我正在尝试计算矩阵(任意大小)的行列式，用于 self 编码/面试练习。我的第一次尝试是使用递归，这使我实现了以下实现:

import java.util.Scanner.*;
public class Determinant {

    double A[][];
    double m[][];
    int N;
    int start;
    int last;

    public Determinant (double A[][], int N, int start, int last){
            this.A = A;
            this.N = N;
            this.start = start;
            this.last = last;
    }

    public double[][] generateSubArray (double A[][], int N, int j1){
            m = new double[N-1][];
            for (int k=0; k<(N-1); k++)
                    m[k] = new double[N-1];

            for (int i=1; i<N; i++){
                  int j2=0;
                  for (int j=0; j<N; j++){
                      if(j == j1)
                            continue;
                      m[i-1][j2] = A[i][j];
                      j2++;
                  }
              }
            return m;
    }
    /*
     * Calculate determinant recursively
     */
    public double determinant(double A[][], int N){
        double res;

        // Trivial 1x1 matrix
        if (N == 1) res = A[0][0];
        // Trivial 2x2 matrix
        else if (N == 2) res = A[0][0]*A[1][1] - A[1][0]*A[0][1];
        // NxN matrix
        else{
            res=0;
            for (int j1=0; j1<N; j1++){
                 m = generateSubArray (A, N, j1);
                 res += Math.pow(-1.0, 1.0+j1+1.0) * A[0][j1] * determinant(m, N-1);
            }
        }
        return res;
    }
}

到目前为止一切都很好，它给了我一个正确的结果。现在我想通过使用多个线程来计算这个行列式值来优化我的代码。 我尝试使用 Java Fork/Join 模型对其进行并行化。这是我的方法:

@Override
protected Double compute() {
     if (N < THRESHOLD) {
         result = computeDeterminant(A, N);
         return result;
     }

     for (int j1 = 0; j1 < N; j1++){
          m = generateSubArray (A, N, j1);
          ParallelDeterminants d = new ParallelDeterminants (m, N-1);
          d.fork();
          result += Math.pow(-1.0, 1.0+j1+1.0) * A[0][j1] * d.join();
     }

     return result;
}

public double computeDeterminant(double A[][], int N){
    double res;

    // Trivial 1x1 matrix
    if (N == 1) res = A[0][0];
    // Trivial 2x2 matrix
    else if (N == 2) res = A[0][0]*A[1][1] - A[1][0]*A[0][1];
    // NxN matrix
    else{
        res=0;
        for (int j1=0; j1<N; j1++){
             m = generateSubArray (A, N, j1);
             res += Math.pow(-1.0, 1.0+j1+1.0) * A[0][j1] * computeDeterminant(m, N-1);
        }
    }
    return res;
}

/*
 * Main function
 */
public static void main(String args[]){
    double res;
    ForkJoinPool pool = new ForkJoinPool();
    ParallelDeterminants d = new ParallelDeterminants();
    d.inputData();
    long starttime=System.nanoTime();
    res = pool.invoke (d);
    long EndTime=System.nanoTime();

    System.out.println("Seq Run = "+ (EndTime-starttime)/100000);
    System.out.println("the determinant valaue is  " + res);
}

但是对比性能后发现，Fork/Join方式的性能很差，矩阵维数越高越慢(相比第一种方式)。开销在哪里？谁能阐明如何改进这一点？

最佳答案

使用此类，您可以计算具有任何维度的矩阵的行列式

此类使用许多不同的方法使矩阵成为三角形，然后计算它的行​​列式。它可用于 500 x 500 甚至更大的高维矩阵。此类的优点是您可以获得BigDecimal 中的结果，因此没有无穷大并且您将始终获得准确的答案。顺便说一句，使用许多不同的方法并避免递归导致更快的方法和更高的答案性能。希望对您有所帮助。

import java.math.BigDecimal;


public class DeterminantCalc {

private double[][] matrix;
private int sign = 1;


DeterminantCalc(double[][] matrix) {
    this.matrix = matrix;
}

public int getSign() {
    return sign;
}

public BigDecimal determinant() {

    BigDecimal deter;
    if (isUpperTriangular() || isLowerTriangular())
        deter = multiplyDiameter().multiply(BigDecimal.valueOf(sign));

    else {
        makeTriangular();
        deter = multiplyDiameter().multiply(BigDecimal.valueOf(sign));

    }
    return deter;
}


/*  receives a matrix and makes it triangular using allowed operations
    on columns and rows
*/
public void makeTriangular() {

    for (int j = 0; j < matrix.length; j++) {
        sortCol(j);
        for (int i = matrix.length - 1; i > j; i--) {
            if (matrix[i][j] == 0)
                continue;

            double x = matrix[i][j];
            double y = matrix[i - 1][j];
            multiplyRow(i, (-y / x));
            addRow(i, i - 1);
            multiplyRow(i, (-x / y));
        }
    }
}


public boolean isUpperTriangular() {

    if (matrix.length < 2)
        return false;

    for (int i = 0; i < matrix.length; i++) {
        for (int j = 0; j < i; j++) {
            if (matrix[i][j] != 0)
                return false;

        }

    }
    return true;
}


public boolean isLowerTriangular() {

    if (matrix.length < 2)
        return false;

    for (int j = 0; j < matrix.length; j++) {
        for (int i = 0; j > i; i++) {
            if (matrix[i][j] != 0)
                return false;

        }

    }
    return true;
}


public BigDecimal multiplyDiameter() {

    BigDecimal result = BigDecimal.ONE;
    for (int i = 0; i < matrix.length; i++) {
        for (int j = 0; j < matrix.length; j++) {
            if (i == j)
                result = result.multiply(BigDecimal.valueOf(matrix[i][j]));

