the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

essentially nonoscillatory spectral fourier methods for shock wave calculations in this paper we present an essentially nonoscillatory spectral fourier method for the solution of hyperbolic partial differential equations the method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the fourier method the high accuracy away from the shock is enhanced by using filters numerical results confirm that essentially no oscillations develop in the solution also the accuracy of the spectral solution of the inviscid burgers equation is shown to be higher than a fixed order

on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function it is well known that the fourier series of an analytic and periodic function truncated after 2n 1 terms converges exponentially with n even in the maximum norm it is also known that if the function is not periodic the rate of convergence deteriorates in particular there is no convergence in the maximum norm although the function is still analytic this is known as the gibbs phenomenon in this paper we show that the first 2n 1 fourier coefficients contain enough information about the function so that an exponentially convergent approximation in the maximum norm can be constructed the proof is a constructive one and makes use of the gegenbauer polynomials c n x it consists of two steps in the first step we show that the first m coefficients of the gegenbauer expansion based on c n x for 0 n m of any l2 function can be obtained within exponential accuracy provided that both and m are proportional to but smaller than n in the second step we construct the gegenbauer expansion based on c n 0 n m from the coefficients found in the first step we show that this series converges exponentially with n provided that the original function is analytic though nonperiodic thus we prove that the gibbs phenomenon can be completely overcome

jacobi approximations in certain hilbert spaces and their applications to singular differential equations jacobi approximations in certain hilbert spaces are investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using jacobi polynomials this method keeps the spectral accuracy some linear problems and a nonlinear logistic equation are considered the stabilities and the convergences of proposed schemes are proved strictly the main idea and techniques used in this paper are also applicable to other singular problems in multiple dimensional spaces

square root rts smoothing algorithms two kinds of new square root algorithms for rauch tung striebel rts smoothing formulas are suggested stable square root algorithms and a fast square root or so called chandrasekhar type algori

gegenbauer approximation in certain hilbert spaces and its applications to singular differential equations gegenbauer approximation in certain hilbert spaces is investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using gegenbauer polynomials this method keeps the spectral accuracy the main idea and techniques used in this paper are also applicable to some singular problems in multiple dimensional spaces

new square root smoothing algorithms this paper presents new square root smoothing algorithms for the three best known smoothing formulas 1 rauch tung striebel rts formulas 2 desai weinert yusypchuk dwy formulas called backward rts formulas and 3 mayne fraser mf formulas called two filter formulas the main feature of the new algorithms is that they use unitary rotations to replace all matrix inversion and backsubstitution steps common in earlier algorithms with unitary operations this feature enables more efficient systolic array and parallel implementations and leads to algorithms with better numerical stability and conditioning properties

paper a new computationally efficient fixed interval discrete time smoother the rauch tung streibel smoother recursion is used to derive a new smoother algorithm based upon a decomposition of the linear model dynamical equation and maximizing use of rank 1 matrix modification this new algorithm it turns out parallels bierman s forward recursive square root information filter backward recursive u d factorized covariance algorithm the new result features computational efficiency reliance on numerically stable matrix modification algorithms and reduced computer storage

on the relationship between the lagrange multiplier method and the two filter smoother some relationships between the lagrange multiplier method and a two filter smoother are explored using orthogonal projection lemmas for linear discrele time systems the obtained expectation results are applied to algebraically derive a forward pass fixed interval smoothing algorithm that utilizes backward pass information filter outputs furthermore a convenient form for numerically efficient and stable algorithms is developed and the derived equation can also be used to quantitatively evaluate the effect of initial statistics on the smoothed estimate numerical results pertaining to a linearized version for the in track motion of a satellite travelling in a circular orbit are included to illustrate the use of the given algorithms

the strong direct product of formal contexts the theory of formal concept analysis is used as a knowledge representation mechanism and as a conceptual clustering method formal context is a basic notion in this theory this paper proposes a kind of formal context called n strong direct product formal context n sdpfc for short by the strong direct product of n formal contexts it is found that the new context is closely related to the n original formal contexts in many aspects for example the concept lattices and the implications between attributes and so on with the help of kronecker product of boolean matrices the main features of an n sdpfc is given and the method how to represent a formal context with such features by the strong direct product of some small formal contexts is brought forward with this method the workload for knowledge acquisition in formal context will be eased enormously

on exponential convergence of gegenbauer interpolation and spectral differentiation this paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the gegenbauer gauss and gegenbauer gauss lobatto points when the underlying function is analytic on and within an ellipse sharp error estimates in the maximum norm are derived

on the gibbs phenomenon v recovering exponential accuracy from collocation point values of a piecewise analytic function this paper presents a method to recover exponential accuracy at all points including at the discontinuities themselves from the knowledge of an approximation to the interpolation polynomial or trigonometrical polynomial we show that if we are given the collocation point values or a highly accurate approximation at the gauss or gauss lobatto points we can reconstruct an uniform exponentially convergent approximation to the function f x in any sub interval of analyticity the proof covers the cases of fourier chebyshev legendre and more general gegenbauer collocation methods a numerical example is also provided

a generalization of the wiener rational basis functions on infinite intervals part i derivation and properties we formulate and derive a generalization of an orthogonal rational function basis for spectral expansions over the infinite or semi infinite interval the original functions first presented by wiener are a mapping and weighting of the fourier basis to the infinite interval by identifying the fourier series as a biorthogonal composition of jacobi polynomials functions we are able to define generalized fourier series which when appropriately mapped to the whole real line and weighted generalize wiener s basis functions it is known that the original wiener rational functions inherit sparse galerkin matrices for differentiation and can utilize the fast fourier transform fft for computation of the expansion coefficients we show that the generalized basis sets also have a sparse differentiation matrix and we discuss connection problems which are necessary theoretical developments for application of the fft

