A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. T...A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. This is the first time that the inverse of this remarkable matrix is determined directly and exactly. Thus, solving 1D Poisson equation becomes very accurate and extremely fast. This method is a very important tool for physics and engineering where the Poisson equation appears very often in the description of certain phenomena.展开更多
This paper establishes an improvement on the QL algorithm for a symmetric tridiagonal matrix T so that we can work out the eigenvalues of T faster. Meanwhile, the new algorithm don’t worsen the stability and precisio...This paper establishes an improvement on the QL algorithm for a symmetric tridiagonal matrix T so that we can work out the eigenvalues of T faster. Meanwhile, the new algorithm don’t worsen the stability and precision of the former algorithm.展开更多
Numeric algorithms for solving the linear systems of tridiagonal type have already existed. The well-known Thomas algorithm is an example of such algorithms. The current paper is mainly devoted to constructing symboli...Numeric algorithms for solving the linear systems of tridiagonal type have already existed. The well-known Thomas algorithm is an example of such algorithms. The current paper is mainly devoted to constructing symbolic algorithms for solving tridiagonal linear systems of equations via transformations. The new symbolic algorithms remove the cases where the numeric algorithms fail. The computational cost of these algorithms is given. MAPLE procedures based on these algorithms are presented. Some illustrative examples are given.展开更多
In the current article we propose a new efficient, reliable and breakdown-free algorithm for solving general opposite-bordered tridiagonal linear systems. An explicit formula for computing the determinant of an opposi...In the current article we propose a new efficient, reliable and breakdown-free algorithm for solving general opposite-bordered tridiagonal linear systems. An explicit formula for computing the determinant of an opposite-bordered tridiagonal matrix is investigated. Some illustrative examples are given.展开更多
In the current paper, the authors present a symbolic algorithm for solving doubly bordered k-tridiagonal linear system having n equations and n unknowns. The proposed algorithm is derived by using partition together w...In the current paper, the authors present a symbolic algorithm for solving doubly bordered k-tridiagonal linear system having n equations and n unknowns. The proposed algorithm is derived by using partition together with UL factorization. The cost of the algorithm is O(n). The algorithm is implemented using the computer algebra system, MAPLE. Some illustrative examples are given.展开更多
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(n2) flops, to find all eigenval...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(n2) 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.展开更多
文摘A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. This is the first time that the inverse of this remarkable matrix is determined directly and exactly. Thus, solving 1D Poisson equation becomes very accurate and extremely fast. This method is a very important tool for physics and engineering where the Poisson equation appears very often in the description of certain phenomena.
文摘This paper establishes an improvement on the QL algorithm for a symmetric tridiagonal matrix T so that we can work out the eigenvalues of T faster. Meanwhile, the new algorithm don’t worsen the stability and precision of the former algorithm.
文摘Numeric algorithms for solving the linear systems of tridiagonal type have already existed. The well-known Thomas algorithm is an example of such algorithms. The current paper is mainly devoted to constructing symbolic algorithms for solving tridiagonal linear systems of equations via transformations. The new symbolic algorithms remove the cases where the numeric algorithms fail. The computational cost of these algorithms is given. MAPLE procedures based on these algorithms are presented. Some illustrative examples are given.
文摘In the current article we propose a new efficient, reliable and breakdown-free algorithm for solving general opposite-bordered tridiagonal linear systems. An explicit formula for computing the determinant of an opposite-bordered tridiagonal matrix is investigated. Some illustrative examples are given.
文摘In the current paper, the authors present a symbolic algorithm for solving doubly bordered k-tridiagonal linear system having n equations and n unknowns. The proposed algorithm is derived by using partition together with UL factorization. The cost of the algorithm is O(n). The algorithm is implemented using the computer algebra system, MAPLE. Some illustrative examples are given.
文摘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(n2) 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.
