Showing papers on "Tridiagonal matrix algorithm published in 2004"
TL;DR: In this paper, the authors have used a range of reaction orders with respect to both gaseous and solid reactants to analyze gas-solid non-catalytic (GSNC) reactions in porous particles.
Abstract: Gas–solid noncatalytic (GSNC) reactions in porous particles are analyzed employing volume reaction model (VRM). Both single-stage and two-stage models have been used in the work. Earlier analysis for such reactions was mostly restricted to the first-order reactions with respect to the gaseous component. The present work has used a range of reaction orders with respect to both gaseous and solid reactants. Front tracking method was employed for solving the moving boundary problem. Finite volume method (FVM) has been used for the first time for the analysis of GSNC reactions in porous particles, based on moving boundary zone models. The discretized equations are solved by tridiagonal matrix algorithm. The results for the first-order reaction agree well with the analytical solution. FVM solution compares well with other numerical method (integral transformation followed by orthogonal collocation). Numerical results have also been validated with reported experimental data for reduction of Magnetite by CO. FVM is thus established to be a suitable numerical method to solve GSNC reactions in porous particles employing VRM. The effects of different process parameters on the progress of reaction have also been assessed.© 2003 Wiley Periodicals, Inc. Int J Chem Kinet 36: 1–11, 2003
49 citations
TL;DR: In this article, it was shown that for quadratic and tridiagonal matrices, the numerical range is an ellipse, and the proof gives an explanation for this phenomenon and, in certain cases, provides an explicit description of the range.
Abstract: It is known that for several classes of matrices, including quadratic and certain tridiagonal matrices, the numerical range is an ellipse. We prove this result for a larger class of matrices, encompassing these published results as well as providing other sufficient conditions for ellipticity. The proof gives an explanation for this phenomenon and, in certain cases, provides an explicit description of the numerical range.
39 citations
TL;DR: In this paper, the facial structure of the polytope Ω t n in R n×n consisting of the tridiagonal doubly stochastic matrices of order n was studied.
35 citations
TL;DR: An exponentially fitted finite difference method for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point approximates the exact solution very well.
26 citations
TL;DR: A block tridiagonalization algorithm is proposed for transforming a sparse (or "effectively" sparse) symmetric matrix into a related block Tridiagonal matrix, such that the eigenvalue error remains bounded by some prescribed accuracy tolerance.
Abstract: A block tridiagonalization algorithm is proposed for transforming a sparse (or "effectively" sparse) symmetric matrix into a related block tridiagonal matrix, such that the eigenvalue error remains bounded by some prescribed accuracy tolerance. It is based on a heuristic for imposing a block tridiagonal structure on matrices with a large percentage of zero or "effectively zero" (with respect to the given accuracy tolerance) elements. In the light of a recently developed block tridiagonal divide-and-conquer eigensolver [Gansterer, Ward, Muller, and Goddard, III, SIAM J. Sci. Comput. 25 (2003), pp. 65--85], for which block tridiagonalization may be needed as a preprocessing step, the algorithm also provides an option for attempting to produce at least a few very small diagonal blocks in the block tridiagonal matrix. This leads to low time complexity of the last merging operation in the block divide-and-conquer method. Numerical experiments are presented and various block tridiagonalization strategies are compared.
21 citations
TL;DR: It is shown that every n by n symmetry matrix over F is orthogonally similar to a tridiagonal symmetric matrix, if further the characteristic is 0.
15 citations
TL;DR: Lower bounds are presented that hold when the number of data items per processor is bounded, are general lower bounds, and for specific data layouts commonly used in designing parallel algorithms to solve tridiagonal linear systems.
Abstract: The problem of solving tridiagonal linear systems on parallel distributed-memory environments is considered in this paper. In particular, two common direct methods for solving such systems are considered: odd-even cyclic reduction and prefix summing. For each method, a variety of lower bounds on execution time for solving tridiagonal linear systems are presented. Specifically, lower bounds are presented that (a) hold when the number of data items per processor is bounded, (b) are general lower bounds, and (c) for specific data layouts commonly used in designing parallel algorithms to solve tridiagonal linear systems. Furthermore, algorithms are presented that have running times within a constant factor of the lower bounds provided. Lastly, a comparison of bounds for odd-even cyclic reduction and prefix summing is given.
