Tridiagonal matrix algorithm

Abstract: The problem of generating a matrix A with specified eigenpairs, where A is a tridiagonal symmetric matrix, is presented. A general expression of such a matrix is provided, and the set of such matrices is denoted by S"E. Moreover, the corresponding least-squares problem under spectral constraint is considered when the set S"E is empty, and the corresponding solution set is denoted by S"L. The best approximation problem associated with S"E(S"L) is discussed, that is: to find the nearest matrix A@^ in S"E(S"L) to a given matrix. The existence and uniqueness of the best approximation are proved and the expression of this nearest matrix is provided. At the same time, we also discuss similar problems when A is a tridiagonal bisymmetric matrix.

Abstract: A real time empirical mode decomposition (EMD) algorithm based ultrasonic imaging system has been developed for non-destructive testing (NDT) applications. It is difficult to implement the EMD based signal processing algorithm in real time because it is totally a data-driven process which comprises numerous sifting operations. In this paper, the EMD algorithm has been implemented in the visual software environment. The EMD implementation encompasses two types of interpolation methods: piecewise linear interpolation (PLI) and cubic spline interpolation (CSI). The cubic spline tridiagonal matrix has been solved by using the Thomas algorithm for real time processing. The total time complexity functions for both the implemented PLI and CSI based EMD methods have been computed. For the signal filtering, the partial reconstruction algorithm has been adopted. The baseline correction and noise filtering applications have been presented using an EMD based visual software. The real time practicability and the efficiency of this method have been validated through ultrasonic NDT experimentation for improvement in the time domain resolution of the ultrasonic A-scan raw data. The practical results show that in the noisy environment, it is possible to enhance the signal-to-noise ratio for the visualization and identification of ultrasonic pulse-echo signals in real time.

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: We extend the elimination method of Chawla and Khazal [1] to uncouple partitioned blocks of "periodic" tridiagonal linear systems. At each step of the elimination stage, we now need three simultaneous eliminations: within each block, one usual forward elimination and one backward elimination from across the succeeding block, and one elimination in the last row of the last block. An interesting feature of the present elimination procedure is that at the end of it, the property of periodicity of the original system is now passed on to the core system . Once the core system is solved, the blocks uncouple and the solution is obtained in parallel from each block by back substitution. For a system of size N , the classical elimination has an arithmetical operations count of 17N . A best serial algorithm, based on the Sherman-Morrison formula, has an operations count of 0 14N . The present parallel elimination algorithm, employing p partition blocks, has an operations count of O 17N p and, in comparison with the...