C# 高斯消元項目運用

時間 2020-05-10

標籤 c# 高斯消元項目運用欄目 C# 简体版

原文原文鏈接

C# 高斯消元項目運用

最近項目涉及到一個需求，須要把指定數量的多個商品，混合裝入到多個不一樣型號的箱子中(每種型號的箱子裝入商品的種類和個數是固定的)。這就涉及到解多元一次方程

針對多元一次方程通常用高斯消元去處理，當消元后仍有不能消掉的元則須要解多元一次方程

1. 將數據轉換爲二維數組，消元代碼以下

樣本數據以下

| 2, 3, 4, 3, 21 | => | 2, 3, 4, 3, 21 | => | 2, 3, 4, 3 , 21 | | 3, 1, 6, 2, 17 | => | 0, 7, 0, 5, 29 | => | 0, 7, 0, 5 , 29 | | 1, 3, 2, 1, 12 | => | 0, -3,0, 1, -3 | => | 0, 0, 0, -22, -66 | html

<div style="font-family: Arial; font-size: 13; color: gainsboro; background: #1e1e1e;">           var rowLength = arr.GetLength(0);//行數             var colLength = arr.GetLength(1);//列數             for (int mainCol = 0; mainCol < colLength - 1; mainCol++) //按照列，一列一列的消元             {                 var mainRow = mainCol; //主元行=主元列                 for (int row = mainRow + 1; row < rowLength; row++) // 從主元的下一行開始 行循環                 {                     if (row >= (colLength - 1))                         break;                     /*                         n1*X + n2*Y = a                            m1*X + m2*Y = b                           =>  n1*m1*X + n2*m1Y = a*m1                         =>  m1*n1*X + m2*n1*Y = b*n1                     */                     //若是主元參數爲0（ 找出此列不爲0的行） 運用行相加  變換0參數                     if (arr[mainRow, mainCol] == 0)                     {                         for (int rowi = mainRow + 1; rowi < rowLength; rowi++)                         {                             if (arr[rowi, mainCol] != 0)                             {                                 for (int colj = mainCol; colj < colLength; colj++)                                 {                                     arr[mainRow, colj] += arr[rowi, colj];                                 }                                 break;                             }                         }                     }                     if (arr[row, mainCol] == 0)                         continue; //當前行 此列已是0 繼續消元下一行                     var m = GetMinCommonMultiple(arr[mainCol, mainCol], arr[row, mainCol]);                     int factorMain = m / arr[mainCol, mainCol];//主列的因子                       int factor = m / arr[row, mainCol]; //待消元列因子                     for (int colk = mainCol; colk < colLength; colk++)                     {                         arr[row, colk] = arr[mainCol, colk] * factorMain - arr[row, colk] * factor;                     }                 }             }</div>數組

2. 消元后，找出仍然沒法消除的元，解一元屢次不定方程

經過消元最終能夠肯定 a[3]=3 ,a[1]=2 a[0]和a[2]解有多個須要解不定方程思路是將多元方程切割成二元方程而後窮舉求解。列入3x+2y+4z+6a=18, 令 w=2y+4z+6a,再令 2w=2(y+2z+3a) ,帶入公式得 3x+2w=16,窮舉求出一個解 x=2 w=6 ，而後再解 y+2z+3a=6。一樣的方法再次處理
代碼以下

