PLU Decomposition
PLU分解的优点是,能够将Ax=b的矩阵,转换成Ly=b, Ux = y
的形式。当我们改变系数矩阵b时,此时因为矩阵L和U均是固定
的,所以总能高效的求出矩阵的解。
// LU.cpp : Defines the entry point for the console application.
//
/************************************************
* Author: JohnsonDu
* From: Institute of Computing Technology
* University of Chinese Academy of Science
* Time: 2014-10-7
* Content: PLU decomposition
*************************************************/ #include "stdafx.h" #define MAXN 5005
#define eps 1e-9 // 精度
int n, m;
double mat[MAXN][MAXN]; // 输入矩阵
double matL[MAXN][MAXN]; // 矩阵L
double matU[MAXN][MAXN]; // 矩阵U
int matP[MAXN][MAXN]; // 矩阵P
int seq[MAXN]; // 记录行变换
//double vecB[MAXN];
//double vecY[MAXN];
//double vecX[MAXN];
//double matPb[MAXN]; void menu()
{
printf("----------------PLU Factorization---------------\n");
printf("| Please follow the instruction |\n");
printf("| to determine the LU decomposition |\n");
printf("| PA = LU |\n");
printf("------------------------------------------------\n\n"); } void initLMatrix()
{
memset(matU, 0, sizeof(matU));
memset(matL, 0, sizeof(matL));
memset(matP, 0, sizeof(matP));
} void padLMatrix()
{
for(int i = 0; i < n; i ++)
matL[i][i] = 1.0;
} inline double Abs(double x)
{
return x < 0 ? -x : x;
} void displayLU()
{
// 输出矩阵L
printf("\n----------------------\n");
printf("Matrix L follows: \n");
for(int i = 0; i < n; i ++)
{
for(int j = 0; j < (n < m ? n : m); j ++)
printf("%.3f ", matL[i][j]);
printf("\n");
} // 输出矩阵U
printf("\nMatrix U follows: \n");
for(int i = 0; i < (n < m ? n : m); i ++)
{
for(int j = 0; j < m; j ++)
printf("%.3f ", matU[i][j]);
printf("\n");
} // 输出矩阵P
printf("\nMatrix P follows: \n");
for(int i = 0; i < n; i ++)
{
for(int j = 0; j < n; j ++)
printf("%d ", matP[i][j]);
printf("\n");
}
printf("----------------------\n");
} /*
// 输出LU的过程及终于解
void displaySolution()
{
// 输出矩阵Pb
printf("\nMatrix Pb follows: \n");
for(int i = 0; i < n; i ++)
{
printf("%.3f\n", matPb[i]);
} // 输出向量y
printf("\nVector Y follows: \n");
for(int i = 0; i < n; i ++)
{
printf("%.3f\n", vecY[i]);
}
printf("\n"); // 输出解向量x
printf("\Vector X follows: \n");
for(int i = 0; i < n; i ++)
{
printf("%.3f\n", vecX[i]);
}
printf("\n");
}
*/ // 交换元素
inline void swap(int &a, int &b)
{
int t = a;
a = b;
b = t;
} // 高斯消元部分
void gauss()
{
int i;
int col;
int max_r; col = 0; //处理的当前列 // 从第一行開始进行消元
// k为处理的当前行
for(int k = 0; k < n && col < min(n, m); k ++, col ++)
{
// 寻找当前col列的绝对值最大值
max_r = k;
for(i = k + 1; i < n; i ++)
if(Abs(mat[i][col]) > Abs(mat[max_r][col]))
max_r = i; // 进行行交换
if(max_r != k)
{
for(int j = col; j < m; j ++)
swap(mat[k][j], mat[max_r][j]);
swap(seq[k], seq[max_r]);
for(int j = 0; j < n; j ++)
swap(matL[k][j], matL[max_r][j]);
} // 当前主元为零, 继续
if(Abs(mat[k][col]) < eps){
continue;
} // 消元部分,并获得L矩阵
for(int i = k + 1; i < n; i ++)
{ double t = mat[i][col] / mat[k][col];
matL[i][col] = t;
for(int j = col; j < m; j ++)
mat[i][j] -= t * mat[k][j];
} } // 为矩阵U进行赋值
for(int i = 0; i < n; i ++)
for(int j = 0; j < m; j ++)
matU[i][j] = mat[i][j]; // 生成矩阵P
for(int i = 0; i < n; i ++) matP[i][seq[i]] = 1.0; // 为矩阵L加入对角线元素
padLMatrix();
} /*
// 计算Pb的值
void calcPb()
{
for(int i = 0; i < n; i ++)
matPb[i] = 0.0;
//cout << "-----------" << endl;
for(int i = 0; i < n; i ++)
{
double t = 0.0;
for(int j = 0; j < n; j ++)
{
t = t + 1.0 * matP[i][j] * vecB[j];
//cout << t << endl;
//cout << matP[i][j] * vecB[j] << "---" << endl;
}
matPb[i] = t;
//cout << matPb[i] << endl;
}
} // 计算Ly = Pb, y向量
void calcY()
{
vecY[0] = matPb[0];
for(int i = 1; i < n; i ++)
{
double t = 0.0;
for(int j = 0; j < i; j ++)
t += vecY[j] * matL[i][j];
vecY[i] = matPb[i] - t;
}
} // 计算Ux = y, y向量
void calcX()
{
vecX[n-1] = vecY[n-1] / matU[n-1][n-1];
for(int i = n-2; i >= 0; i --)
{
double t = 0.0;
for(int j = n-1; j > i; j --)
t += vecX[j] * matU[i][j];
vecX[i] = (vecY[i] - t) / matU[i][i];
}
}
*/ int _tmain(int argc, _TCHAR* argv[])
{
menu();
while(true)
{
printf("Please input the matrix's dimension n & m: ");
// 输入矩阵的行n和列m
scanf("%d%d", &n, &m);
printf("Please input the matrix: \n"); // 输入矩阵
for(int i = 0; i < n; i ++)
{
for(int j = 0; j < m; j ++)
cin >> mat[i][j];
seq[i] = i;
} // 初始化为0
initLMatrix(); // 高斯消元
gauss(); // 输出P, L, U矩阵
displayLU();
system("pause");
system("cls");
menu(); /*
//此处是输入b,求取x, y 和 pb
while(true){
printf("please input vector b(whose length equals to %d): \n", n);
for(int i = 0; i < n; i ++) cin >> vecB[i];
calcPb();
calcY();
calcX();
displaySolution();
}
*/
} return 0;
}
当中stdafx.h的头文件:
// stdafx.h : include file for standard system include files,
// or project specific include files that are used frequently, but
// are changed infrequently
// #pragma once
#define _CRT_SECURE_NO_WARNINGS #include "targetver.h" #include <stdio.h>
#include <tchar.h>
#include <iostream>
using namespace std;
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