Java中两个矩阵相乘

Question

我目前正在开发一个表示矩阵的类，它表示任何通用的 mxn 矩阵。我已经计算出加法和标量乘法，但我正在努力开发两个矩阵的乘法。矩阵的数据保存在一个二维数组中。

方法看起来有点像这样：

   public Matrix multiply(Matrix A) {
            ////code
   }

它将返回产品矩阵。这是右边的乘法。所以，如果我调用 A.multiply(B) 那么它将返回矩阵 AB，B 在右边。

我还不需要担心检查是否在给定矩阵上定义了乘法，我可以假设我会得到正确维度的矩阵。

有没有人知道一个简单的算法，甚至可能用伪代码来执行乘法过程？

提前致谢。

Answer 1

矩阵 A (l x m) 和 B (m x n) 的乘积在数学上定义为由元素组成的矩阵 C (l x n)：

        m
c_i_j = ∑  a_i_k * b_k_j
       k=1

因此，如果您对速度不太感兴趣，您可能会对直接的 O(n^3) 实现感到满意：

  for (int i=0; i<l; ++i)
    for (int j=0; j<n; ++j)
      for (int k=0; k<m; ++k)
        c[i][j] += a[i][k] * b[k][j]  

如果您想加快速度，您可能需要检查其他替代方案，例如 Strassen 算法（请参阅：Strassen 算法）。

但请注意 - 特别是当您在现代处理器架构上乘以小矩阵时，速度在很大程度上取决于矩阵数据和乘法顺序的排列方式，以充分利用缓存行。

我非常怀疑是否有机会通过使用虚拟机来影响这个因素，所以我不确定是否要考虑这一点。

Answer 2

Java。矩阵乘法。

这里是“执行乘法过程的代码”。使用不同大小的矩阵进行测试。

public class Matrix {

/**
 * Matrix multiplication method.
 * @param m1 Multiplicand
 * @param m2 Multiplier
 * @return Product
 */
    public static double[][] multiplyByMatrix(double[][] m1, double[][] m2) {
        int m1ColLength = m1[0].length; // m1 columns length
        int m2RowLength = m2.length;    // m2 rows length
        if(m1ColLength != m2RowLength) return null; // matrix multiplication is not possible
        int mRRowLength = m1.length;    // m result rows length
        int mRColLength = m2[0].length; // m result columns length
        double[][] mResult = new double[mRRowLength][mRColLength];
        for(int i = 0; i < mRRowLength; i++) {         // rows from m1
            for(int j = 0; j < mRColLength; j++) {     // columns from m2
                for(int k = 0; k < m1ColLength; k++) { // columns from m1
                    mResult[i][j] += m1[i][k] * m2[k][j];
                }
            }
        }
        return mResult;
    }

    public static String toString(double[][] m) {
        String result = "";
        for(int i = 0; i < m.length; i++) {
            for(int j = 0; j < m[i].length; j++) {
                result += String.format("%11.2f", m[i][j]);
            }
            result += "\n";
        }
        return result;
    }

    public static void main(String[] args) {
        // #1
        double[][] multiplicand = new double[][] {
                {3, -1, 2},
                {2,  0, 1},
                {1,  2, 1}
        };
        double[][] multiplier = new double[][] {
                {2, -1, 1},
                {0, -2, 3},
                {3,  0, 1}
        };
        System.out.println("#1\n" + toString(multiplyByMatrix(multiplicand, multiplier)));
        // #2
        multiplicand = new double[][] {
                {1, 2, 0},
                {-1, 3, 1},
                {2, -2, 1}
        };
        multiplier = new double[][] {
                {2},
                {-1},
                {1}
        };
        System.out.println("#2\n" + toString(multiplyByMatrix(multiplicand, multiplier)));
        // #3
        multiplicand = new double[][] {
                {1, 2, -1},
                {0,  1, 0}
        };
        multiplier = new double[][] {
                {1, 1, 0, 0},
                {0, 2, 1, 1},
                {1, 1, 2, 2}
        };
        System.out.println("#3\n" + toString(multiplyByMatrix(multiplicand, multiplier)));
    }
}

