Abstract: Even though Gaussian elimination with partial pivoting is very widely used, $n \times n$ matrices can be constructed where the error growth in the algorithm is proportional to $2^{n-1}$. Thus for moderate or large $n$, in theory, there is a potential for disastrous error growth. However, prior to 1993 no reports of such an example in a practical application had appeared in the literature. Examples are presented that arise naturally from integral and differential equations and that lead to disastrous error growth in Gaussian elimination with partial pivoting.

Abstract: In this paper, a fast algorithm for solving the special tridiagonal system is presented. This special tridiagonal system is a symmetric diagonally dominant and Toeplitz system of linear equations. The error analysis is also given. Our algorithm is quite competitive with the Gaussian elimination, cyclic reduction, specialLU factorization, reversed triangular factorization, and Toeplitz factorization methods. In addition, our result can be applied to solve the near-Toeplitz tridiagonal system. Some examples demonstrate the good efficiency and stability of our algorithm.

Abstract: This paper presents an algorithm for the eigenvalue problem of symmetric tridiagonal matrices. The algorithm employs the determinant evaluation, split-and-merge strategy, and the Laguerre iteration. The method directly evaluates eigenvalues and uses inverse iteration as an option when eigenvectors are needed. This algorithm combines the advantages of existing algorithms such as QR, bisection/multisection, and Cuppen’s divide-and-conquer method. It is fully parallel and competitive in speed with the most efficient QR algorithm in serial mode. On the other hand, the algorithm is as accurate as any standard algorithm for the symmetric tridiagonal eigenproblem and enjoys the flexibility in evaluating partial spectrum.

Abstract: Tridiagonal systems play a fundamental role in matrix computation. In particular, in recent years parallel algorithms for the solution of tridiagonal systems have been developed. Among these, the cyclic reduction algorithm is particularly interesting. Here the stability of the cyclic reduction method is studied under the assumption of diagonal dominance. A backward error analysis is made, yielding a representation of the error matrix for the factorization and for the solution of the linear system. The results are compared with those for LU factorization.

Abstract: This paper presents a new finite-difference formulation of the multidimensional phase change problems involving unique phase change temperature. The solutions obtained with this formulation show that the problem of “waviness” of the temperature histories encountered with the conventional enthalpy formulation is now removed. The formulation derived provides a simple method for “local” tracking of the interface using the enthalpy variable in a novel way. During the solution of the finite-difference equations, the present formulation obviates the need for “book-keeping” of the phase-change nodes, and hence allows solution of the equations by tridiagonal matrix algorithm. It is argued that the benefits of enthalpy formulation can be extended to phase-change problems involving convection by solving the equations of motion on non-staggered grid.

Abstract: In this paper the Gaussian Elimination and Implicit Matrix Elimination methods are compared. Timings on a shared memory computer confirm the superiority of the new method for both the sequential and parallel implementations.

Abstract: An algorithm for the determination of the eigenvalues of tridiagonal symmetric interval matrices is presented. The intervals of the eigenvalues are not overestimated, but exactly calculated. The algorithm is an efficient one, because it only needs twice as many operations as the Sturm algorithm, which is used for real matrices. >

Abstract: This paper is concerned with the solution of block tridiagonal linear systems by the preconditioned conjugate gradient (PCG) method. If we consider a block AGE splitting of the coefficient matrix, it is possible to derive an additive polynomial preconditioner and to give conditions for such preconditioner to be symmetric positive definite. Numerical experiments on diffusion problem are carried out on Cray Y-MP in order to evaluate the effectiveness of the parallel polynomial preconditioner.

Abstract: An adaptive Richardson iteration method is described for the solution of large sparse symmetric positive definite linear systems of equations with multiple right-hand side vectors. This scheme ``learns'' about the linear system to be solved by computing inner products of residual matrices during the iterations. These inner products are interpreted as block modified moments. A block version of the modified Chebyshev algorithm is presented which yields a block tridiagonal matrix from the block modified moments and the recursion coefficients of the residual polynomials. The eigenvalues of this block tridiagonal matrix define an interval, which determines the choice of relaxation parameters for Richardson iteration. Only minor modifications are necessary in order to obtain a scheme for the solution of symmetric indefinite linear systems with multiple right-hand side vectors. We outline the changes required.

