用参数法求一些特殊的线性代数方程组的数值解

PARAMETRIC METHODS FOR SOME SPECIAL SYSTEM OF LINEAR ALGEBRAIC EQUATIONS

本文将求解线性方程组数值解的双参数法进行推广,得到(?)种求解一些特殊的线性方程组的较为(?)般的方法-参数法,并具体给出利用三组参数求解拟二对角方程组和拟Hessen-berg方程组的算法.此算法具有明显的优越性.比如,在求解拟二对角方程组时,和利用LU分解法相比,乘除运算的次数由11n-16变为9n+20,所需要设定的向量组由5个降为4个.在求解拟Hessenberg方程组时,和Gauss消去法相比,除法运算的次数由1/2n(n+1)变为3n-4.这对求解大型的拟三对角方程组和拟Hessenberg方程组非常有利.当然,此种方法还可以用来求解其它一些方程组。

In this paper, biparametric methods for system of linear algebraic equations are popularized and more commonly methods, parametric methods, are derived for some special system of linoar equations. Meanwhile, the concrete algorithms, which are used to solve systems of quasi-tridiagonal equations and quasi-Hesscnberg equations are suggested. These methods have many advantages. For example, when they are used to solve system of quasi-tridiagonal equations, the number of multiplication and division operation changes form 11n - 16 to 9n + 20 comparing with LU decomposition method. Moreover, the number of vectors need to be set in program is reduced from 5 to 4. When they are used to solve system of quasi-tridiagonal equations, the number of division operation is reduced form 1/2n (n + 1) to 3n - 1 comparing with the Gaussian elimination. The methods in this paper are beneficial to solve large scale systems of quasi-tridiagonal equations and quasi-Hessenberg equations. Of course, these methods can also be used to solve other systems of linear equations.