正定可对称化矩阵与预对称迭代算法

１．问题的提出 我们引入正定可对称化矩阵定义的背景是为了研究求解二阶椭圆型非自共轭方程的离散迭代有效算法、这类方程的椭圆型是本质的分析性质。是由二阶项决定的，在离散方程中表现为正定性；非自共轭性则是由方程中的一阶项引起的，在相当广泛一类问题中可通过变量代换化为自共轭。因此，我们称这类问题为正定可对称化问题。 例１．高维二阶常系数椭圆型方程其中 Ａ为常系数正定对称（ｓ．ｐ．ｄ）阵， 为正交阵， Ｄ是对角元素为正的对角阵。 先作变量代换，通过演算，偏微分方程对于新变量变成这里进而令可将原非自共轭偏微分算子…

ON POSITIVE SYMMETRIZABLE MATRICES AND PRE-SYMMETRY ITERATION ALGORITHMS

Second order elliptic equation is a class of mathematical model for scientific computing, such as convex-diffusion, oil-reservoir simulation, etc. Based on intrinsic symmetrizable property, a new concept on positively symmetrizable matrix is proposed in this paper. We point that for such kind of equation systems, it is possible to adopt special preconditioning CG algorithm, e.g. 1]-3], instead of the usual iteration procedure for general non-symmetry systems, such as GMRES 3]-4] ) BiCGSTAB 5]. Numerical tests show the new algorithm is effective for solving this kind of second order elliptic discrete systems.