解线性方程组的直接方法——高斯消去和LU消去法

高斯消去法：

/** *  * 高斯消元算法 * @author luzi * */public class guess {double[] x2 = {0,0,0,0,0,0,0,0};//列主元高斯消去法public void guess1(double[][] A,double[] b,int n,int target){//选主元，消元double max = Double.MIN_VALUE;for(int k = 0; k < n - 1; k++){int t = k;int p = k;//选主元for(; t < n;t++){if(A[t][k] > max){max = A[t][k];p = t;}}//主元交换if(p != k){double temp = 0;for(int i = 0;i < n;i++){temp = A[p][i];A[p][i] = A[k][i];A[k][i] = temp;}temp = b[p];b[p] = b[k];b[k] = temp;}for(int i = k + 1; i < n; i++){double tp = A[i][k]/A[k][k]; b[i] = b[i] - tp*b[k]; //A[i][k] = 0; for(int j = k + 1;j < n;j++){ A[i][j] = A[i][j] - tp*A[k][j]; }}}//回代x2[n - 1] = b[n - 1]/A[n - 1][n - 1];for(int i = n-2;i >= 0;i--){double sum = 0;for(int j = i + 1; j < n;j++){sum += A[i][j]*x2[j];}x2[i] = (b[i] - sum)/A[i][i];}for(int i = 0; i < n; i++)System.out.println(x2[i]);}//顺序高斯消去法public double[] guess2(double[][] A,double[] b,int n,int target){double[] x = new double[n];for(int k = 0; k < n - 1; k++){for(int i = k + 1; k < n; k++){double tp = A[i][k]/A[k][k];b[i] = b[i] - tp*b[k];for(int j = k ; j < n; j++){A[i][j] = A[i][j] - tp*A[k][j];}}}x[n - 1] = b[n - 1]/A[n - 1][n - 1];for(int i = n - 2; i >= 0; i--){double sum = 0;for(int j = i + 1; j < n; j++){sum += A[i][j]*x[j];}x[i] = (b[i] - sum)/A[i][i];}return x;}}

直接LU消去法：

package jisuanfangfashiyan;/** *  * @author luzi * */public class LU {//直接lU分解消去法public double[] getLU(double[][] A,double[] x,double[] b,int n){double[][] u = new double[n][n];double[][] l = new double[n][n];double[] y = new double[n];double sum = 0;double temp = 0;for(int k = 0; k < n; k++){//先求  u 的第  k 行的值for( int j = k; j < n; j++){for(int m = 0; m < k; m++){sum += (l[k][m] * u[m][j]);}u[k][j] = A[k][j] - sum;sum = 0;}//求 l  的第 k 的值for(int i = k + 1; i < n; i++){for(int m = 0; m < k; m++){sum += (l[i][m] * u[m][k]);}l[i][k] = ( A[i][k] - sum ) /u[k][k];sum = 0;}}y[0] = b[0];for(int i = 1; i < n; i++){for(int j = 0; j < i; j++){sum += (l[i][j] * y[j]);}y[i] = b[i] - sum;sum = 0;}x[n - 1] = y[n - 1]/u[n - 1][n - 1];for(int i = n - 2; i >= 0; i--){for(int j = i + 1; j < n; j++){sum += (u[i][j] * x[j]);}x[i] = (y[i] - sum)/u[i][i];sum = 0; }return x;}public static void main(String args[]){double[][] A =  {{2,-3,2},{-4,2,-6},{2,2,4}};double[] x = new double[3];double[] b = {5,14,8};LU lu = new LU();x = lu.getLU(A, x, b, 3);for(double ele : x)System.out.println(ele);}}

LU消去法例子运行结果：