Abstract: In this paper, a fully parallel method for finding all eigenvalues of a real matrix pencil (A,B) is given, where A and B are real symmetric tridiagonal and B is positive definite. The method is based on the homotopy continuation coupled with the strategy ?Divide-Conquer? and Laguerre iterations. The numerical results obtained from implementation of this method on both single and multiprocessor computers are presented. It appears that our method is strongly competitive with other methods. The natural parallelism of our algorithm makes it an excellent candidate for a variety of advanced architectures.

Abstract: In this paper, we consider the problem of solving tridiagonal linear systems on distributed-memory multiprocessors. We present a new parallel partition-based tridiagonal solver that is suitable for such parallel computers. As compared with the well representative algorithm-cyclic reduction and its parallel variant-cyclic elimination, our algorithm exploits sufficient high of parallelism throughout without incurring more data traffic.

Abstract: A cyclic reduction method is described for the fast numerical solution of constant tridiagonal Toeplitz linear systems which occur repeatedly in the solution of the implicit finite difference equations derived from linear first order hyperbolic equations, i.e. the Transport equation, under a variety of boundary conditions. In this paper, we show that the linear systems can be solved efficiently by the Stride of 3 reduction algorithm.

Abstract: For the direct solution of tridiagonal linear systems Ax = d, the best known serial algorithm is based on LU-factorization of the coefficient matrix A. In the present paper we consider extending the idea to partitioned tridiagonal matrices. Let A be partitioned: A = (A (i, j)) so that the diagonal blocks A( i,i )are tridiagonal. We seek a factorization of A into L = (L (i j) and U = (U (i j) ), partitioned conformally. For the diagonal blocks of A we require the classical factorization: A( ii ) = L (i,i) U (i,i) , L (i,i) unit lower bidiagonal and U (i,i) upper bidiagonal. But, because of the presence of a non-zero element in each of the off-diagonal blocks of A, it is necessary to have Lupper block bidiagonal and U lower block bidiagonal, with only last row of L(i,i+1) and last column of U(i,i-1) filled. To avoid any interlocking/updating during/after the factorization stage, each of these last row and column in each block are required to have their last elements as zeros. On completion of the determinat...

Abstract: This paper considers elimination methods to solve dense linear systems, in particular a variant due to Huard of Gaussian elimination [13]. This variant reduces the system to an equivalent diagonal system just as GaussJordan elimination, but does not require more floating-point operations than Gaussian elimination. Huard's method may be advantageous for use in computers with hierarchical memory, such as cache, and in distributedmemory systems. An error analysis is given showing that Huard's elimination method is as stable as Gauss-Jordan elimination with appropriate pivoting strategy. This result was announced in [5] and is proven in a similar way as the proof of stability for Gauss-Jordan given in [4].

Abstract: A fast and efficient parallel algorithm for tridiagonal matrix inversion is presented. The algorithm is based on partitioning and reordering the initial tridiagonal matrix which allows us to parallelize it efficiently. The algorithm is implemented on a linear processor array. For interprocessor communication we consider three types of network components: dual-port RAM, FIFO RAM and a router. We derive the performances of the algorithm and the system which relates to the number of calculation steps, speedup and efficiency. The obtained results show that the method is highly valuable.

Abstract: In this paper, the parallelisation of the stride of three method for the solution of a tridiagonal system of equations for P processors is investigated. The presented algorithm is organised in such a way that all processors are fully operational at every stage of the solution process. The results of experiments carried out on the Sequent Balance 8000 multiprocessor are presented.