输出：

#1
      12.00      -1.00       2.00
       7.00      -2.00       3.00
       5.00      -5.00       8.00

#2
       0.00
      -4.00
       7.00

#3
       0.00       4.00       0.00       0.00
       0.00       2.00       1.00       1.00

Answer 3

在这个答案中，我创建了一个名为 Matrix 的类，另一个名为 MatrixOperations 的类定义了可以对矩阵执行的各种操作（当然，行操作除外）。但我将从 MatrixOperations 中提取乘法代码。完整的项目可以在我的GitHub页面here找到。

下面是Matrix类的定义。

package app.matrix;

import app.matrix.util.MatrixException;

public class Matrix {

private double[][] entries;

public void setEntries(double[][] entries) {
    this.entries = entries;
}

private String name;

public double[][] getEntries() {
    return entries;
}

public String getName() {
    return name;
}

public void setName(String name) {
    this.name = name;
}

public class Dimension {
    private int rows;
    private int columns;

    public int getRows() {
        return rows;
    }

    public void setRows(int rows) {
        this.rows = rows;
    }

    public int getColumns() {
        return columns;
    }

    public void setColumns(int columns) {
        this.columns = columns;
    }

    public Dimension(int rows, int columns) {
        this.setRows(rows);
        this.setColumns(columns);
    }

    @Override
    public boolean equals(Object obj) {
        if(obj instanceof Dimension){
            return (this.getColumns() == ((Dimension) obj).getColumns()) && (this.getRows() == ((Dimension) obj).getRows());
        }
        return false;
    }
}

private Dimension dimension;

public Dimension getDimension() {
    return dimension;
}

public void setDimension(Dimension dimension) {
    this.dimension = dimension;
}

public Matrix(int dimension, String name) throws MatrixException {
    if (dimension == 0) throw new MatrixException(ZERO_UNIT_DIMENSION);
    else this.setEntries(new double[Math.abs(dimension)][Math.abs(dimension)]);
    this.setDimension(new Dimension(dimension, dimension));
    this.setName(name);
}

public Matrix(int dimensionH, int dimensionV, String name) throws MatrixException {
    if (dimensionH == 0 || dimensionV == 0) throw new MatrixException(ZERO_UNIT_DIMENSION);
    else this.setEntries(new double[Math.abs(dimensionH)][Math.abs(dimensionV)]);
    this.setDimension(new Dimension(dimensionH, dimensionV));
    this.setName(name);

}

private static final String OVERFLOW_ITEMS_MSG = "The values are too many for the matrix's specified dimensions";
private static final String ZERO_UNIT_DIMENSION = "Zero cannot be a value for a dimension";

public Matrix(int dimensionH, int dimensionV, String name, double... values) throws MatrixException {
    if (dimensionH == 0 || dimensionV == 0) throw new MatrixException(ZERO_UNIT_DIMENSION);
    else if (values.length > dimensionH * dimensionV) throw new MatrixException(Matrix.OVERFLOW_ITEMS_MSG);
    else this.setEntries(new double[Math.abs(dimensionH)][Math.abs(dimensionV)]);
    this.setDimension(new Dimension(dimensionH, dimensionV));
    this.setName(name);

    int iterator = 0;
    int j;
    for (int i = 0; i < dimensionH; i++) {
        j = 0;
        while (j < dimensionV) {
            this.entries[i][j] = values[iterator];
            j++;
            iterator++;
        }
    }
}

public Matrix(Dimension dimension) throws MatrixException {
    this(dimension.getRows(), dimension.getColumns(), null);
}

public static Matrix identityMatrix(int dim) throws MatrixException {
    if (dim == 0) throw new MatrixException(ZERO_UNIT_DIMENSION);

    double[] i = new double[dim * dim];
    int constant = dim + 1;
    for (int j = 0; j < i.length; j = j + constant) {
        i[j] = 1.0;
    }

