The Eigenspace Spectral Regularization Method for solving Discrete Ill-Posed Systems

In this paper, it is shown that discrete equations with Hilb ert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, and sparse matrix operator are ill-posed in the sense of Hadamard. These ill-posed problems cannot be regularized by Gauss Least Square Method (GLSM), QR Factorization Method (QRFM), Cholesky Decomposition Method (CDM) and Singular Value Decomposition (SVDM). To overcome the limitations of these methods of regularization, an Eigenspace Spectral Regularization Method (ESRM) is introduced which solves ill-p os ed discrete equations with Hilb ert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, and sparse matrix operator. Unlike GLSM, QRFM, CDM, and SVDM, the ESRM regularize such a system. In addition, the ESRM has a unique property, the norm of the eigenspace spectral matrix operator κ (K) = ||K − 1K|| = 1. Thus, the condition number of ESRM is bounded by unity unlike the other regularization methods such as SVDM, GLSM, CDM, and QRFM.

Study on Banded Implicit Runge–Kutta Methods for Solving Stiff Differential Equations

The implicit Runge–Kutta method with A-stability is suitable for solving stiff differential equations. However, the fully implicit Runge–Kutta method is very expensive in solving large system problems. Although some implicit Runge–Kutta methods can reduce the cost of computation, their accuracy and stability are also adversely affected. Therefore, an effective banded implicit Runge–Kutta method with high accuracy and high stability is proposed, which reduces the computation cost by changing the Jacobian matrix from a full coefficient matrix to a banded matrix. Numerical solutions and results of stiff equations obtained by the methods involved are compared, and the results show that the banded implicit Runge–Kutta method is advantageous to solve large stiff problems and conducive to the development of simulation.

Matrices whose inverses are tridiagonal, band or block-tridiagonal and their relationship with the covariance matrices of a random Markov process

The article discusses the matrices of the form A1n, Amn, AmN, whose inverses are: tridiagonal matrix A-1n (n - dimension of the A-mn matrix), banded matrix A-mn (m is the half-width band of the matrix) or block-tridiagonal matrix A-m N (N = n x m - full dimension of the block matrix; m - the dimension of the blocks) and their relationships with the covariance matrices of measurements with ordinary (simple) Markov Random Processes (MRP), multiconnected MRP and vector MRP, respectively. Such covariance matrices frequently occur in the problems of optimal filtering, extrapolation and interpolation of MRP and Markov Random Fields (MRF). It is shown, that the structures of the matrices A1n, Amn, AmN have the same form, but the matrix elements in the first case are scalar quantities; in the second case matrix elements represent a product of vectors of dimension m; and in the third case, the off-diagonal elements are the product of matrices and vectors of dimension m. The properties of such matrices were investigated and a simple formulas of their inverses were found. Also computational efficiency in the storage and the inverse of such matrices have been considered. To illustrate the acquired results, an example on the covariance matrix inversions of two-dimensional MRP is given.