Applications In Optimization And Highly Oscillatory Integrals For Special Matrices

Special matrices is one of the important research topics in matrix analysis and numerical algebra, the research results have a wide range of applications in many fields such as optimization theory, computational mathematics, cybernetics, management science and engineering and so on. But in practice, it is difficult to determine a matrix, especially for a large scale matrix, is a special matrix or not, so to study the criteria for special matrices has important theoretical and practical significance. The linear complementarity problems are of an important branch in the optimization field, which are widely applied in many areas such as economics, game theory, engineering and traffic control. Numerical computation of highly oscillatory integrals encounters in a wide range of applications ranging from quantum chemistry, medical imaging, signal processing, fluid mechanics, etc, and are widely recognized as difficult problems.In this paper, we are mainly concerned with the criteria for H-matri-ces and generalized block diagonally dominant matrices under matrix norm, P-matrix and H-matrix linear cmplementarity problems, numerical calculation of highly oscillatory Fourier-type integrals. Our main results are as follows.In Chapter 1, we show the background and significance and the main work of this paper.In Chapter 2, we devote our attention to the criteria for H-matrices. By using the equivalence of generalized strictly?-diagonally dominant matrices and H-matrices, we define a new subclass of H-matrices and present some criteria for H-matrices. By applying Schur complements, we reduce the order of matrices and obtain some equivalent conditions for H-matrices. By improving iteration factor and applying interleaved iterative method, we provide an iterative algorithm for H-matrices.In Chapter 3, we focus on the criteria for generalized block diagonal-ly dominant matrices under matrix norm. By applying inequality scaling techniques, from the whole and the any partition to the subscript set of the elements of the partitioned matrix with different row types, we give some criteria for generalized block diagonally dominant matrices under matrix norm.when the block matrix degenerates into a pointwise matrix, these criteria namely become the criteria for generalized diagonally dominant matrices.In Chapter 4, we mainly research the P-matrix and H-matrix linear complementarity problems. Based on the expression of the solution of LCPs, we present sensitivity analysis for P-matrix LCPs, establish a computational error bound for H-matrix LCPs, and establish the convergence of the block splitting methods for H-matrix with positive diagonal entries LCPs, particularly the convergence rates for projected block accelerated over-relaxation method (PBAOR) and projected block symmetric accelerated over-relaxation method (PBSAOR).These results show that the block splitting methods for LCPs have the same convergen-ce rates as those for corresponding systems of linear equations.In Chapter 6, we concentrate on the numerical computation of highly oscillatory integrals. Firstly, we explore related special matrices problems in efficient methods for computing highly oscillatory integrals. Secondly, for highly oscillatory Fourier-type integrals, we present a new efficient Levin-type method, which can get higher accuracy than the Levin's collocation method and can be easily implemented. Lastly, for highly oscillatory integrals with an algebraic singularity, we first expand such integrals into asymptotic series in inverse powers of??and then give the asymptotic order of the Filon-type methods.