线性方程组的直接解法

第六章线性方程组的直接解法第六章线性方程组的直接解法本节课的目的学习求解线性方程组的Gauss消去法,主元消去法,特殊矩阵消去法和矩阵的三角分解等直接解法.并掌握MATHMATICA和MATLAB软件的使用方法.问题的提出考察由图所示的由一个电池电源和一些电阻组成的简单电网络。若用和表示其环路电流,利用克希霍夫定律和欧姆定律，可得到下列线性方程组:1012725740.unknown1012725786.unknown1012726109.unknown1019367717.unknown研究国民经济的数学模型下面是某城镇的一个经济系统:1012726109.unknown生产价值消费一元煤1一元电x2一元运输服务x3订单煤0.400.4540000电0.250.050.1024000运输费0.350.100.10310001015658168.unknown试问这三个企业在这星期各应生产多少产值才能满足内外需求？分配平衡方程组:1012812176.unknown1012812362.unknown1012812418.unknown1012811961.unknown或写为:1012812176.unknown1012812362.unknown1012812418.unknown1012811961.unknown本节的目的是.介绍求解方程组6.1.4)的直接解法.1012812362.unknown1012812418.unknownGauss消去法考虑一个线性方程组的求解：Matlab1019368078.unknown1019909896.unknown1019371311.unknown对于上三角方程组，如果,求解此三角形方程组是容易的.为此,我们可以先从最后一个方程求得,1012855659.unknown1012856164.unknown将xn代入第n1个方程求得EMBEDEquation.3,,这样逐次用后面方程的结果代入前一方程，即可求得解的所有分量:385272100.unknown1012856230.unknown讨论题顺序Gauss消去法能否进行下去）1.1012815559.unknown讨论题2.用顺序Gauss消去法并分别用4位有效数字及8位有效数字求解1012815695.unknown3.请用顺序Gauss消去法求解②1015923054.unknown1015923063.unknown列主元消去法1.对于做到第7步；1012817307.unknown1012817450.unknown1012817883.unknown1012817984.unknown1012818133.unknown1012818134.unknown1012817935.unknown1012817595.unknown1012817778.unknown1012817562.unknown1012817355.unknown1012817383.unknown1012817330.unknown1012817053.unknown1012817282.unknown1012816982.unknown2.按列选主元素1012817307.unknown1012817355.unknown1012817383.unknown1012817450.unknown1012817330.unknown1012817282.unknown3.若则计算停止；1012817330.unknown1012817383.unknown1012817450.unknown1012817355.unknown1012817307.unknown4.若则转5，否则换行：1012817383.unknown1012817450.unknown1012817355.unknown列主元消去法续）5.计算乘数1012817307.unknown1012817450.unknown1012817883.unknown1012817984.unknown1012818133.unknown1012818134.unknown1012817935.unknown1012817595.unknown1012817778.unknown1012817562.unknown1012817355.unknown1012817383.unknown1012817330.unknown1012817053.unknown1012817282.unknown1012816982.unknown6.消元计算1012817883.unknown1012817984.unknown1012818133.unknown1019373871.unknown1012817935.unknown1012817778.unknown7.1012817984.unknown1012818133.unknown1019373871.unknown1012817935.unknown8.代求解对于1012818133.unknown1019373871.unknown1012817984.unknown9.1012818133.unknown顺序Gauss消去法与列主元消去法的比较用列主元消去法可得解为。1015925396.unknown1015925400.unknown计算剩余向量1015925395.unknown385174299.unknown1015925405.unknown用顺序Gauss消去法可得解,1015925396.unknown6.4矩阵的三角分解下面,我们按矩阵变换的观点来描述消去过程,由此,我们可以得到矩阵的三角分解.记方程组(6.2.3)的增广矩阵为[A,b].设，若记Gauss矩阵:385174702.unknown6.4.1）385174853.unknown1019385986.unknown其中由6.2.4）表示。则将方程组(6.2.3)变化为(6.2.5)的过程，相当于用的逆左乘增广矩阵[A,b]一般的,如果已经有Gauss矩阵,1012859881.unknown1012860084.unknown1019386033.unknown1012860037.unknown1012859823.unknown6.4矩阵的三角分解则当时,从方程组(6.2.8)变化为(6.2.10)的过程,相当于用(6.2.9)构成的Gauss矩阵:385273004.unknown1012860084.unknown　　　　　(6.4.2)1015570264.unknown1019385958.unknown385174966.unknown的逆左乘(6.2.10)的增广矩阵如此继续，最后我们可得到由(6.2.12)表示的上三角矩阵U.