Doolittle 算法：LU 分解

在数值分析和线性代数中，LU 分解(其中“LU”代表“下上”，也称为 LU 分解)将矩阵分解为下三角矩阵和上三角矩阵的乘积。计算机通常使用 LU 分解来求解线性方程组的平方系统，这也是矩阵求逆或矩阵行列式计算的关键步骤。 LU 分解是由数学家 Tadeusz Banachiewicz 在 1938 年引入的。

设 A 为方阵。 LU 分解是指将 A 分解为两个因子，下三角矩阵 L 和上三角矩阵 U， A=LU 。

杜立特算法：
总是可以将方阵分解为下三角矩阵和上三角矩阵。即， [A] = [L][U]
Doolittle 的方法提供了一种将 A 因子分解为 LU 分解的替代方法，而无需经历高斯消除的麻烦。
对于一般的 n×n 矩阵 A，我们假设存在 LU 分解，并明确写出 L 和 U 的形式。然后，我们从 A=LU 所需的乘法产生的方程中系统地求解 L 和 U 中的条目。

U 矩阵的项由下式给出：

$\forall j\\ i=0 \rightarrow U_{ij} = A_{ij}\\ i>0 \rightarrow U_{ij} = A_{ij} - \sum_{k=0}^{i-1}L_{ik}U_{kj}$

L 矩阵的项：
$\forall i\\ j=0 \rightarrow L_{ij} = \frac{A_{ij}}{U_{jj}}\\ j>0 \rightarrow L_{ij} = \frac{A_{ij} - \sum_{k=0}^{j-1}L_{ik}U_{kj}}{U_{jj}}$

示例：

Input : 

Output :

C++

// C++ Program to decompose a matrix into
// lower and upper triangular matrix
#include 
using namespace std;
 
const int MAX = 100;
 
void luDecomposition(int mat[][MAX], int n)
{
    int lower[n][n], upper[n][n];
    memset(lower, 0, sizeof(lower));
    memset(upper, 0, sizeof(upper));
 
    // Decomposing matrix into Upper and Lower
    // triangular matrix
    for (int i = 0; i < n; i++)
    {
        // Upper Triangular
        for (int k = i; k < n; k++)
        {
            // Summation of L(i, j) * U(j, k)
            int sum = 0;
            for (int j = 0; j < i; j++)
                sum += (lower[i][j] * upper[j][k]);
 
            // Evaluating U(i, k)
            upper[i][k] = mat[i][k] - sum;
        }
 
        // Lower Triangular
        for (int k = i; k < n; k++)
        {
            if (i == k)
                lower[i][i] = 1; // Diagonal as 1
            else
            {
                // Summation of L(k, j) * U(j, i)
                int sum = 0;
                for (int j = 0; j < i; j++)
                    sum += (lower[k][j] * upper[j][i]);
 
                // Evaluating L(k, i)
                lower[k][i]
                    = (mat[k][i] - sum) / upper[i][i];
            }
        }
    }
 
    // setw is for displaying nicely
    cout << setw(6)
         << "      Lower Triangular"
         << setw(32)
         << "Upper Triangular" << endl;
 
    // Displaying the result :
    for (int i = 0; i < n; i++)
    {
        // Lower
        for (int j = 0; j < n; j++)
            cout << setw(6) << lower[i][j] << "\t";
        cout << "\t";
 
        // Upper
        for (int j = 0; j < n; j++)
            cout << setw(6) << upper[i][j] << "\t";
        cout << endl;
    }
}
 
// Driver code
int main()
{
    int mat[][MAX]
        = { { 2, -1, -2 }, { -4, 6, 3 }, { -4, -2, 8 } };
 
    luDecomposition(mat, 3);
    return 0;
}

Java

// Java Program to decompose a matrix into
// lower and upper triangular matrix
class GFG {
 
    static int MAX = 100;
    static String s = "";
    static void luDecomposition(int[][] mat, int n)
    {
        int[][] lower = new int[n][n];
        int[][] upper = new int[n][n];
 
