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Abstract: In this paper, two efficient iterative methods are presented to solve the symmetric and skew symmetric solutions of a linear matrix equation , respectively, with real pair matrices and . By these two iterative methods, the solvability of the symmetric and skew symmetric solutions for the matrix equation can be determined automatically. When the matrix equation has symmetric and skew symmetric solutions, then, for any initial pair matrices and , symmetric and skew symmetric solutions can be obtained within finite iteration steps in the absence of roundoff errors, and the minimum norm of the symmetric and skew symmetric solutions can be obtained by choosing a special kind of initial pair matrices. In addition, the unique optimal approximation pair solution and to the given matrices and in Frobenius norm can be obtained by finding the minimum norm solution of a new matrix equation , where . The given numerical examples demonstrate that the iterative methods are quite efficient. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of boundary value problems in the reproducing kernel space. The analytical solution and approximate solution are represented in terms of series. For any initial function , we prove , , . Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective. [Copyright &y& Elsevier]