Some Problems Of Matrix Computing And Applications In Image Recognition

This thesis investigates two problems: the inverses of tridiagonal,block tridiagonal and banded matrices, which are a class of important sparse matrices; the investigation of image recognition based on matrix computing.Tridiagonal and banded matrices are a class of important sparse matrices. They often arise in numerical analysis,image processing and signal processing. How to invert the matrices is an important topic. The inverses are necessary in many problems, such as the computation of the condition number of this matrix, the computation of the "partial inverse elements" in some engineering problems and the solution of linear system whose coefficient matrix is tridiagonal or block tridiagonal.Image processing and recognition is an applied discipline, which is related to many disciplines and basic knowledge. As an important tool, the theory and method of matrix computation has some important applications in image processing and recognition.The thesis proceeds along with the above two parts, the main results are as follows:1. The inverses of tridiagonal matrices are investigated. The investigation is based on two methods: the LU factorization of tridiagonal matrix and expression the inverse matrix with four column vectors. And then two simple algorithms are derived. The former is applicable to a general tridiagonal matrix without any additional conditions. The theory analysis indicates the computing complexity of the proposed algorithms is remarkably lower than the classical algorithms; and experiments indicate the computing time of the proposed algorithms is about 0.75～0.85 time of that of the algorithm proposed recently by Nabben, 0.40～0.60 time of that of "Catch up method".2. The inverses of block tridiagonal matrices are investigated. The investigation is based on three methods: the twisted factorization,LU factorization of tridiagonal matrix and expression the block inverse matrix with four block column vectors. And then three simple algorithms are derived. The theory analysis indicates the computing complexity of the proposed algorithms is remarkably lower than existed algorithms; and...