approximation of circular arcs by geometrically smooth parametric polynomials abstract the quality of the approximation of circular arcs by parametric polynomials is usually measured by the hausdorff distance it is sometimes important that a parametric polynomial approximant additionally preserves some particular geometric property in this paper we study the circular sector area preserving parametric polynomial approximants of circular arcs a general approach to this problem is considered and corresponding nonlinear equations are derived for the approximants possessing the maximal order of geometric smoothness a scalar nonlinear equation is analyzed in detail for the parabolic the cubic and the quartic case the existence of the admissible solution is confirmed moreover the uniqueness of the solution with the optimal approximation order with respect to the radial distance is proved theoretical results are confirmed by numerical examples

quintic polynomial approximation of log aesthetic curves by curvature deviation log aesthetic curves lacs possessing monotone curvature and including many classical curves have been widely used to describe fair shapes in geometric modeling however they are generally represented in non polynomial form and are thus not compatible with current cad systems in this paper we present quintic polynomial approximation of lac segments for a given lac segment a quintic g 2 interpolating bezier curve is obtained by minimizing a curvature based error metric with the advantage of being more likely to preserve the monotone curvature property numerical experiments demonstrate that our method can usually generate better results than the previous methods in terms of the deviation in positions and curvatures

a preconditioned conjugate gradient method for nonselfadjoint or indefinite orthogonal spline collocation problems we study the computation of the orthogonal spline collocation solution of a linear dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form lu sum a ij x u x ix j sum b i x u x i c x u we apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a separable operator and prove that the resulting algorithm has a convergence rate independent of the partition step size we solve a problem with the preconditioner using an efficient direct matrix decomposition algorithm on a uniform n n partition the cost of the algorithm for computing the collocation solution within a tolerance epsilon is o n 2 ln n ln epsilon

efficient orthogonal spline collocation methods for solving linear second order hyperbolic problems on rectangles piecewise hermite bicubic orthogonal spline collocation laplace modified and alternating direction schemes for the approximate solution of linear second order hyperbolic problems on rectangles are analyzed the schemes are shown to be unconditionally stable and of optimal order accuracy in the h 1 and discrete maximum norms for space and time respectively implementations of the schemes are discussed and numerical results presented which demonstrate the accuracy and rate of convergence using various norms

an alternating direction implicit backward differentiation orthogonal spline collocation method for linear variable coefficient parabolic equations we present a new fully discrete method to solve a linear variable coefficient parabolic equation on a rectangular region the scheme is an alternating direction implicit method which uses the third order backward differentiation formula for the time discretization and piecewise hermite bicubic orthogonal spline collocation for the spatial discretization the l 2 norm stability and convergence analysis is carried out for the heat equation the stability analysis reveals that the scheme is stable with respect to the right hand sides but it is unstable with respect to the initial conditions we explain using a damping property of the scheme and round off error analysis why this instability is harmless we also show how a different choice of initial values leads to optimal convergence orders numerical results demonstrate the third order accuracy in time and optimal order accuracy in space using maximum and sobolev norms respectively

approximation of circular arcs by bezier curves abstract for the circular arc of angle 0 we present the explicit form of the best gc 3 quartic approximation and the best gc 2 quartic approximations of various types and give the explicit form of the hausdorff distance between the circular arc and the approximate bezier curves for each case we also show the existence of the gc 4 quintic approximations to the arc and find the explicit form of the best gc 3 quintic approximation in certain constraints and their distances from the arc all approximations we construct in this paper have the optimal order of approximation twice of the degree of approximate bezier curves

orthogonal spline collocation methods for biharmonic problems orthogonal spline collocation methods are formulated and analyzed for the solution of certain biharmonic problems in the unit square particular attention is given to the dirichlet biharmonic problem which is solved using capacitance matrix techniques

multilevel preconditioners for non self adjoint or indefinite orthogonal spline collocation problems efficient numerical algorithms are developed and analyzed that implement symmetric multilevel preconditioners for the solution of an orthogonal spline collocation osc discretization of a dirichlet boundary value problem with a non self adjoint or an indefinite operator the osc solution is sought in the hermite space of piecewise bicubic polynomials it is proved that the proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the normal osc equation the preconditioners are used with the preconditioned conjugate gradient method and numerical results are presented that demonstrate their efficiency

spectral analysis of hermite cubic spline collocation systems in this paper we present an eigenvalue analysis of the first order and second order hermite cubic spline collocation differentiation matrices with arbitrary collocation points some important features are explored and compared with some other discrete methods such as finite difference methods

a new method of dynamical stability i e fractional generalized hamiltonian method and its applications fractional generalized hamiltonian method of dynamical stability is presented six criterions for the new method of dynamical stability are given the new method s applications are studied five kinds of actual fractional dynamical models are constructed dynamical stability of the five actual fractional dynamical models is explored in the paper we present fractional generalized hamiltonian method of dynamical stability in terms of riesz riemann liouville derivative and study its applications for an actual dynamical system the fractional generalized hamiltonian method of constructing a fractional dynamical model is given and then the six criterions for fractional generalized hamiltonian method of dynamical stability are presented as applications by using the fractional generalized hamiltonian method we construct five kinds of actual fractional dynamical models which include a fractional euler poinsot model of rigid body that rotates with respect to a fixed point a fractional hojman urrutia model a fractional lorentz dirac model a fractional whittaker model and a fractional robbins lorenz model and we explore the dynamical stability of these models respectively this work provides a general method for studying the dynamical stability of an actual fractional dynamical system that is related to science and engineering