Java——(高斯消去法/列主元消去法/LU消去)求解方程

Java 高斯消去 / 列主元消去 / LU消去

• 首先输入方程未知数个数
• 输入系数矩阵，按顺序(↖为头)，每个数字输入后回车
• 输入常量向量，按顺序(↑为头)，每个数字输入后回车
• 选择计算方法/输入0退出

优化结果计算：
x[i] % 1 > 0.999999999 || x[i] % 1 < 0.000000001
本项结果将近似为整数！

• 源代码只包含高斯&列主元方法
  (点击此处)下载源代码

import java.util.Scanner;

package one;

import java.util.Arrays;
import java.util.List;
import java.util.Scanner;

public class Q2 {
	static double A[][];
	static double b[];
	static double x[];

	static int n; // n表示未知数的个数
	static int n_2; // 记录换行的次数

	public static void main(String[] args) {

		while (true) {
			System.out.println("请输入数字选择计算方法：\n1.Gauss 2.LineGuess 3.LU 0.退出");
			@SuppressWarnings("resource")
			Scanner sn = new Scanner(System.in);
			int c = sn.nextInt();
			System.out.println("--------------输入方程组未知数的个数---------------");
			@SuppressWarnings("resource")
			Scanner sc = new Scanner(System.in);
			n = sc.nextInt();

			A = new double[n][n];
			b = new double[n];
			x = new double[n];

			System.out.println("--------------输入方程组的系数矩阵A:---------------");
			for (int i = 0; i < n; i++) {
				for (int j = 0; j < n; j++) {
					A[i][j] = sc.nextDouble();
				}
			}

			System.out.println("--------------输入方程组的常量向量b:---------------");
			for (int i = 0; i < n; i++) {
				b[i] = sc.nextDouble();
			}
			if (1 == c) {
				Gauss();
			} else if (2 == c) {
				LineGuess();
			} else if (3 == c) {
				LU();
			} else if (0 == c) {
				System.out.println("Bye!");
				System.exit(0);
			}
			
		}
	}

	// 1
	public static void Gauss() {
		Elimination();
		BackSubstitution();
		PrintRoot();
	}

	// 2
	public static void LineGuess() {
		Elimination2();
		BackSubstitution();
		PrintRoot();
	}

	// 3
	public static void LU() {
		double[] x = solve(A, b);
		System.out.println("结果为：");
		for (int z = 0; z < x.length; z++) {
			System.out.println("X" + z + " = " + x[z]);
		}
	}

	// 高斯
	public static void Elimination() {
		PrintA();
		for (int k = 0; k < n;) {
			for (int i = k + 1; i < n; i++) {
				double l = A[i][k] / A[k][k];
				A[i][k] = 0;

				for (int j = k + 1; j < n; j++) {
					A[i][j] = A[i][j] - l * A[k][j];
				}
				b[i] = b[i] - l * b[k];
			}
			System.out.println("第" + ++k + "次消元后:");
			PrintA();
		}
	}

	// 列主元
	public static void Elimination2() {
		PrintA();
		for (int k = 0; k < n;) {
			WrapRow(k);
			for (int i = k + 1; i < n; i++) {
				double l = A[i][k] / A[k][k];
				A[i][k] = 0;

				for (int j = k + 1; j < n; j++) {
					A[i][j] = A[i][j] - l * A[k][j];
				}
				b[i] = b[i] - l * b[k];
			}
			System.out.println("第" + ++k + "次消元后:");
			PrintA();
		}
	}

	// 交换矩阵的行
	public static void WrapRow(int k) { // k表示第k+1轮消元
		double max = Math.abs(A[k][k]);
		int Line = k; // 记住要交换的行
		for (int i = k + 1; i < n; i++) {
			if (Math.abs(A[i][k]) > max) {
				Line = i;
				max = A[i][k];
				break;
			}
		}
		if (Line != k) { // 交换求得最大主元
			n_2 += 1;
			System.out.println("k = " + k + "时，" + "要交换的行为" + k + "和" + Line);
			// 先交换A
			for (int j = k; j < n; j++) {
				double[] arr = { A[k][j], A[Line][j] };
				Swap(arr, 0, 1);
				A[k][j] = arr[0];
				A[Line][j] = arr[1];

			}
			// 再交换b
			double[] arr = { b[k], b[Line] };
			Swap(arr, 0, 1);
			b[k] = arr[0];
			b[Line] = arr[1];
			System.out.println("--------------交换后---------------");
			PrintA();
		}
	}