    return new Matrix(dim, dim, null, i);
}

public String toString() {

    StringBuilder builder = new StringBuilder("Matrix \"" + (this.getName() == null ? "Null Matrix" : this.getName()) + "\": {\n");

    for (int i = 0; i < this.getDimension().getRows(); i++) {
        for (int j = 0; j < this.getDimension().getColumns(); j++) {
            if (j == 0) builder.append("\t");
            builder.append(this.entries[i][j]);
            if (j != this.getDimension().getColumns() - 1)
                builder.append(", ");
        }
        if (i != this.getDimension().getRows()) builder.append("\n");
    }

    builder.append("}");

    return builder.toString();
}

public boolean isSquare() {
    return this.getDimension().getColumns() == this.getDimension().getRows();
}

}

这是MatrixOperations中矩阵乘法的代码方法

public static Matrix multiply(Matrix matrix1, Matrix matrix2) throws MatrixException {

    if (matrix1.getDimension().getColumns() != matrix2.getDimension().getRows())
        throw new MatrixException(MATRIX_MULTIPLICATION_ERROR_MSG);

    Matrix retVal = new Matrix(matrix1.getDimension().getRows(), matrix2.getDimension().getColumns(), matrix1.getName() + " x " + matrix2.getName());


    for (int i = 0; i < matrix1.getDimension().getRows(); i++) {
        for (int j = 0; j < matrix2.getDimension().getColumns(); j++) {
            retVal.getEntries()[i][j] = sum(arrayProduct(matrix1.getEntries()[i], getColumnMatrix(matrix2, j)));
        }
    }

    return retVal;
}

下面是 sum、arrayProduct 和 getColumnMatrix 方法的代码

private static double sum(double... values) {
    double sum = 0;
    for (double value : values) {
        sum += value;
    }
    return sum;
}

private static double[] arrayProduct(double[] arr1, double[] arr2) throws MatrixException {
    if (arr1.length != arr2.length) throw new MatrixException("Array lengths must be the same");
    double[] retVal = new double[arr1.length];
    for (int i = 0; i < arr1.length; i++) {
        retVal[i] = arr1[i] * arr2[i];
    }

    return retVal;
}


private static double[] getColumnMatrix(Matrix matrix, int col) {
    double[] ret = new double[matrix.getDimension().getRows()];
    for (int i = 0; i < matrix.getDimension().getRows(); i++) {
        ret[i] = matrix.getEntries()[i][col];
    }
    return ret;
}

Answer 4

对多个任意维数组尝试此代码并打印它。觉得这个比较简单，任何人都可以理解。

public class Test
{

  public static void main(String[] args)
  {
    int[][] array1 = {
        { 1, 4, -2 },
        { 3, 5, -6 },
        { 4, 5, 2 }
    };

    int[][] array2 = {
        { 5, 2, 8, -1 },
        { 3, 6, 4, 5 },
        { -2, 9, 7, -3 }
    };
    Test test = new Test();
    test.printArray(test.multiplication(array1, array2));
  }

  private int[][] multiplication(int[][] array1, int[][] array2)
  {
    int r1, r2, c1, c2;
    r1 = array1.length;
    c1 = array1[0].length;

    r2 = array2.length;
    c2 = array2[0].length;

    int[][] result;
    if (c1 != r2)
    {
      System.out.println("Error!");
      result = new int[0][0];
    }
    else
    {
      result = new int[r1][c2];

      for (int i = 0; i < r1; i++)//2
      {
        for (int j = 0; j < c2; j++)//4
        {
          for (int k = 0; k < c1; k++)
          {
            result[i][j] += array1[i][k] * array2[k][j];
          }
        }
      }
    }

    return result;
  }

  private void printArray(int[][] array)
  {
    for (int[] arr : array)
    {
      for (int element : arr)
      {
        System.out.print(element + " ");
      }
      System.out.println();
    }
  }
}