Abstract: In this paper, we propose an algorithm for finding eigenvalues of symmetric tridiagonal matrices based on Laguerre''s iteration The algorithm is fully parallelizable and has been parallelized on CM5 at University of California at Berkeley We''ve achieved best possible speedup when matrix dimension is large enough Besides, we have a well-written serial code which works very efficient in pathologically close eigenvalue cases

Abstract: Many scientific computer codes involve linear systems of equations which are coupled only between nearest neighbors in a single dimension. The most common situation can be formulated as a tridiagonal matrix relating source terms and unknowns. This system of equations is commonly solved using simple forward and back substitution. The usual algorithm is spectacularly ill suited for parallel processing with distributed data, since information must be sequentially communicated across all domains. Two new tridiagonal algorithms have been implemented in FORTRAN 77. The two algorithms differ only in the form of the unknown which is to be found. The first and simplest algorithm solves for a scalar quantity evaluated at each point along the single dimension being considered. The second algorithm solves for a vector quantity evaluated at each point. The solution method is related to other recently published approaches, such as that of Bondeli. An alternative parallel tridiagonal solver, used as part of an Alternating Direction Implicit (ADI) scheme, has recently been developed at LLNL by Lambert. For a discussion of useful parallel tridiagonal solvers, see the work of Mattor, et al. Previous work appears to be concerned only with scalar unknowns. This paper presents a new technique which treatsmore » both scalar and vector unknowns. There is no restriction upon the sizes of the subdomains. Even though the usual tridiagonal formulation may not be theoretically optimal when used iteratively, it is used in so many computer codes that it appears reasonable to write a direct substitute for it. The new tridiagonal code can be used on parallel machines with a minimum of disruption to pre-existing programming. As tested on various parallel computers, the parallel code shows efficiency greater than 50% (that is, more than half of the available computer operations are used to advance the calculation) when each processor is given at least 100 unknowns for which to solve.« less

Abstract: In this paper, a fully parallel method for finding all eigenvalues of a real matrix pencil (A, B) is given, where A and B are real symmetric tridiagonal and B is positive definite. The method is based on the homotopy continuation coupled with the strategy Divide-Conquer and Laguerre iterations. The numerical results obtained from implementation of this method on both single and multiprocessor computers are presented. It appears that our method is strongly competitive with other methods. The natural parallelism of our algorithm makes it an excellent candidate for a variety of advanced

Abstract: The statement of Proposition 2.2 (p.437) published by the author in Bull. Austral. Math. Soc. Vol. 46 (1992) pp.435–440 is incorrect. The correct version follows.

Abstract: The algoritm of Chawla et al [1] for tridiagonal linear systems is generalized to obtain a parallel elimination method for the solution of banded linear systems suitable for a multi-processor machine.

Abstract: In this paper, the normative matrices and their doubleLR transformation with origin shifts are defined, and the essential relationship between the doubleLR transformation of a normative matrix and theQR transformation of the related symmetric tridiagonal matrix is proved. We obtain a stable doubleLR algorithm for doubleLR transformation of normative matrices and give the error analysis of our algorithm. The operation number of the stable doubleLR algorithm for normative matrices is only four sevenths of the rationalQR algorithm for real symmetric tridiagonal matrices.

Abstract: The solution of linear, tridiagonal systems having real, symmetric, diagonally dominant coefficient matrices with constant diagonals is considered. Details of cyclic reduction to solve such systems are discussed. It is proved that the sequence of the diagonal elements produced by the reduction phase of cyclic reduction converges quadratically. This fact is exploited to reduce the number of steps of the reduction phase (special cyclic reduction). An estimate of the rate of convergence of the diagonal elements will be proved, which can be used to determine the number of steps of the reduction phase. Several possibilities to compute the diagonal elements are discussed and compared.

Abstract: This paper gives a simple algorithm for finding the explicit inverse of a general Jacobi tridiagonal matrix. A few illustrations are also provided to demonstrate the applicability of our algorithm.

Abstract: In this paper we present an algorithm, parallel in nature, for finding eigenvalues of a symmetric definite tridiagonal matrix pencil. Our algorithm employs the determinant evaluation, split-and-merge strategy and Laguerre's iteration. Numerical results on both single and multiprocessor computers are presented which show that our algorithm is reliable, efficient and accurate. It also enjoys flexibility in evaluating a partial spectrum.