如何计算矩阵行列式？ n * n或仅5 * 5

Question

大家。我需要找到矩阵n*n（或5*5）行列式。我有一个从Pascal翻译过的函数，但有INDEX OUT OF RANGE EXCEPTION。有人能帮助我吗？

这是我的代码：

public static double DET(double[,] a, int n)
    {
        int i, j, k;
        double det = 0;
        for (i = 0; i < n - 1; i++)
        {   
            for (j = i + 1; j < n + 1; j++)
            {
                det = a[j, i] / a[i, i];
                for (k = i; k < n; k++)
                    a[j, k] = a[j, k] - det * a[i, k]; // Here's exception
            }
        }
        det = 1;
        for (i = 0; i < n; i++)
            det = det * a[i, i];
            return det;
    }

Thanx寻求帮助。

Answer 1

计算n * n行列式的工作解决方案如下：

using System;

internal class MatrixDecompositionProgram
{
    private static void Main(string[] args)
    {
        float[,] m = MatrixCreate(4, 4);
        m[0, 0] = 3.0f; m[0, 1] = 7.0f; m[0, 2] = 2.0f; m[0, 3] = 5.0f;
        m[1, 0] = 1.0f; m[1, 1] = 8.0f; m[1, 2] = 4.0f; m[1, 3] = 2.0f;
        m[2, 0] = 2.0f; m[2, 1] = 1.0f; m[2, 2] = 9.0f; m[2, 3] = 3.0f;
        m[3, 0] = 5.0f; m[3, 1] = 4.0f; m[3, 2] = 7.0f; m[3, 3] = 1.0f;

        int[] perm;
        int toggle;

        float[,] luMatrix = MatrixDecompose(m, out perm, out toggle);

        float[,] lower = ExtractLower(luMatrix);
        float[,] upper = ExtractUpper(luMatrix);

        float det = MatrixDeterminant(m);

        Console.WriteLine("Determinant of m computed via decomposition = " + det.ToString("F1"));
    }

    // --------------------------------------------------------------------------------------------------------------
    private static float[,] MatrixCreate(int rows, int cols)
    {
        // allocates/creates a matrix initialized to all 0.0. assume rows and cols > 0
        // do error checking here
        float[,] result = new float[rows, cols];
        return result;
    }

    // --------------------------------------------------------------------------------------------------------------
    private static float[,] MatrixDecompose(float[,] matrix, out int[] perm, out int toggle)
    {
        // Doolittle LUP decomposition with partial pivoting.
        // rerturns: result is L (with 1s on diagonal) and U; perm holds row permutations; toggle is +1 or -1 (even or odd)
        int rows = matrix.GetLength(0);
        int cols = matrix.GetLength(1);

        //Check if matrix is square
        if (rows != cols)
            throw new Exception("Attempt to MatrixDecompose a non-square mattrix");

        float[,] result = MatrixDuplicate(matrix); // make a copy of the input matrix

        perm = new int[rows]; // set up row permutation result
        for (int i = 0; i < rows; ++i) { perm[i] = i; } // i are rows counter

        toggle = 1; // toggle tracks row swaps. +1 -> even, -1 -> odd. used by MatrixDeterminant

        for (int j = 0; j < rows - 1; ++j) // each column, j is counter for coulmns
        {
            float colMax = Math.Abs(result[j, j]); // find largest value in col j
            int pRow = j;
            for (int i = j + 1; i < rows; ++i)
            {
                if (result[i, j] > colMax)
                {
                    colMax = result[i, j];
                    pRow = i;
                }
            }

            if (pRow != j) // if largest value not on pivot, swap rows
            {
                float[] rowPtr = new float[result.GetLength(1)];

                //in order to preserve value of j new variable k for counter is declared
                //rowPtr[] is a 1D array that contains all the elements on a single row of the matrix
                //there has to be a loop over the columns to transfer the values
                //from the 2D array to the 1D rowPtr array.
                //----tranfer 2D array to 1D array BEGIN

                for (int k = 0; k < result.GetLength(1); k++)
                {
                    rowPtr[k] = result[pRow, k];
                }

                for (int k = 0; k < result.GetLength(1); k++)
                {
                    result[pRow, k] = result[j, k];
                }

                for (int k = 0; k < result.GetLength(1); k++)
                {
                    result[j, k] = rowPtr[k];
                }

                //----tranfer 2D array to 1D array END

                int tmp = perm[pRow]; // and swap perm info
                perm[pRow] = perm[j];
                perm[j] = tmp;

                toggle = -toggle; // adjust the row-swap toggle
            }

            if (Math.Abs(result[j, j]) < 1.0E-20) // if diagonal after swap is zero . . .
                return null; // consider a throw

            for (int i = j + 1; i < rows; ++i)
            {
                result[i, j] /= result[j, j];
                for (int k = j + 1; k < rows; ++k)
                {
                    result[i, k] -= result[i, j] * result[j, k];
                }
            }
        } // main j column loop

        return result;
    } // MatrixDecompose

    // --------------------------------------------------------------------------------------------------------------
    private static float MatrixDeterminant(float[,] matrix)
    {
        int[] perm;
        int toggle;
        float[,] lum = MatrixDecompose(matrix, out perm, out toggle);
        if (lum == null)
            throw new Exception("Unable to compute MatrixDeterminant");
        float result = toggle;
        for (int i = 0; i < lum.GetLength(0); ++i)
            result *= lum[i, i];

        return result;
    }

    // --------------------------------------------------------------------------------------------------------------
    private static float[,] MatrixDuplicate(float[,] matrix)
    {
        // allocates/creates a duplicate of a matrix. assumes matrix is not null.
        float[,] result = MatrixCreate(matrix.GetLength(0), matrix.GetLength(1));
        for (int i = 0; i < matrix.GetLength(0); ++i) // copy the values
            for (int j = 0; j < matrix.GetLength(1); ++j)
                result[i, j] = matrix[i, j];
        return result;
    }