6 citations
TL;DR: Two exact algorithms are proposed to solve the steady state probability distributions of irreducible Markov chains whose generator matrices have tridiagonal structure based on divide-and-conquer procedure and a parallel algorithm.
5 citations
TL;DR: The performances show that this algorithm is a good compromise between a direct method and other iterative methods as block SOR, and suggests its use as inner solver in the solution of problems derived by application of a decomposition domain method.
5 citations
Journal Article•
TL;DR: In this paper, the upper bound for inverse elements of strictly diagonally dominant periodic tridiagonal matrices is given. But the upper bounds are not applicable to the case of strictly dominant periodic matrices.
Abstract: In this paper, we give the upper bounds for inverse elements of strictly diagonally dominant periodic tridiagonal matrices.
4 citations
26 Apr 2004
TL;DR: This work presents two parallel QR factorization algorithms used to solve Toeplitz tridiagonal systems that exhibit high scalability and near linear to superlinear speedup on large system sizes when implemented on a distributed system.
Abstract: Summary form only given. QR methods for solving Toeplitz tridiagonal systems are well developed with applications in numerous interdisciplinary fields. There is a strong motivation to develop faster, more efficient and, more importantly, scalable algorithms to factor such systems due to their significance in many scientific applications. We present two parallel QR factorization algorithms used to solve Toeplitz tridiagonal systems. QR factorization is accomplished using Householder reflections and Givens rotations. These parallel algorithms exhibit high scalability and near linear to superlinear speedup on large system sizes when implemented on a distributed system.
TL;DR: In this paper, a cross product function of the elements in the tridiagonal matrix is obtained by using the right-angled triangle property among the coefficients. But the lower triangle of the inverse of a general tridimensional matrix can be computed in a non-block case.
Abstract: In this study a further relationship is extended to the elements in the lower triangle of the inverse of a general tridiagonal matrix for a non‐block case. Once the upper triangle of the inverse is determined based on Huang and McColl's analytical inversion formula, the corresponding lower triangle can be calculated efficiently using two proposed theorems. Each element in the lower triangle is decomposed into two parts: one is the coefficient; the other the counterpart element in the upper triangle. The coefficient is a cross product function of the elements in the tridiagonal matrix and can be easily obtained by using the right‐angled triangle property among the coefficients. This results in a faster computation of the lower triangle of the inverse of a general tridiagonal matrix. Several examples are given to demonstrate the superiority of two theorems developed by the author to Huang and McColl's algorithm. It is shown that the algorithm based on the right‐angled triangle property outperforms ...
TL;DR: These algebras are naturally implemented upon various elliptic curves and are designed to describe (after quantization) generalized eigenvalue problems for two tridiagonal matrices.
Abstract: We introduce particular Poisson algebras designed to describe (after quantization) generalized eigenvalue problems for two tridiagonal matrices. These algebras are naturally implemented upon various elliptic curves.
Journal Article•
TL;DR: This article gave explicit formulas for caluctating the elements of the inverse of a general nonsingular tridiagonal matrix, which improved some well-known results concerning the inverse for general tridimensional matrices.
Abstract: We give explicit formulas for caluctating the elements of the inverse of a general nonsingular tridiagonal matrix which improve some well-known results concerning the inverse of a nonsingular tridiagonal matrix, and the inverse of a nonsingular general tridiagonal matrix can be easily computed.
TL;DR: Five problems are put forward: Convergence and convergence rate; The convergence of diagonal elements; Shift designed to produce the eigenvalues in monotone order; QL algorithm with multi-shift; and Error bound.
Abstract: QL(QR) method is an efficient method to find eigenvalues of a matrix. Especially we use QL(QR) method to find eigenvalues of a symmetric tridiagonal matrix. In this case it only costs O(n 2) flops, to find all eigenvalues. So it is one of the most efficient method for symmetric tridiagonal matrices. Many experts have researched it. Even the method is mature, it still has many problems need to be researched. We put forward five problems here. They are: (1) Convergence and convergence rate; (2) The convergence of diagonal elements; (3) Shift designed to produce the eigenvalues in monotone order; (4) QL algorithm with multi-shift; (5) Error bound. We intoduce our works on these problems, some of them were published and some are new.