因此，方程组(6.2.3)的Gauss消去过程,用矩阵变换的观点,表示为:1015515545.unknown其中是一个单位下三角矩阵.因此,对方程组(6.2.3)的Gauss消去过程实现了矩阵A的一个三角分解,(6.4.3)式称为A的LU分解.1015515641.unknownLU分解定理定理6.1设A为n阶矩阵，如果，,dt(A)≠0385175339.unknown385175619.unknown385175275.unknown定理6.2设A为非奇异矩阵,1015515927.unknown则A可分解为一个下三角矩阵L和一个上三角矩阵U的乘积，385175619.unknown6.4.4）且这种分解是唯一的。385175619.unknown则存在置换矩阵P以及元素的绝对值不大于1的单位下三角矩阵L和上三角矩阵U,使得1015515927.unknown(6.4.8)并且这种三角分解可由列主元消去法得到。1015515927.unknown特殊矩阵消去法A是对称正定矩阵1.1012821932.unknown1012894059.unknown1012894063.unknown1012894067.unknown1012894053.unknown385176643.unknown1012821923.unknown385176547.unknown2.385176643.unknown1012821932.unknown1012894059.unknown1012894063.unknown1012821923.unknown385176547.unknown3.求解385176643.unknown1012821923.unknown1012821932.unknown385176547.unknown4.求解385176643.unknown1012821923.unknown(1)分解(Doolittl分解)1012775360.unknown1012775724.unknown1012775975.unknown1012777094.unknown1015516884.unknown1015516894.unknown1012776167.unknown1012775790.unknown1012775672.unknown1012775710.unknown1012775631.unknown1012775242.unknown1012775306.unknown385276206.unknown2）解方程组(”追”的过程)EMBEDEquation.3EMBEDEquation.31012775710.unknown1012775790.unknown1012776167.unknown1012777094.unknown1015516894.unknown1012775975.unknown1012775724.unknown1012775631.unknown1012775672.unknown385276206.unknown3）解(“赶”的过程)1012775975.unknown1012776167.unknown1012777094.unknown385276206.unknown是三对角矩阵1019909285.unknown病态问题和算法的数值稳定性,如果把(6.6.1)的右端项作微小扰动则(6.7.1)变为1012861374.unknown1012861692.unknown1015517943.unknown1012861391.unknown1012840480.unknown求解如下的线形方程组(6.7.1)1012840323.unknown1012861374.unknown1012861692.unknown1015517943.unknown1012861391.unknown1012840480.unknown1012778992.unknown容易看出,它的解为1012840480.unknown1012861391.unknown1012861692.unknown1015517943.unknown1012861374.unknown1012840323.unknown(6.7.2)1012861391.unknown1012861692.unknown1015517943.unknown1012861374.unknown1012861391.unknown1012861692.unknown1012861374.unknown直接计算可知，(6.6.2)的解为EMBEDEquation.3.因此有1012861374.unknown1012861391.unknown病态问题和算法的数值稳定性定义6.1：一个算法，如果计算过程的舍入误差的积累不大，就说这一算法具有较好的数值稳定性；定义7.2对非奇异矩阵Ａ，量称为矩阵Ａ的条件数．记为Cond(A).1012849985.unknown反之，如果计算过程的舍入误差的积累很大，就说该算法的数值稳定性不好，或者说其是不稳定的。病态问题和算法的数值稳定性例1．设，试计算。1012824861.unknown1012824862.unknown1012824863.unknown1012824860.unknown例2．1012825391.unknown1012825441.unknown1012825529.unknown1012825257.unknown得:，，1012825441.unknown1012825529.unknown1012825391.unknown因，。1012824860.unknown1012824861.unknown讨论Mathmatica下列方程组并考察n分别取时，计算解与精确解，的误差。用笔算的情形以弄清楚计算机中出现了什么困难。使用《Mathmatica》软件，求解例6.5，能否得到精确解。1012825934.unknown1012825972.unknown1012826681.unknown1012825906.unknown