    // --------------------------------------------------------------------------------------------------------------
    private static float[,] ExtractLower(float[,] matrix)
    {
        // lower part of a Doolittle decomposition (1.0s on diagonal, 0.0s in upper)
        int rows = matrix.GetLength(0); int cols = matrix.GetLength(1);
        float[,] result = MatrixCreate(rows, cols);
        for (int i = 0; i < rows; ++i)
        {
            for (int j = 0; j < cols; ++j)
            {
                if (i == j)
                    result[i, j] = 1.0f;
                else if (i > j)
                    result[i, j] = matrix[i, j];
            }
        }
        return result;
    }

    // --------------------------------------------------------------------------------------------------------------
    private static float[,] ExtractUpper(float[,] matrix)
    {
        // upper part of a Doolittle decomposition (0.0s in the strictly lower part)
        int rows = matrix.GetLength(0); int cols = matrix.GetLength(1);
        float[,] result = MatrixCreate(rows, cols);
        for (int i = 0; i < rows; ++i)
        {
            for (int j = 0; j < cols; ++j)
            {
                if (i <= j)
                    result[i, j] = matrix[i, j];
            }
        }
        return result;
    }
}

Answer 2

for (j = i + 1; j < n + 1; j++)

最后J值将大于数组大小。因此，您必须重新检查数组大小以及所有索引是如何从pascal转换的。

Answer 3

为什么在下载可用的C＃代码

时会费心去翻译

• http://www.codeproject.com/KB/cs/CSML.aspx
• http://www.codeproject.com/KB/recipes/AdvancedMatrixLibrary.aspx
• http://www.codeproject.com/KB/recipes/psdotnetmatrix.aspx

Answer 4

对于更大的矩阵，您可能需要运行 Bareiss算法来计算行列式：

• http://en.wikipedia.org/wiki/Bareiss_algorithm

Answer 5

我认为这个算法不好，至少对于5x5矩阵的计算。即使我们纠正了这个

  

for（j = i + 1; j

就像这样

  

表示（j = i + 1; j

然后编写完整的代码，例如：

using System;

public class Matrix
{
    private int row_matrix; //number of rows for matrix
    private int column_matrix; //number of colums for matrix
    private double[,] matrix; //holds values of matrix itself

    //create r*c matrix and fill it with data passed to this constructor
    public Matrix(double[,] double_array)
    {
        matrix = double_array;
        row_matrix = matrix.GetLength(0);
        column_matrix = matrix.GetLength(1);
        Console.WriteLine("Contructor which sets matrix size {0}*{1} and fill it with initial data executed.", row_matrix, column_matrix);
    }

    //returns total number of rows
    public int countRows()
    {
        return row_matrix;
    }

    //returns total number of columns
    public int countColumns()
    {
        return column_matrix;
    }

    //returns value of an element for a given row and column of matrix
    public double readElement(int row, int column)
    {
        return matrix[row, column];
    }

    //sets value of an element for a given row and column of matrix
    public void setElement(double value, int row, int column)
    {
        matrix[row, column] = value;
    }

    public double deterMatrix()
    {
        double det = 0;
        double value = 0;
        int i, j, k;

        i = row_matrix;
        j = column_matrix;
        int n = i;

        if (i != j)
        {
            Console.WriteLine("determinant can be calculated only for sqaure matrix!");
            return det;
        }

        for (i = 0; i < n - 1; i++)
        {
            for (j = i + 1; j < n; j++)
            {
                det = (this.readElement(j, i) / this.readElement(i, i));

                for (k = i; k < n; k++)
                {
                    value = this.readElement(j, k) - det * this.readElement(i, k);

                    this.setElement(value, j, k);
                }
            }
        }
        det = 1;
        for (i = 0; i < n; i++)
            det = det * this.readElement(i, i);

        return det;
    }
}

internal class Program
{
    private static void Main(string[] args)
    {
        Matrix mat03 = new Matrix(new[,]
        {
            {1.0, 2.0, -1.0},
            {-2.0, -5.0, -1.0},
            {1.0, -1.0, -2.0},
        });

        Matrix mat04 = new Matrix(new[,]
        {
            {1.0, 2.0, 1.0, 3.0},
            {-2.0, -5.0, -2.0, 1.0},
            {1.0, -1.0, -3.0, 2.0},
            {4.0, -1.0, -3.0, 1.0},
        });

        Matrix mat05 = new Matrix(new[,]
        {
            {1.0, 2.0, 1.0, 2.0, 3.0},
            {2.0, 1.0, 2.0, 2.0, 1.0},
            {3.0, 1.0, 3.0, 1.0, 2.0},
            {1.0, 2.0, 4.0, 3.0, 2.0},
            {2.0, 2.0, 1.0, 2.0, 1.0},
        });

        double determinant = mat03.deterMatrix();
        Console.WriteLine("determinant is: {0}", determinant);

        determinant = mat04.deterMatrix();
        Console.WriteLine("determinant is: {0}", determinant);

        determinant = mat05.deterMatrix();
        Console.WriteLine("determinant is: {0}", determinant);
    }
}

结果是： 决定因素是：-8 决定因素是：-142 决定因素是：-NaN

因为除零而发生NaN（我调试了它） 对于某些非常具体的输入，这可能是正常的，但在一般情况下，这不是一个好的算法。

因此，它适用于3x3和4x4但不适用于5x5

我写这篇文章给任何可能会遇到这个问题的人，以避免在尝试实施或修复错误算法的时候失去几个小时。