        }

    }
    return result;
}


// when matrix[i][j] = 0 it makes it's value non-zero
public void makeNonZero(int rowPos, int colPos) {

    int len = matrix.length;

    outer:
    for (int i = 0; i < len; i++) {
        for (int j = 0; j < len; j++) {
            if (matrix[i][j] != 0) {
                if (i == rowPos) { // found "!= 0" in it's own row, so cols must be added
                    addCol(colPos, j);
                    break outer;

                }
                if (j == colPos) { // found "!= 0" in it's own col, so rows must be added
                    addRow(rowPos, i);
                    break outer;
                }
            }
        }
    }
}


//add row1 to row2 and store in row1
public void addRow(int row1, int row2) {

    for (int j = 0; j < matrix.length; j++)
        matrix[row1][j] += matrix[row2][j];
}


//add col1 to col2 and store in col1
public void addCol(int col1, int col2) {

    for (int i = 0; i < matrix.length; i++)
        matrix[i][col1] += matrix[i][col2];
}


//multiply the whole row by num
public void multiplyRow(int row, double num) {

    if (num < 0)
        sign *= -1;


    for (int j = 0; j < matrix.length; j++) {
        matrix[row][j] *= num;
    }
}


//multiply the whole column by num
public void multiplyCol(int col, double num) {

    if (num < 0)
        sign *= -1;

    for (int i = 0; i < matrix.length; i++)
        matrix[i][col] *= num;

}


// sort the cols from the biggest to the lowest value
public void sortCol(int col) {

    for (int i = matrix.length - 1; i >= col; i--) {
        for (int k = matrix.length - 1; k >= col; k--) {
            double tmp1 = matrix[i][col];
            double tmp2 = matrix[k][col];

            if (Math.abs(tmp1) < Math.abs(tmp2))
                replaceRow(i, k);
        }
    }
}


//replace row1 with row2
public void replaceRow(int row1, int row2) {

    if (row1 != row2)
        sign *= -1;

    double[] tempRow = new double[matrix.length];

    for (int j = 0; j < matrix.length; j++) {
        tempRow[j] = matrix[row1][j];
        matrix[row1][j] = matrix[row2][j];
        matrix[row2][j] = tempRow[j];
    }
}


//replace col1 with col2
public void replaceCol(int col1, int col2) {

    if (col1 != col2)
        sign *= -1;

    System.out.printf("replace col%d with col%d, sign = %d%n", col1, col2, sign);
    double[][] tempCol = new double[matrix.length][1];

    for (int i = 0; i < matrix.length; i++) {
        tempCol[i][0] = matrix[i][col1];
        matrix[i][col1] = matrix[i][col2];
        matrix[i][col2] = tempCol[i][0];
    }
} }

此类从用户那里接收一个 n x n 的矩阵，然后计算它的行​​列式。它还显示了解决方案和最终的三角矩阵。

 import java.math.BigDecimal;
 import java.text.NumberFormat;
 import java.util.Scanner;


public class DeterminantTest {

public static void main(String[] args) {

    String determinant;

    //generating random numbers
    /*int len = 300;
    SecureRandom random = new SecureRandom();
    double[][] matrix = new double[len][len];

    for (int i = 0; i < len; i++) {
        for (int j = 0; j < len; j++) {
            matrix[i][j] = random.nextInt(500);
            System.out.printf("%15.2f", matrix[i][j]);
        }
    }
    System.out.println();*/

    /*double[][] matrix = {
        {1, 5, 2, -2, 3, 2, 5, 1, 0, 5},
        {4, 6, 0, -2, -2, 0, 1, 1, -2, 1},
        {0, 5, 1, 0, 1, -5, -9, 0, 4, 1},
        {2, 3, 5, -1, 2, 2, 0, 4, 5, -1},
        {1, 0, 3, -1, 5, 1, 0, 2, 0, 2},
        {1, 1, 0, -2, 5, 1, 2, 1, 1, 6},
        {1, 0, 1, -1, 1, 1, 0, 1, 1, 1},
        {1, 5, 5, 0, 3, 5, 5, 0, 0, 6},
        {1, -5, 2, -2, 3, 2, 5, 1, 1, 5},
        {1, 5, -2, -2, 3, 1, 5, 0, 0, 1}
    };
    */

    double[][] matrix = menu();

    DeterminantCalc deter = new DeterminantCalc(matrix);

    BigDecimal det = deter.determinant();

    determinant = NumberFormat.getInstance().format(det);

    for (int i = 0; i < matrix.length; i++) {
        for (int j = 0; j < matrix.length; j++) {
            System.out.printf("%15.2f", matrix[i][j]);
        }
        System.out.println();
    }

    System.out.println();
    System.out.printf("%s%s%n", "Determinant: ", determinant);
    System.out.printf("%s%d", "sign: ", deter.getSign());

}


public static double[][] menu() {

    Scanner scanner = new Scanner(System.in);

    System.out.print("Matrix Dimension: ");
    int dim = scanner.nextInt();

    double[][] inputMatrix = new double[dim][dim];

    System.out.println("Set the Matrix: ");
    for (int i = 0; i < dim; i++) {
        System.out.printf("%5s%d%n", "row", i + 1);
        for (int j = 0; j < dim; j++) {

            System.out.printf("M[%d][%d] = ", i + 1, j + 1);
            inputMatrix[i][j] = scanner.nextDouble();
        }
        System.out.println();
    }
    scanner.close();

    return inputMatrix;
}}

关于java - 计算矩阵行列式，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/16602350/