数学：《线性代数》矩阵初等行变换及其应用（线性方程求解）

背景

高斯小朋友确实聪明，发明了高斯消元法，进而引入了线性代数，本文给出矩阵的初等行变换代码实现及其应用（线性方程求解）。

实现

代码

  1 using System;

  2 using System.Collections.Generic;

  3 using System.Linq;

  4 using System.Text;

  5 using System.Threading.Tasks;

  6 

  7 namespace DataStuctureStudy.Matrixes

  8 {

  9     class MatrixTest

 10     {

 11         public static void Test()

 12         {

 13             var matrix =

 14                 new Matrix(3, 4)

 15                 .SetRowData(0, 1, 2, 3, 2)

 16                 .SetRowData(1, 1, 2, 7, 5)

 17                 .SetRowData(2, 4, 9, 2, 6);

 18 

 19             matrix.Dispaly();

 20 

 21             matrix.Simplify().Dispaly();

 22         }

 23         class Matrix

 24         {

 25             private readonly int _m;

 26             private readonly int _n;

 27             private readonly double[][] _data;

 28 

 29             public Matrix(int m, int n)

 30             {

 31                 _m = m;

 32                 _n = n;

 33                 _data = new double[m][];

 34                 for (var i = 0; i < m; i++)

 35                 {

 36                     _data[i] = new double[n];

 37                 }

 38             }

 39 

 40             public Matrix SetRowData(int row, params double[] values)

 41             {

 42                 Array.Copy(values, _data[row], _n);

 43 

 44                 return this;

 45             }

 46 

 47             public Matrix SwapRow(int rowX, int rowY)

 48             {

 49                 var temp = _data[rowX];

 50                 _data[rowX] = _data[rowY];

 51                 _data[rowY] = temp;

 52 

 53                 return this;

 54             }

 55 

 56             public Matrix MultiplyRow(int row, double operand)

 57             {

 58                 for (var i = 0; i < _n; i++)

 59                 {

 60                     _data[row][i] *= operand;

 61                 }

 62 

 63                 return this;

 64             }

 65 

 66             public Matrix AddSourceRowToTargetRow(int sourceRow, double operand, int targetRow)

 67             {

 68                 for (var i = 0; i < _n; i++)

 69                 {

 70                     _data[targetRow][i] += _data[sourceRow][i] * operand;

 71                 }

 72 

 73                 return this;

 74             }

 75 

 76             public Matrix Simplify()

 77             {

 78                 for (var col = 0; col < _n; col++)

 79                 {

 80                     if (col >= _m)

 81                     {

 82                         break;

 83                     }

 84 

 85                     if (_data[col][col] == 0)

 86                     {

 87                         var nonZeroRowIndex = this.FindNonZeroRowIndex(col, col + 1);

 88                         if (nonZeroRowIndex == -1)

 89                         {

 90                             break;

 91                         }

 92                         this.SwapRow(col, nonZeroRowIndex);

 93                     }

 94 

 95                     this.MultiplyRow(col, 1 / _data[col][col]);

 96 

 97                     for (var row = 0; row < _m; row++)

 98                     {

 99                         if (row == col)

100                         {

101                             continue;

102                         }

103                         this.AddSourceRowToTargetRow(col, -1 * _data[row][col], row);

104                     }

105                 }

106 

107                 return this;

108             }

109 

110             public int FindNonZeroRowIndex(int col, int startRow)

111             {

112                 if (startRow >= _m)

113                 {

114                     return -1;

115                 }

116 

117                 for (var i = startRow; i < _m; i++)

118                 {

119                     if (_data[i][col] != 0)

120                     {

121                         return i;

122                     }

123                 }

124 

125                 return -1;

126             }

127 

128             public void Dispaly()

129             {

130                 Console.WriteLine();

131                 foreach (var row in _data)

132                 {

133                     Console.WriteLine("[ " + String.Join(" , ", row) + " ]");

134                 }

135                 Console.WriteLine();

136             }

137         }

138     }

139 }

输出结果