求解拟五对角线性方程组的四参数法

 基于五对角线性方程组的追赶法，给出了拟五对角线性方程组的四参数求解方法。算法的基本思想是，将方程组的前2个未知量x1，x2和最后2个未知量xn－1,xn看作参数，这4个未知量正好对应于拟五对角方程组边角位置上的非零元素。然后通过特殊的矩阵分解将方程组解向量中的其他n－4个未知量用x1，x2，xn－1和xn 4个参数表示，从而形成标准的五对角线性方程组，可以方便地利用求解标准五对角线性方程组的追赶法进行求解。被看作参数的4个未知量可以利用原方程组中的前后两个方程及中间变量求出。最后，将已经求出的4个参数再代入分解矩阵形成的方程组中求得其余分量。鉴此，本文给出了两种不同的实现方法，其主要区别在于求解4个参数的过程不同。一种方法是将解向量的全部分量用参数线性表出，然后取出前后各2个式子组成参数方程，求出4个参数。另一种方法是将4个参数作为已知量先代入第3～n－2个方程中，整理后得到一个n－4阶的方程组，解出第3～n－2个解分量的参数表达式，再将x3,x4,xn－3,xn－2回代到前2个方程和最后2个方程中组成参数方程，求出4个参数。对于规模较大的拟五对角线性方程组而言，这两种算法的计算量几乎一样。该算法的数值稳定性分析结果表明，系数矩阵在满足严格对角占优的条件下，该算法是稳定的。数值实验结果表明，两种算法的实际计算时间与算法的理论分析相符合。

Four Parameter Algorithm for Solving the Quasi-pentadiagonal Linear Equations

Quasi-pentadiagonal linear equations are important in computational mathematics and scientific/engineering computing, which would arise during the solution of boundary value problems for elliptical or parabolic partial differential equation(s) with periodicalboundary conditions, and the quintic interpolating splines with periodical boundary conditions. Based on the ideas of the forward elimination and backward substitution algorithm for pentadiagonal linear equations and the matrix decomposition, a four parameter algorithm for quasi-pentadiagonal linear equations is proposed in this paper. It involves four steps. In the first step, the first two unknowns x1, x2 and the last two unknowns xn-1, xn are taken as the four parameters, which are responsible for the non-zero elements at corners of the quasi-pentadiagonal matrix. In the second step, other unknowns of the equations are expressed explicitly by the four parameters. Then the standard pentadiagonal linear equations can be formed with the help of special matrix decomposition and solved conveniently with the help of the forward elimination and backward substitution algorithm. In the third step, the four parameters are solved with the help of the first two equations and the last two equations with decomposed matrixes. At last, all unknowns are solved efficiently when the four parameters are substituted into the equations with the decomposed matrix. With this algorithm, two methods of solving the four parameters are presented in this paper. One is that all other unknowns are expressed explicitly by the four parameters. Then the four parameters are solved with the help of the first two equations and the last two equations. The other is that the four parameters are regarded as known quantities and are substituted into the 3~n-2 equations, which are to be solved. Then the four parameters can be solved when x3, x4, xn-3, xn-2 are substituted into the first two equations and the last two equations. The computation cost of the two methods is almost the same for large scale pentadiagonal equations. The stability analysis shows that the four parameter algorithm is stable if the coefficient matrix is a strictly diagonally dominant matrix. The numerical example indicates that the computational time of the two algorithms is consistent with the theoretical result.