	// 交换Swap函数
	public static void Swap(double[] ar, int x, int y) {
		Double tmp = ar[x];
		ar[x] = ar[y];
		ar[y] = tmp;
	}

	// 回代法
	public static void BackSubstitution() {
		x[n - 1] = b[n - 1] / A[n - 1][n - 1];
		for (int i = n - 2; i >= 0; i--) {
			x[i] = (b[i] - solve(i)) / A[i][i];
		}
	}

	public static double solve(int i) {
		double result = 0.0;
		for (int j = i; j < n; j++)
			result += A[i][j] * x[j];
		return result;
	}

	// 输出方程组的根
	public static void PrintRoot() {
		System.out.println("--------------方程组的根为---------------");
		for (int i = 0; i < n; i++) {
			if (x[i] % 1 > 0.999999999 || x[i] % 1 < 0.000000001) {
				x[i] = (int) x[i];
				System.out.println("X" + i + " = " + x[i]);
			} else {
				System.out.println("X" + i + " = " + x[i]);
			}
		}
	}

	// 输出A的增广矩阵
	public static void PrintA() {
		System.out.println("--------------增广矩阵---------------");
		for (int i = 0; i < n; i++) {
			for (int j = 0; j < n; j++) {
				System.out.print(+A[i][j] + "  ");
			}
			System.out.println(b[i]);
		}
	}

	// LU
	public static double[] solve(double[][] a, double[] b) {
		List<double[][]> LAndU = decomposition(a); // LU decomposition
		double[][] L = LAndU.get(0);
		double[][] U = LAndU.get(1);
		double[] UMultiX = getUMultiX(a, b, L); // 前代
		return getSolution(a, U, UMultiX); // 回代
	}

	private static double[] getSolution(double[][] a, double[][] U, double[] UMultiX) {
		double[] solutions = new double[a[0].length];
		for (int i = U.length - 1; i >= 0; i--) {
			double right_hand = UMultiX[i];
			for (int j = U.length - 1; j > i; j--) {
				right_hand -= U[i][j] * solutions[j];
			}
			solutions[i] = right_hand / U[i][i];
		}
		return solutions;
	}

	private static double[] getUMultiX(double[][] a, double[] b, double[][] L) {
		double[] UMultiX = new double[a.length];
		for (int i = 0; i < a.length; i++) {
			double right_hand = b[i];
			for (int j = 0; j < i; j++) {
				right_hand -= L[i][j] * UMultiX[j]; //
			}
			UMultiX[i] = right_hand / L[i][i];
		}
		return UMultiX;
	}

	private static List<double[][]> decomposition(double[][] a) {
		double[][] U = a; // a是要分解的矩阵
		double[][] L = createIndentityMatrix(a.length);

		for (int j = 0; j < a[0].length - 1; j++) {
			if (a[j][j] == 0) {
				throw new IllegalArgumentException("zero pivot encountered.");
			}

			for (int i = j + 1; i < a.length; i++) {
				double mult = a[i][j] / a[j][j];
				for (int k = j; k < a[i].length; k++) {
					U[i][k] = a[i][k] - a[j][k] * mult;
					// 得出上三角矩阵U,通过减去矩阵的第一行,第二行,第一行(第二行)得到上三角矩阵
				}
				L[i][j] = mult; // 得到下三角矩阵是得出上三角矩阵的乘积因子
			}
		}
		System.out.println("--------------L矩阵---------------");
		PrintLU(L);
		System.out.println("--------------U矩阵---------------");
		PrintLU(U);
		return Arrays.asList(L, U);

	}
	
	public static void PrintLU(double [][]z){
		for (int i = 0; i < n; i++) {
			for (int j = 0; j < n; j++) {
				System.out.print(z[i][j] + " ");
			}
			System.out.println("\n");
		}
	}

	private static double[][] createIndentityMatrix(int n) {
		double[][] identityMatrix = new double[n][n];
		for (int i = 0; i < identityMatrix.length; i++) {
			for (int j = i; j < identityMatrix[i].length; j++) {
				if (j == i) {
					if (j == i) {
						identityMatrix[i][j] = 1;
					} else {
						identityMatrix[i][j] = 0;
					}
				}
			}
		}
		return identityMatrix;
	}

}

本文地址：https://blog.csdn.net/GodOuO/article/details/109555999