        // Decomposing matrix into Upper and Lower
        // triangular matrix
        for (int i = 0; i < n; i++)
        {
            // Upper Triangular
            for (int k = i; k < n; k++)
            {
                // Summation of L(i, j) * U(j, k)
                int sum = 0;
                for (int j = 0; j < i; j++)
                    sum += (lower[i][j] * upper[j][k]);
 
                // Evaluating U(i, k)
                upper[i][k] = mat[i][k] - sum;
            }
 
            // Lower Triangular
            for (int k = i; k < n; k++)
            {
                if (i == k)
                    lower[i][i] = 1; // Diagonal as 1
                else
                {
                    // Summation of L(k, j) * U(j, i)
                    int sum = 0;
                    for (int j = 0; j < i; j++)
                        sum += (lower[k][j] * upper[j][i]);
 
                    // Evaluating L(k, i)
                    lower[k][i]
                        = (mat[k][i] - sum) / upper[i][i];
                }
            }
        }
 
        // setw is for displaying nicely
        System.out.println(setw(2) + "     Lower Triangular"
                           + setw(10) + "Upper Triangular");
 
        // Displaying the result :
        for (int i = 0; i < n; i++)
        {
            // Lower
            for (int j = 0; j < n; j++)
                System.out.print(setw(4) + lower[i][j]
                                 + "\t");
            System.out.print("\t");
 
            // Upper
            for (int j = 0; j < n; j++)
                System.out.print(setw(4) + upper[i][j]
                                 + "\t");
            System.out.print("\n");
        }
    }
    static String setw(int noOfSpace)
    {
        s = "";
        for (int i = 0; i < noOfSpace; i++)
            s += " ";
        return s;
    }
 
    // Driver code
    public static void main(String arr[])
    {
        int mat[][] = { { 2, -1, -2 },
                        { -4, 6, 3 },
                        { -4, -2, 8 } };
 
        luDecomposition(mat, 3);
    }
}
 
/* This code contributed by PrinciRaj1992 */

Python3

# Python3 Program to decompose
# a matrix into lower and
# upper triangular matrix
MAX = 100
 
 
def luDecomposition(mat, n):
 
    lower = [[0 for x in range(n)]
             for y in range(n)]
    upper = [[0 for x in range(n)]
             for y in range(n)]
 
    # Decomposing matrix into Upper
    # and Lower triangular matrix
    for i in range(n):
 
        # Upper Triangular
        for k in range(i, n):
 
            # Summation of L(i, j) * U(j, k)
            sum = 0
            for j in range(i):
                sum += (lower[i][j] * upper[j][k])
 
            # Evaluating U(i, k)
            upper[i][k] = mat[i][k] - sum
 
        # Lower Triangular
        for k in range(i, n):
            if (i == k):
                lower[i][i] = 1  # Diagonal as 1
            else:
 
                # Summation of L(k, j) * U(j, i)
                sum = 0
                for j in range(i):
                    sum += (lower[k][j] * upper[j][i])
 
                # Evaluating L(k, i)
                lower[k][i] = int((mat[k][i] - sum) /
                                  upper[i][i])
 
    # setw is for displaying nicely
    print("Lower Triangular\t\tUpper Triangular")
 
    # Displaying the result :
    for i in range(n):
 
        # Lower
        for j in range(n):
            print(lower[i][j], end="\t")
        print("", end="\t")
 
        # Upper
        for j in range(n):
            print(upper[i][j], end="\t")
        print("")
 
 
# Driver code
mat = [[2, -1, -2],
       [-4, 6, 3],
       [-4, -2, 8]]
 
luDecomposition(mat, 3)
 
# This code is contributed by mits

C#

// C# Program to decompose a matrix into
// lower and upper triangular matrix
using System;
 
class GFG {
 
    static int MAX = 100;
    static String s = "";
    static void luDecomposition(int[, ] mat, int n)
    {
        int[, ] lower = new int[n, n];
        int[, ] upper = new int[n, n];
 
        // Decomposing matrix into Upper and Lower
        // triangular matrix
        for (int i = 0; i < n; i++)
        {
            // Upper Triangular
            for (int k = i; k < n; k++)
            {
                // Summation of L(i, j) * U(j, k)
                int sum = 0;
                for (int j = 0; j < i; j++)
                    sum += (lower[i, j] * upper[j, k]);
 