TL;DR: It is shown that an upper triangular matrix equivalent to a non-singular tridiagonal matrix is easily found.
Journal Article•
TL;DR: In this article, two kind of inverse eigenvalue problems are proposed which are the reconstruction of Persymmetric Tridiagonal matrix from their mixed-eigenpairs, and necessary and sufficient conditions for the existence and uniqueness of solutions are given.
Abstract: In this paper ,two kind of inverse eigenvalue problems are proposed which are the reconstruction of Persymmetric Tridiagonal matrix from theirs mixed-eigenpairs.The necessary and sufficient conditions for the existence and uniqueness of solutions are given.
TL;DR: Using orthogonal polynomials, this work gives a different approach to some recent results on tridiagonal matrices.
Abstract: Using orthogonal polynomials, we give a different approach to some
recent results on tridiagonal matrices.
Posted Content•
27 Dec 2004
TL;DR: In this paper, the authors introduced bidiagonal coordinates, a new set of spectral coordinates on open dense charts covering the space of real symmetric tridiagonal matrices, which are convenient for the study of isospectral dynamics, both for continuous and discrete time.
Abstract: We introduce bidiagonal coordinates, a new set of spectral coordinates on open dense charts covering the space of real symmetric tridiagonal matrices. In contrast to the standard Jacobi inverse variables, reduced tridiagonal matrices now lie in the interior of some chart. Bidiagonal coordinates are thus convenient for the study of asymptotics of isospectral dynamics, both for continuous and discrete time. In particular, we study the rate of convergence of Wilkinson’s shift iteration. For AP-free spectra (i.e., simple spectra containing no arithmetic progression with 3 terms), convergence is cubic. In order 3, for AP-spectra, however, there exists a matrix P0 such that if Wilkinson’s iteration converges to P0 then convergence is strictly quadratic. Near p0 ∈ R , the bidiagonal coordinates of P0, the set X of initial conditions with convergence to p0 is a union of disjoint arcs Xs meeting at p0, where s ranges over the Cantor set of sign sequences s : N → {1,−1}. Wilkinson’s step takes Xs to X s ′ , where s′ is the left shift of s. The set X is rather thin and for initial conditions near p0 but not in X , cubic convergence still applies.
TL;DR: In this article, it was shown that the Hadamard product of oscillatory tridiagonal matrices of the same order is a basic oscillatory matrix as a factor, which explains the role of the class of basic oscillator matrices within a class of nonnegative and oscillatory matrices.
15 Aug 2004
TL;DR: The improved version of P-scheme works well for smaller problems on distributed environment like PC cluster systems and linear and super-linear speedups can be achieved for 8194 x 8194 and 16386 x 16386 problems, respectively.
Abstract: We propose ?P-scheme? for solving recurrence equations for a tridiagonal linear system of equations on distributed-memory parallel computers, but its effectiveness is limited to the case where the problem is enough large. The limitation is mainly due to the communication cost of propagation phase of P-scheme. In order to overcome the difficulty, we use ?message vectorization?, which aggregates several communication messages into one, to alleviates the communication cost of P-scheme and evaluate the effectiveness of message vectorization for tridiagonal matrix solver. Our experiments prove that the improved version of P-scheme works well for smaller problems on distributed environment like PC cluster systems and show linear and super-linear speedups can be achieved for 8194 x 8194 and 16386 x 16386 problems, respectively.
TL;DR: A new efficient computational algorithm to find the inverse of a general tridiagonal matrix is presented and is suited for implementation using computer algebra systems such as MAPLE, MATHEMATICA, MATLAB and MACSYMA.
Posted Content•
TL;DR: In this paper, the authors introduced a mild generalization of a Leonard pair called a tridiagonal pair, which is the same thing as a tridimensional pair such that for each transformation all eigenspaces have dimension one.
Abstract: Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. Consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy both conditions below:
(i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal, and the matrix representing $A^*$ is irreducible tridiagonal.
(ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal, and the matrix representing $A$ is irreducible tridiagonal.
Such a pair is called a Leonard pair on $V$. In this paper we introduce a mild generalization of a Leonard pair called a tridiagonal pair. A Leonard pair is the same thing as a tridiagonal pair such that for each transformation all eigenspaces have dimension one.