特殊矩阵乘积在非线性数值计算中的应用

本文将Hadamard矩阵乘积引入到非线性数值计算,获得了简单的矩阵形式的非线性代数模拟方程,利用Hadamard矩阵乘积和Hadamard矩阵函数的方法,我们能够容易地构造快速收敛的简单迭代法解非线性代数方程组的迭代公式,使该法成为与Newton-Raphson法相比有竞争力的方法,我们也首次定义了一种新的特殊矩阵乘积—SJT矩阵乘积。运用SJT积,我们能够方便高效的计算Newton-Raphson法中Jacobi导数矩阵的精确解,利用Hadamard矩阵乘积的范数性质,我们也导出了非线性计算摄动误差的分析公式,此外,Hadamard积和SJT积能够被用于非线性数值解耦计算,这极大地减少了求解耦合的非线性偏微分方程组的计算工作量和内存需要量。

Application of Special Matrix Product to Nonlinear Numerical Computations

In this paper, we introduce the Hadamard product into nonlinear numerical computations to obtain nonliner analog equation in simple matrix form. By using the Hadamard matrix product and function approach, we can very easily construct highly efficient iteration formuias of simple iteration method for solving the nonlinear formulations in Hadamard product, form. Consequently, the simple iteration method becomes very competitive to the Newton-Baphson method. We also first defined it now kind of special matrix product-SJT product. Applying SJT product, the exact solution of the Jacobian derivative matrix in the Newton-Raphson method can be co nputed very easily and efficiently. In addition, the Hadamard product and SJT product can be utilized for nonlinear decoupling computation to greatly reduce the computer time and storage requirements in the solution of nonlinear coupled partial differential equations.