                // Evaluating U(i, k)
                upper[i, k] = mat[i, k] - sum;
            }
 
            // Lower Triangular
            for (int k = i; k < n; k++)
            {
                if (i == k)
                    lower[i, i] = 1; // Diagonal as 1
                else
                {
                    // Summation of L(k, j) * U(j, i)
                    int sum = 0;
                    for (int j = 0; j < i; j++)
                        sum += (lower[k, j] * upper[j, i]);
 
                    // Evaluating L(k, i)
                    lower[k, i]
                        = (mat[k, i] - sum) / upper[i, i];
                }
            }
        }
 
        // setw is for displaying nicely
        Console.WriteLine(setw(2) + "     Lower Triangular"
                          + setw(10) + "Upper Triangular");
 
        // Displaying the result :
        for (int i = 0; i < n; i++)
        {
            // Lower
            for (int j = 0; j < n; j++)
                Console.Write(setw(4) + lower[i, j] + "\t");
            Console.Write("\t");
 
            // Upper
            for (int j = 0; j < n; j++)
                Console.Write(setw(4) + upper[i, j] + "\t");
            Console.Write("\n");
        }
    }
    static String setw(int noOfSpace)
    {
        s = "";
        for (int i = 0; i < noOfSpace; i++)
            s += " ";
        return s;
    }
 
    // Driver code
    public static void Main(String[] arr)
    {
        int[, ] mat = { { 2, -1, -2 },
                        { -4, 6, 3 },
                        { -4, -2, 8 } };
 
        luDecomposition(mat, 3);
    }
}
 
// This code is contributed by Princi Singh

Javascript

 
// Javascript Program to decompose a matrix
// into lower and upper triangular matrix   
// function MAX = 100;
var s = "";
 
function luDecomposition(mat, n)
{
    var lower = Array(n).fill(0).map(
           x => Array(n).fill(0));
    var upper = Array(n).fill(0).map(
           x => Array(n).fill(0));
 
    // Decomposing matrix into Upper and
    // Lower triangular matrix
    for(var i = 0; i < n; i++)
    {
         
        // Upper Triangular
        for(var k = i; k < n; k++)
        {
             
            // Summation of L(i, j) * U(j, k)
            var sum = 0;
            for(var j = 0; j < i; j++)
                sum += (lower[i][j] * upper[j][k]);
 
            // Evaluating U(i, k)
            upper[i][k] = mat[i][k] - sum;
        }
 
        // Lower Triangular
        for(var k = i; k < n; k++)
        {
            if (i == k)
             
                // Diagonal as 1
                lower[i][i] = 1;
            else
            {
                 
                // Summation of L(k, j) * U(j, i)
                var sum = 0;
                for(var j = 0; j < i; j++)
                    sum += (lower[k][j] * upper[j][i]);
 
                // Evaluating L(k, i)
                lower[k][i] = parseInt((mat[k][i] - sum) /
                                      upper[i][i]);
            }
        }
    }
 
    // Setw is for displaying nicely
    document.write(setw(2) + "Lower Triangular" +
                   setw(10) + "Upper Triangular<br>");
 
    // Displaying the result :
    for(var i = 0; i < n; i++)
    {
        
        // Lower
        for(var j = 0; j < n; j++)
            document.write(setw(4) +
                           lower[i][j] + "\t");
                            
        document.write(setw(10));
 
        // Upper
        for(var j = 0; j < n; j++)
            document.write(setw(4) +
                           upper[i][j] + "\t");
                            
        document.write("<br>");
    }
}
 
function setw(noOfSpace)
{
    var s = "";
    for(i = 0; i < noOfSpace; i++)
        s += " ";
         
    return s;
}
 
// Driver code
var mat = [ [ 2, -1, -2 ],
            [ -4, 6, 3 ],
            [ -4, -2, 8 ] ];
 
luDecomposition(mat, 3);
 
// This code is contributed by Amit Katiyar
 

输出：

Lower Triangular    Upper Triangular
1    0      0       2    -1     -2    
-2   1      0       0     4     -1    
-2  -1      1       0     0      3