用稀疏矩阵解决联立方程组
用稀疏矩阵解决联立方程组为了解说明稀疏矩阵在MATLAB中应用，我们将用全矩阵和稀疏矩阵来解决下面的联立方程组。1.0x1 + 0.0x2 + 1.0x3+ 0.0x4 + 0.0x5 + 2.0x6 + 0.0x7- 1.0x8 = 3.00.0x1 + 1.0x2 + 0.0x3+ 0.4x4 + 0.0x5 + 0.0x6 + 0.0x7+ 0.0x8 = 2.00.5x1 + 0.0x2 + 2.0x3+ 0.0x4 + 0.0x5 + 0.0x6 - 1.0x7+ 0.0x8 = -1.50.0x1 + 0.0x2 + 0.0x3+ 2.0x4 + 0.0x5 + 1.0x6 + 0.0x7+ 0.0x8 = 1.00.0x1 + 0.0x2 + 1.0x3+ 1.0x4 + 1.0x5 + 0.0x6 + 0.0x7+ 0.0x8 = -2.00.0x1 + 0.0x2 + 0.0x3+ 1.0x4 + 0.0x5 + 1.0x6 + 0.0x7+ 0.0x8 = 1.00.5x1 + 0.0x2 + 0.0x3+ 0.0x4 + 0.0x5 + 0.0x6 + 1.0x7+ 0.0x8 = 1.00.0x1 + 1.0x2 + 0.0x3+ 0.0x4 + 0.0x5 + 0.0x6 + 0.0x7+ 1.0x8 = 1.0答案为了解决这一问题，我们将创建一个方程系数的全矩阵，并用sparse函数把他转化为稀疏矩阵。我们用两种方法解这个方程组，比较它们的结果和所需的内存。代码如下：% Script file: simul.m%% Purpose:% This program solves a system of 8 linear equations in 8% unknowns (a*x = b), using both full and sparse matrices.%% Record of revisions:% Date Programmer Description of change% ==== ======== ===============% 10/14/98 S. J. Chapman Originalcode%% Define variables:% a --Coefficients of x (full matrix) % as --Coefficients of x (sparse matrix)% b --Constantcoefficients (full matrix)% bs --Constantcoefficients (sparse matrix)% x --Solution(full matrix)% xs --Solution(sparse matrix)% Define coefficients of the equation a*x = b for% the full matrix solution.a = [ 1.0 0.0 1.0 0.0 0.0 2.0 0.0 -1.0; ...0.0 1.0 0.0 0.4 0.0 0.0 0.0 0.0; ...0.5 0.0 2.0 0.0 0.0 0.0 -1.0 0.0; ...0.0 0.0 0.0 2.0 0.0 1.0 0.0 0.0; ...0.0 0.0 1.0 1.0 1.0 0.0 0.0 0.0; ...0.0 0.0 0.0 1.0 0.0 1.0 0.0 0.0; ...0.5 0.0 0.0 0.0 0.0 0.0 1.0 0.0; ...0.0 1.0 0.0 0.0 0.0 0.0 0.0 1.0];b = [ 3.0 2.0 -1.5 1.0 -2.0 1.0 1.0 1.0]';% Define coefficients of the equation a*x = b for% the sparse matrix solution.as = sparse(a);bs = sparse(b);% Solve the system both waysdisp ('Full matrix solution:');x = a\bdisp ('Sparse matrix solution:');xs = as\bs% Show workspacedisp('Workspace contents after the solutions:')whos
运行这个程序，结果如下>> simulFull matrix solution:x = 0.5000 2.0000 -0.5000 -0.0000 -1.5000 1.0000 0.7500 -1.0000Sparse matrix solution:xs = (1,1) 0.5000 (2,1) 2.0000 (3,1) -0.5000 (5,1) -1.5000 (6,1) 1.0000 (7,1) 0.7500 (8,1) -1.0000Workspace contents after the solutions: Name Size Bytes Class a 8x8 512 double array as 8x8 276 double array (sparse) b 8x1 64 double array bs 8x1 104 double array (sparse) x 8x1 64 double array xs 8x1 92 double array (sparse)Grand total is 115 elements using 1112 bytes两种算法得到了相同的答案。注意用稀疏矩阵产生的结果不包含x4，因为它的值为0。注意b的稀疏形式占的内存空间比全矩阵形式还要大。这种情况是因为稀疏矩阵除了元素值之外必须存储它的行号和列号，所以当一个矩阵的大部分元素都是非零元素，用稀疏矩阵将降低运算效率。
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