矩阵求逆

首先要知道LU分解：

package com.zhp.third.LU分解;

import java.util.ArrayList;

import com.zhp.common.Methods;

/**
 * @author 郑海鹏
 * @since 2014/10/18 21:23
 * @email 284967632@qq.com
 */
public class LU {
	
	public static void main(String[] args) {
		double[][] matrix = new double[][] { { 2, -1, 1, 1 }, { 4, 1, -1, 5 }, { 1, 1, 1, 0 } };

		ArrayList<double[][]> lu = getLU(matrix);

		Methods.print(lu.get(0));
		System.out.println();
		Methods.print(lu.get(1));
	}

	/**
	 * @return 第0个是L， 第1个是U;
	 */
	public static ArrayList<double[][]> getLU(double matrix[][]) {
		int row = matrix.length;
		int col = matrix[0].length;

		double[][] matrixL = new double[row][col];
		double[][] matrixU = Methods.copy(matrix);

		for (int i = 0; i < row; i++)
			matrixL[i][i] = 1;

		for (int rowIndex = 0; rowIndex < row; rowIndex++) {
			int maxRow = rowIndex;

			// 选主元
			for (int selectRow = rowIndex + 1; selectRow < row; selectRow++) {
				if (Math.abs(matrix[selectRow][rowIndex]) > Math.abs(matrix[maxRow][rowIndex])) {
					maxRow = selectRow;
				}
			}

			// 交换
			for (int colIndex = rowIndex; colIndex < col; colIndex++) {
				double temp = matrixU[rowIndex][colIndex];
				matrixU[rowIndex][colIndex] = matrixU[maxRow][colIndex];
				matrixU[maxRow][colIndex] = temp;
			}

			// 高斯消去
			for (int rollRow = rowIndex + 1; rollRow < row; rollRow++) {
				double scale = matrixU[rollRow][rowIndex] / matrixU[rowIndex][rowIndex];
				matrixL[rollRow][rowIndex] = scale;
				// 等比缩放
				for (int colIndex = rowIndex; colIndex < col; colIndex++) {
					matrixU[rollRow][colIndex] = matrixU[rollRow][colIndex]
							- matrixU[rowIndex][colIndex] * scale;
				}
			}

		}

		ArrayList<double[][]> lu = new ArrayList<>();
		lu.add(matrixL);
		lu.add(matrixU);
		return lu;
	}

}

其次要利用LU分解解线性方程组：

package com.zhp.third.解线性方程组;

import java.util.ArrayList;

import com.zhp.common.Methods;
import com.zhp.third.LU分解.LU;

/**
 * @author 郑海鹏
 * @since 2014/10/18 21:23
 * @email 284967632@qq.com
 */
public class LinearEquation {

	public static void main(String[] args) {
		double[][] matrix = new double[][] { { 2, -1, 1, 1 }, { 4, 1, -1, 5 }, { 1, 1, 1, 0 } };		
		double[] result = getResult(matrix);
		Methods.print(result);
	}
	
	/**
	 *	获得一个线性方程组的解 
	 */
	public static double[] getResult(double[][] matrix){
		ArrayList<double[][]> lu = LU.getLU(matrix);
		return getResultByU(lu.get(1));
	}

	/**
	 * @param U是一个上三角矩阵
	 */
	private static double[] getResultByU(double[][] U) {
		int row = U.length;

		double[] result = new double[row];

		// 最后一个答案
		result[row - 1] = U[row - 1][row] / U[row - 1][row - 1];

		for(int i = row - 2; i >= 0; i--){
			double sum = 0;
			int startCol = i + 1;
			int endCol = row;
			
			for(int j = startCol; j < endCol; j++){
				sum += U[i][j] * result[j];
			}
			
			result[i] = (U[i][endCol] - sum) / U[i][i];
		}
		return result;
	}
	
	
}

然后就是求逆矩阵啦！

package com.zhp.third.矩阵的逆;

import com.zhp.common.Methods;
import com.zhp.third.LU分解.LU;
import com.zhp.third.解线性方程组.LinearEquation;

/**
 * @author 郑海鹏
 * @since 2014/10/18 21:23
 * @email 284967632@qq.com
 */
public class Inverse {

	public static void main(String[] args) {
		double[][] a = new double[][] { { 1, 1, 2 }, { 1, 2, 1 }, { 2, 1, 1 } };
		double[][] result = getInverseMatrix(a);
		Methods.round(result); // 四舍五入掉舍入误差。
		Methods.print(result);
	}

	/**
	 * 获得逆矩阵
	 * 
	 * @exception 本方法不检查传入的矩阵有没有逆矩阵
	 */
	public static double[][] getInverseMatrix(double[][] matrix) {
		// 因为 A = LU, 得 I = AA' = LUA', 得 L(UA') = LB = I
		// 利用getRightMatrix() 求得B；
		// 又：UA' = B
		// 再次利用 getRightMatrix() 求得A';
		
		double[][] L = LU.getLU(matrix).get(0);
		double[][] U = LU.getLU(matrix).get(1);
		
		// 构建单位矩阵
		double[][] I = new double[matrix.length][matrix.length];
		for(int i = 0; i < matrix.length; i++){
			I[i][i] = 1;
		}
		
		double[][] B = getRightMatrix(L, I);
		
		double[][] inverse = getRightMatrix(U, B);
		
		return inverse;
	}

	/**
	 * [leftMatrix][rightMatrix] = [resultMatrix]
	 * 
	 * @return rightMatrix
	 */
	public static double[][] getRightMatrix(double[][] leftMatrix, double[][] resultMatix) {
		// 构成等式的前提是：左边矩阵的行数 == 结果矩阵的行数
		if (leftMatrix.length != resultMatix.length) {
			System.out.println("错误！左边矩阵的行数不等于结果矩阵的行数！");
			return null;
		}

		int row = leftMatrix[0].length; // 行数等于左边矩阵的列数
		int col = resultMatix[0].length; // 列数等于结果矩阵的列数
		double[][] rightMatrix = new double[row][col];

		double[] colMatrix = new double[row]; // 这是右边矩阵的一列，col个它构成rightMatrix
		double[] colResult = new double[row]; // 这是结果矩阵的一列

		// 按列求得colMatrix
		for (int colIndex = 0; colIndex < col; colIndex++) {

			// 取出这一列的结果矩阵
			for (int i = 0; i < row; i++) {
				colResult[i] = resultMatix[colIndex][i];
			}

			// 获得这一列的右矩阵
			colMatrix = getRightMatrix(leftMatrix, colResult);

			// 把这一列的值放到右矩阵中
			for (int i = 0; i < row; i++) {
				rightMatrix[colIndex][i] = colMatrix[i];
			}
		}

		return rightMatrix;
	}

	/**
	 * [leftMatrix][rightMatrix] = [resultMatrix] 意义是：一个二维矩阵 x 一个列向量 = 矩阵在列向量上的值
	 * 
	 * @return rightMatrix
	 */
	public static double[] getRightMatrix(double[][] leftMatrix, double[] resultMatrix) {
		// 构建增广矩阵
		int row = leftMatrix.length;
		int col = leftMatrix[0].length + 1;
		double[][] augMatrix = new double[row][col];

		for (int i = 0; i < row; i++) {
			for (int j = 0; j < col - 1; j++) {
				augMatrix[i][j] = leftMatrix[i][j];
			}
			augMatrix[i][col - 1] = resultMatrix[i];
		}

		// 增广矩阵的解就是所求
		double[] rightMatrix = LinearEquation.getResult(augMatrix);

		return rightMatrix;
	}


}