An inversion algorithm for general tridiagonal matrix

An algorithm for the inverse of a general tridiagonal matrix is presented. For a tridiagonal matrix having the Doolittle factorization, an inversion algorithm is established. The algorithm is then generalized to deal with a general tridiagonal matrix without any restriction. Comparison with other methods is provided, indicating low computational complexity of the proposed algorithm, and its applicability to general tridiagonal matrices.

QL Method for Symmetric Tridiagonal Matrices

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.

Efficient BTCS + CTCS Finite Difference Scheme for General Linear Second Order PDE

This work deals with a second order linear general equation with partial derivatives for a two-variable function. It covers a wide range of applications. This equation is solved with a finite difference hybrid method: BTCS + CTCS. This scheme is simple, precise, and economical in terms of time and space occupancy in memory.

An Incomplete Splitting-up Conjugate Gradient Method for Parallel Computing

This paper describes several variants of SPCG （splitting up conjugate gradient） method suitable for parallel computing and evaluates the performance and the speed of convergence on a distributed-memory multicomputer. SP （splitting-up） preconditioner can be easily parallelized because other dimensions except for one dimension are independent. Among the variants, one of incomplete SPCG method, which does not carry out one of three Widiagonal matrix solvers, achieves the best performance, and this method is about 20 times faster than one-process version of the SPCG method on 32 CPU cores of the multicomputer.

Generate mesh with shape parameters

In CAD/CAM, mesh rather than smooth surface is only needed sometimes. A mesh-generating method from permanence principle of Coons patch is developed. A new mesh point is defined through local small subpatch and all mesh points are computed by a linear system with special symmetric block tridiagonal coefficient matrix. By simplification, the determinant of coefficient matrix is determined by determinants of submatrices. Condition of existence of solution is given. Whether coefficient matrix is singular can be judged by a simple polynomial function with the eigenvalue of submatrix as variable. Numerical examples demonstrate the effects of shape parameters.

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk+1,nA new inverse eigenvalues problem has put forward as follows: How do we construct anunreduced symmetric tridiagonal matrix T1,n, if we only know the spectral data: theeigenvalues of T1,n, the eigenvalues of Ti,k-1 and the eigenvalues of Tk+1,n?Namely if we only know the data: A1, A2, An,how do we find the matrix T1,n? A necessary and sufficient condition and an algorithm ofsolving such problem, are given in this paper.

Criteria for Discrete Spectrum of 1D Operators

For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators.Interestingly,these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier,simply replacing the‘finite supremum’by‘vanishing at infinity’.Except a dual technique,the main tool used here is a transform in terms of the harmonic function,to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context.All of them are illustrated by examples.The main body of the paper is devoted to the hard part of the story,the easier part but powerful one is delayed to the end of the paper.

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

In this paper, we discuss an inverse eigenvalue problem for constructing a 2n × 2n Jacobi matrix T such that its 2n eigenvalues are given distinct real values and its leading principal submatrix of order n is a given Jacobi matrix. A new sufficient and necessary condition for the solvability of the above problem is given in this paper. Furthermore, we present a new algorithm and give some numerical results.

Efficient initials for computing maximal eigenpair

This paper introduces some efficient initials for a well-known algorithm （an inverse iteration） for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also unexpectedly efficient. The initials presented here are based on our analytic estimates of the maximal eigenvalue and a mimic of its eigenvector for many years of accumulation in the study of stochastic stability speed. In parallel, the same problem for computing the next to the maximal eigenpair is also studied.