两类五对角行列式的计算研究

Determinant of Two Types of Five-Diagonal Matrices

五对角行列式的求解问题在偏微分方程的数值求解、图像处理、电路分析、生物信息学等方面具有非常广泛的应用.本文将两类五对角行列式通过Laplace展开,将各余子式与其自身表示为一阶线性递推方程组,在此基础上将行列式随阶数变化的值表示为系数矩阵的幂方与初值向量乘积的形式.并通过LU分解将两类行列式的值表示为二元递推方程组的解的连乘积的形式.

The value of five-diagonal determinants has very broad applications in numerical solutions of partial differential equations,image processing,circuit analysis,and bioinformatics.Firstly,in this paper,the two types of five-diagonal determinants are expanded through Laplace,and each cofactor and itself are represented as the solution of a first-order linear recursive equation system.Further,the values of the determinant are represented as the product of the power of the coefficient matrix and the initial value vector.Secondly,the values of the two types of determinants are represented as the continuous product of the solutions of the binary recursive equation system through LU decomposition.