a rational approximation and its applications to nonlinear partial differential equations on the whole line an orthogonal system of rational functions is discussed some inverse inequalities imbedding inequalities and approximation results are obtained two model problems are considered the stabilities and convergences of proposed rational spectral schemes and rational pseudospectral schemes are proved the techniques used in this paper are also applicable to other problems on the whole line numerical results show the efficiency of this approach

brief square root information filtering and fixed interval smoothing with singularities the square root information filter algorithm and the related covariance smoother algorithm have been generalized to handle singular state transition matrices and perfect measurements this allows the use of srif techniques for problems with delays and state constraints the generalized algorithms use complete qr factorization to isolate deterministically known parts of the state and nonsingular parts of the state transition and disturbance influence matrices these factorizations and the corresponding changes of coordinates are used to solve the recursive least squares problems that are basic to the srif technique numerical stability computation time and storage requirements are comparable to the traditional srif algorithms

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

concept analysis via rough set and afs algebra formal concept analysis fca was originally proposed by wille 1982 which is an important theory for data analysis and knowledge discovery afs axiomatic fuzzy set algebra was proposed by liu x liu the fuzzy theory based on afs algebras and afs structure journal of mathematical analysis and applications 217 1998 459 478 x liu the topology on afs algebra and afs structure journal of mathematical analysis and applications 217 1998 479 489 which is a semantic methodology relating to the fuzzy theory combining above two theories we propose afs formal concept which can be viewed as the generalization and development of monotone concept proposed by deogun and saquer 2003 moreover we show that the set of all afs formal concepts forms a complete lattice afs formal concept can be applied to represent the logic operations of queries in information retrieval furthermore we give an approach to find the afs formal concepts whose intents extents approximate any element of afs algebra by virtue of rough set theory

square root bryson frazier smoothing algorithms some new square root algorithms for bryson frazier smoothing formulas are suggested square root algorithms and a fast square root or so called chandrasekhar type algorithm the new square root algorithms use square root arrays composed of smoothed estimates and their error covariances these algorithms provide many advantages over the conventional algorithms with respect to systolic array and parallel implementations as well as numerical stability and conditioning for the case of constant parameter systems a fast square root algorithm is suggested which requires less computation than others

covariance factorization algorithms for fixed interval smoothing of linear discrete dynamic systems efficient factorized covariance smoothers designed to work with factorized covariance filters are derived for linear discrete dynamic systems the approach to factorized covariance smoothers either u d or square root uses outputs from factorized covariance filters and is closely derived from the g j bierman s earlier algorithm 1974 the dyer mcreynolds covariance smoother these algorithms are more efficient than the bierman s newer smoother 1983 based upon rank 1 process noise updates the efficiency of the new algorithms increases significantly as the order of process noise increases for full process noise they can be implemented in a way that avoids the inverse of the transition matrix

jacobi rational approximation and spectral method for differential equations of degenerate type we introduce an orthogonal system on the half line induced by jacobi polynomials some results on the jacobi rational approximation are established which play important roles in designing and analyzing the jacobi rational spectral method for various differential equations with the coefficients degenerating at certain points and growing up at infinity the jacobi rational spectral method is proposed for a model problem appearing frequently in finance its convergence is proved numerical results demonstrate the efficiency of this new approach

a new forward pass fixed interval smoother using the u d information matrix factorization abstract a new u d factorized smoothing algorithm that is numerically stable and reliable is developed by using a forward pass fixed interval smoother recursion introducing a gain for a backward pass information filter and the notion of an input to the smoother and decomposing the system noises into each element it is shown that three famous u d algorithms can be naturally applied to construct such a smoother the new result does not necessitate two burdensome matrix inversions or any algebraic computations equivalent to them for the transition matrix and the predicted error covariance which are at present necessary for the bierman s u d smoother consequently compared with the result of bierman the present algorithm can deal with a broader system which may cover a time delay system and provide an improvement in computation speed and computer storage

chebyshev rational spectral and pseudospectral methods on a semi infinite interval a weighted orthogonal system on the half line based on the chebyshev rational functions is introduced basic results on chebyshev rational approximations of several orthogonal projections and interpolations are established to illustrate the potential of the chebyshev rational spectral method a model problem is considered both theoretically and numerically error estimates for the chebyshev rational spectral and pseudospectral methods are established preliminary numerical results agree well with the theoretical estimates and demonstrate the effectiveness of this approach

an efficient jacobi pseudospectral approximation for nonlinear complex generalized zakharov system in this paper we derive an efficient spectral collocation algorithm to solve numerically the nonlinear complex generalized zakharov system gzs subject to initial boundary conditions the jacobi pseudospectral approximation is investigated for spatial approximation of the gzs it possesses the spectral accuracy in space the jacobi gauss lobatto quadrature rule is established to treat the boundary conditions and then the problem with its boundary conditions is reduced to a system of ordinary differential equations in time variable this scheme has the advantage of allowing us to obtain the spectral solution in terms of the jacobi parameters and which therefore means that the algorithm holds a number of collocation methods as special cases finally two illustrative examples are implemented to assess the efficiency and high accuracy of the jacobi pseudo spectral scheme