<div style="font-family: Arial; font-size: 13; color: gainsboro; background: #1e1e1e;">      /// <summary>         /// 解不定方程         /// </summary>         public static bool ResoveIndefiniteEquation(int[] arr, int value, int startIndex, ref int[] result, ref int count)         {             count++;//遞歸計數，防止無限遞歸             if (arr == null || arr.Length == 0)                 return false;             if (arr.Length == 1)             {                 var reValue = value / arr[0];                 if (value % arr[0] == 0 && reValue >= MinResove && reValue <= MaxResove)                 {                     result[startIndex] = reValue;                     return true;                 }                 return false;             }             //獲取第二部分元的最大公約數             var commonDivisor = arr[1];             for (int i = 1; i < (arr.Length - 1); i++)             {                 commonDivisor = GetCommonDivisor(commonDivisor, arr[i + 1]);             }             for (int i = MinResove; i <= MaxResove; i++)             {                 var currValue = value - i * arr[0];                 if (currValue % commonDivisor == 0)                 {                     var newArr = new int[arr.Length - 1];                     Array.Copy(arr, 1, newArr, 0, arr.Length - 1);                     result[startIndex] = i;                     var currArr = ResoveIndefiniteEquation(newArr, currValue, startIndex + 1, ref result, ref count);                     if (currArr)                         return true;                     if (count > 1000000)                     {                         Console.WriteLine("遞歸太深沒法求解");                         return false;                     }                 }             }             return false;         }</div>ide

