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1、10-11-2几何与代数数学实验报告学号：06010329 姓名：周志浩 得分： .实验一：平板的稳态温度分布问题（线性方程组应用）在热传导的研究中，一个重要的问题是确定一块平板的稳态温度分布。假定下图中的平板代表一条金属梁的截面，并忽略垂直于该截面方向上的热传导。已知平板内部有9个节点，每个节点的温度近似等于与它相邻的四个节点温度的平均值，例如,；为避免出现分数，可写成。设4条边界上的温度分别等于每位同学学号的后四位的4倍，例如学号为16308209的同学计算时，选择、。求：（1）建立可以确定平板内节点温度的线性方程组；（2）用MATLAB软件的三种方法求解该线性方程组 (请输出精确解（分数
2、形式）) ； 方法一：利用Cramer法则求解； 方法二：作为逆矩阵的方法求解； 方法三：利用Gauss消元法即通过初等行变换求解。实验部分构造的线性方程：方法一：Cramer法则 format rata1=4,-1,0,-1,0,0,0,0,0,;a2=-1,4,-1,0,-1,0,0,0,0;a3=0,-1,4,0,0,-1,0,0,0;a4=-1,0,0,4,-1,0,-1,0,0;a5=0,-1,0,-1,4,-1,0,-1,0;a6=0,0,-1,0,-1,4,0,0,-1;a7=0,0,0,-1,0,0,4,-1,0;a8=0,0,0,0,-1,0,-1,4,-1;a9=0,0,0
3、,0,0,-1,0,-1,4;b=12,0,4,12,0,4,24,12,16; D=det(a1,a2,a3,a4,a5,a6,a7,a8,a9),D = 100352 D1=det(b,a2,a3,a4,a5,a6,a7,a8,a9),D2=det(a1,b,a3,a4,a5,a6,a7,a8,a9),D3=det(a1,a2,b,a4,a5,a6,a7,a8,a9),D4=det(a1,a2,a3,b,a5,a6,a7,a8,a9),D5=det(a1,a2,a3,a4,b,a6,a7,a8,a9),D6=det(a1,a2,a3,a4,a5,b,a7,a8,a9),D7=det(a1,
4、a2,a3,a4,a5,a6,b,a8,a9),D8=det(a1,a2,a3,a4,a5,a6,a7,b,a9),D9=det(a1,a2,a3,a4,a5,a6,a7,a8,b),D1 = 630784 D2 = 419328 D3 =344064 D4 =899584 D5 =702464 D6 =555520 D7 =1060864 D8 =935424 D9 =774144 T1=D1/D,T2=D2/D,T3=D3/D,T4=D4/D,T5=D5/D,T6=D6/D,T7=D7/D,T8=D8/D,T9=D9/DT1 = 44/7 T2 = 117/28 T3 = 24/7 T4
5、= 251/28 T5 = 7 T6 = 155/28 T7 = 74/7 T8 = 261/28 T9 =54/7 方法二：逆矩阵a1=4,-1,0,-1,0,0,0,0,0,;a2=-1,4,-1,0,-1,0,0,0,0;a3=0,-1,4,0,0,-1,0,0,0;a4=-1,0,0,4,-1,0,-1,0,0;a5=0,-1,0,-1,4,-1,0,-1,0;a6=0,0,-1,0,-1,4,0,0,-1;a7=0,0,0,-1,0,0,4,-1,0;a8=0,0,0,0,-1,0,-1,4,-1;a9=0,0,0,0,0,-1,0,-1,4; b=12,0,4,12,0,4,24,
6、12,16; A=a1,a2,a3,a4,a5,a6,a7,a8,a9,B=12,0,4,12,0,4,24,12,16,A = Columns 1 through 5 4 -1 0 -1 0 -1 4 -1 0 -1 0 -1 4 0 0 -1 0 0 4 -1 0 -1 0 -1 4 0 0 -1 0 -1 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 Columns 6 through 9 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 -1 0 -1 0 4 0 0 -1 0 4 -1 0 0 -1 4 -1 -1 0 -1 4 B = 12 0
7、4 12 0 4 24 12 16 T=inv(A)*BT = 44/7 117/28 24/7 251/28 7 155/28 74/7 261/28 54/7 方法三：Gauss消元法a1=4,-1,0,-1,0,0,0,0,0,;a2=-1,4,-1,0,-1,0,0,0,0;a3=0,-1,4,0,0,-1,0,0,0;a4=-1,0,0,4,-1,0,-1,0,0;a5=0,-1,0,-1,4,-1,0,-1,0;a6=0,0,-1,0,-1,4,0,0,-1;a7=0,0,0,-1,0,0,4,-1,0;a8=0,0,0,0,-1,0,-1,4,-1;a9=0,0,0,0,0,-1
8、,0,-1,4; b=12,0,4,12,0,4,24,12,16; A=a1,a2,a3,a4,a5,a6,a7,a8,a9,b,A = Columns 1 through 7 4 -1 0 -1 0 0 0 -1 4 -1 0 -1 0 0 0 -1 4 0 0 -1 0 -1 0 0 4 -1 0 -1 0 -1 0 -1 4 -1 0 0 0 -1 0 -1 4 0 0 0 0 -1 0 0 4 0 0 0 0 -1 0 -1 0 0 0 0 0 -1 0 Columns 8 through 10 0 0 12 0 0 0 0 0 4 0 0 12 -1 0 0 0 -1 4 -1 0