3.完整代碼

<div style="font-family: Arial; font-size: 13; color: gainsboro; background: #1e1e1e;">    public class GaussHelper     {         /// <summary>         /// 不定方程解的最小值         /// </summary>         private const int MinResove = 1;         /// <summary>         /// 不定方程解的最大值         /// </summary>         private const int MaxResove = 100;         /// <summary>         /// 利用高斯消元求解一元屢次方程組         /// </summary>         /// <param name="arr"></param>         public static int?[] ResoveGauss(int[,] arr)         {             if (arr == null) throw new ArgumentNullException("arr");             var rowLength = arr.GetLength(0);//行數             var colLength = arr.GetLength(1);//列數             if (colLength < 2) throw new IndexOutOfRangeException("arr.GetLength(1)的值必須大於等於2");             var result = new int?[colLength - 1]; #if DEBUG             Console.WriteLine("解一元屢次方程：");             PrintArr(arr); #endif             for (int mainCol = 0; mainCol < colLength - 1; mainCol++) //按照列，一列一列的消元             {                 var mainRow = mainCol; //主元行=主元列                 for (int row = mainRow + 1; row < rowLength; row++) // 從主元的下一行開始 行循環                 {                     if (row >= (colLength - 1))                         break;                     /*                         n1*X + n2*Y = a                            m1*X + m2*Y = b                           =>  n1*m1*X + n2*m1Y = a*m1                         =>  m1*n1*X + m2*n1*Y = b*n1                     */                     //若是主元參數爲0（ 找出此列不爲0的行） 運用行相加  變換0參數                     if (arr[mainRow, mainCol] == 0)                     {                         for (int rowi = mainRow + 1; rowi < rowLength; rowi++)                         {                             if (arr[rowi, mainCol] != 0)                             {                                 for (int colj = mainCol; colj < colLength; colj++)                                 {                                     arr[mainRow, colj] += arr[rowi, colj];                                 }                                 break;                             }                         }                     }                     if (arr[row, mainCol] == 0)                         continue; //當前行 此列已是0 繼續消元下一行                     var m = GetMinCommonMultiple(arr[mainCol, mainCol], arr[row, mainCol]);                     int factorMain = m / arr[mainCol, mainCol];//主列的因子                       int factor = m / arr[row, mainCol]; //待消元列因子                     for (int colk = mainCol; colk < colLength; colk++)                     {                         arr[row, colk] = arr[mainCol, colk] * factorMain - arr[row, colk] * factor;                     }                 } #if DEBUG                 if (rowLength > mainCol + 1)                     PrintArr(arr); #endif             }             //回代過程              //回代行 判斷 行數是否大於列數減-1  取到能夠計算結果的回代行             var backRow = rowLength > (colLength - 1) ? colLength - 1 : rowLength;             var backRowIndex = backRow - 1;             //回代行的結果值             var backValue = arr[backRowIndex, colLength - 1];             //判斷消解行 有多少個變元 ，存儲 變元的 索引             var listColIndex = new List<int>();             for (int col = 0; col < colLength - 1; col++)             {                 if (arr[backRowIndex, col] != 0)                 {                     listColIndex.Add(col);                 }             }             if (listColIndex.Count == 0)                 return null;             //根據須要求解的個數 解不定方程處理             if (listColIndex.Count > 1)             {                 var resoveArr = listColIndex.Select(x => arr[backRowIndex, x]).ToArray();                 var resoveResultArr = new int[listColIndex.Count];                 var count = 0;                 var flag = ResoveIndefiniteEquation(resoveArr, backValue, 0, ref resoveResultArr, ref count); #if DEBUG                 Console.WriteLine();                 Console.WriteLine("--解不定方程");                 var sb = new StringBuilder();                 for (int i = 0; i < colLength; i++)                 {                     sb.Append("********");                 }                 Console.WriteLine(sb.ToString());                 Console.WriteLine($"方程: {string.Join(",\t", resoveArr)} = {backValue}");                 Console.WriteLine($"結果: {string.Join(",\t", resoveResultArr)}");                 Console.WriteLine($"計算: {string.Join(",\t", resoveResultArr.Select((x, index) => $"{x * resoveArr[index]}"))}");                 Console.WriteLine($"遞歸次數: {count}");                 Console.WriteLine(sb.ToString());                 Console.WriteLine(); #endif                 if (!flag)                     return null;                 for (int i = 0; i < resoveResultArr.Length; i++)                 {                     result[listColIndex[i]] = resoveResultArr[i];                 }             }             if (listColIndex.Count == 1)             {                 if (backValue % arr[backRowIndex, listColIndex[0]] == 0)                 {                     result[listColIndex[0]] = backValue / arr[backRowIndex, listColIndex[0]];                 }                 else                 { #if DEBUG                     var number = backValue * 1.0 / arr[backRowIndex, listColIndex[0]];                     Console.WriteLine($"第{backRow}行 {backValue}/{arr[backRowIndex, listColIndex[0]]} = {number}, 不能被被整除計算失敗"); #endif                     return null;                 }             }             for (int row = backRowIndex - 1; row >= 0; row--) //從倒數第二行開始往前迭代             {                 if (arr[row, row] == 0)                     continue;                 int addResult = 0;                 var currlist = new List<int>();                 for (int j = row; j < colLength - 2; j++)//j=2   j 最大值爲2，每行未知數可能不止一個，故須要遍歷已知的未知數並代入                 {                     if (!result[j + 1].HasValue && arr[row, j + 1] != 0)                         currlist.Add(j + 1);//把沒有計算出結果的列的索引存入                     else                         addResult += result[j + 1].GetValueOrDefault() * arr[row, j + 1];//k表明計算的行，j+1表明的列，係數與解要對應，故都爲 j+1                 }                 var calculateValue = arr[row, colLength - 1] - addResult;                 //發現沒有計算出結果的列 解不定方程                 if (currlist.Count > 0)                 {                     currlist.Add(row);                     var resoveArr = currlist.Select(x => arr[row, x]).ToArray();                     var resoveResultArr = new int[currlist.Count];                     var count = 0;                     var flag = ResoveIndefiniteEquation(resoveArr, calculateValue, 0, ref resoveResultArr, ref count); #if DEBUG                     Console.WriteLine();                     Console.WriteLine($"----發現第{row + 1}行 第({string.Join(",", currlist)})列 沒有計算出結果，須要解不定方程");                     var sb = new StringBuilder();                     for (int i = 0; i < colLength; i++)                     {                         sb.Append("********");                     }                     Console.WriteLine(sb.ToString());                     Console.WriteLine($"方程: {string.Join(",\t", resoveArr)} = {calculateValue}");                     Console.WriteLine($"結果: {string.Join(",\t", resoveResultArr)}");                     Console.WriteLine($"計算: {string.Join(",\t", resoveResultArr.Select((x, index) => $"{x * resoveArr[index]}"))}");                     Console.WriteLine($"遞歸次數: {count}");                     Console.WriteLine(sb.ToString());                     Console.WriteLine(); #endif                     if (!flag)                         return null;                     for (int i = 0; i < resoveResultArr.Length; i++)                     {                         result[currlist[i]] = resoveResultArr[i];                     }                 }                 else                 {                     if (calculateValue % arr[row, row] == 0)                     {                         result[row] = calculateValue / arr[row, row];//本行的未知數用本行最右邊數-本行已知未知數代入係數之差 再除以本未知數係數                     }                     else                     { #if DEBUG                         var number = calculateValue * 1.0 / arr[row, row];                         Console.WriteLine($"第{row + 1}行 {calculateValue}/{arr[row, row]} = {number}, 不能被被整除計算失敗"); #endif                         return null;                     }                 }             } #if DEBUG             Console.WriteLine("----------結果------------");             Console.Write($"結果: {string.Join(",", result)}"); #endif             return result;         }         /// <summary>         /// 解不定方程         /// </summary>         public static bool ResoveIndefiniteEquation(int[] arr, int value, int startIndex, ref int[] result, ref int count)         {             count++;//遞歸計數，防止無限遞歸             if (arr == null || arr.Length == 0)                 return false;             if (arr.Length == 1)             {                 var reValue = value / arr[0];                 if (value % arr[0] == 0 && reValue >= MinResove && reValue <= MaxResove)                 {                     result[startIndex] = reValue;                     return true;                 }                 return false;             }             //獲取第二部分元的最大公約數             var commonDivisor = arr[1];             for (int i = 1; i < (arr.Length - 1); i++)             {                 commonDivisor = GetCommonDivisor(commonDivisor, arr[i + 1]);             }             for (int i = MinResove; i <= MaxResove; i++)             {                 var currValue = value - i * arr[0];                 if (currValue % commonDivisor == 0)                 {                     var newArr = new int[arr.Length - 1];                     Array.Copy(arr, 1, newArr, 0, arr.Length - 1);                     result[startIndex] = i;                     var currArr = ResoveIndefiniteEquation(newArr, currValue, startIndex + 1, ref result, ref count);                     if (currArr)                         return true;                     if (count > 1000000)                     {                         Console.WriteLine("遞歸太深沒法求解");                         return false;                     }                 }             }             return false;         }         private static void PrintArr(int[,] arr)         {             var rowLength = arr.GetLength(0);//行數             var colLength = arr.GetLength(1);//列數             var sb = new StringBuilder();             for (int i = 0; i < colLength; i++)             {                 sb.Append("--------");             }             Console.WriteLine(sb.ToString());             for (int i = 0; i < rowLength; i++)             {                 for (int j = 0; j < colLength; j++)                 {                     Console.Write(arr[i, j]);                     var str = j == (colLength - 1) ? "" : ",";                     Console.Write($" {str}\t ");                 }                 Console.WriteLine();             }         }         /// <summary>         /// 求最大公約數         /// </summary>         /// <param name="a"></param>         /// <param name="b"></param>         /// <returns></returns>         public static int GetCommonDivisor(int a, int b)         {             if (a == 0 || b == 0) return 0;             if (Math.Abs(a) < Math.Abs(b))             {                 var temp = a;                 a = b;                 b = temp;             }             return (a % b == 0) ? b : GetCommonDivisor(a % b, b);         }         /// <summary>         /// 求最小公倍數          /// </summary>         /// <param name="a"></param>         /// <param name="b"></param>         /// <returns></returns>         public static int GetMinCommonMultiple(int a, int b)         {             var commonDivisor = GetCommonDivisor(a, b);             if (commonDivisor == 0)                 return 0;             return a * b / commonDivisor;         }     }</div>ui

4. 運行結果

<div style="font-family: Arial; font-size: 13; color: gainsboro; background: #1e1e1e;">        static void Main(string[] args)         {             var arr = new int[,]             {                 { 2, 3, 4, 3, 21 },                 { 3, 1, 6, 2, 17 },                 { 1, 3, 2, 1, 12},             };             GaussHelper.ResoveGauss(arr);             Console.ReadKey();         }</div>spa

原文出處：https://www.cnblogs.com/castyuan/p/10251885.htmlxml

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