線性方程組數(shù)值解法

1、計算方法實驗題目:班級:學號:目錄計算方法實驗仁1實驗目的3.2實驗步驟3.2.1環(huán)境配置：3.2.2添加頭文件3.2.3主要模塊3.3代碼4.3.1主程序部分4.3.2多項式方程部分4.3.3核心算法部分8.3.4數(shù)據(jù)結(jié)構部分1.24運行結(jié)果144.1列主元高斯消去法運行結(jié)果 144.2LU三角分解法運行結(jié)果 154.3雅克比迭代法運行結(jié)果 16邊界情況調(diào)試165總結(jié)1.8輸入輸出18列主元高斯消元法1.8雅克比迭代法186參考資料181實驗目的1. 通過編程加深對列主元高斯消去法、LU三角分解法和雅克比迭代法等求解多 項式方程方法的理解2. 觀察上述三種方法的計算穩(wěn)定性和求解精度并比較各種

2、方法利弊2實驗步驟2.1環(huán)境配置：VS2013, C+控制臺程序2.2添加頭文件#i nclude "stdio.h"#i nclude "stdlib.h"#include "stdafx.h"#in clude<iostream>2.3主要模塊程序一共分成三層，最底層是數(shù)據(jù)結(jié)構部分，負責存儲數(shù)據(jù)，第二層是交互 部分，即多項式方程部分，負責輸入輸出獲得數(shù)據(jù)，最上層是核心的算法部分， 負責處理已獲得的數(shù)據(jù)。具體功能如下：數(shù)據(jù)結(jié)構部分數(shù)據(jù)結(jié)構部分是整個程序的最底層，負責存儲部分。因數(shù)組作為數(shù)據(jù)元素插 入和刪除操作較少，而順序

3、表空間利用率大且查看方便，故此程序選用二維 順序表保存系數(shù)。數(shù)據(jù)結(jié)構文件中寫的是有關其的所有基本操作以供其他文 件調(diào)用。多項式方程部分多項式方程部分是程序的第二層，容是有關方程組的所有函數(shù)、構建方程、 輸出方程等等，同時在此文件中獲得方程系數(shù)并儲存，同時此文件還負責顯 示菜單部分。算法部分此文件負責核心算法，處于整個程序最上層部分，負責列主元高斯消去法、LU三角分解法和雅克比迭代法的具體實現(xiàn)過程。 通過調(diào)用方程文件的函數(shù), 將獲得的數(shù)據(jù)進行處理運算，可以得到結(jié)果返回給方程主函數(shù)和輸出的第二總結(jié)：主函數(shù)負責獲取方程系數(shù)并顯示，算法和方程作為后臺程序，順序表 作為存儲手段。3代碼3.1主程序部分/

4、 Solutionoflinearquations.cpp :定義控制臺應用程序的入口點。/#include "stdio.h"#i nclude "stdlib.h"#include "stdafx.h"#i nclude "sque ncelist.h"#i nclude "equati on .h"#in clude "algorithm.h"#in clude<iostream>int _tmain(int argc, _TCHAR* argv)while

5、 (Exflag)GetEquatio n(); ShowMe nu();return 0;3.2多項式方程部分方程部分頭文件#ifndef _EQUATION_H#defi ne _EQUATION_H #include "stdio.h"#i nclude "stdlib.h" #in clude "sque ncelist.h" extern int Xnumbers; extern int Fnumber; extern int Exflag;extern datacoa *A;void GetEquation(void);v

6、oid ShowMenu(void);void printfunction(datacoa *A);void printresx(datacoa *A);void Tip(void);#endif方程部分CPP文件#include "stdafx.h"#include "equation.h"#include "math.h"#include "algorithm.h"#include "stdio.h"#include "stdlib.h"#include<iostr

7、eam>#include <iomanip> using namespace std; / 全局變量 int Xnumbers = 0; int Fnumber = 0; int Exflag = 1; datacoa *A;/ 多項式函數(shù)系數(shù) / void GetEquation(void)int i, j,flag=1;float x;A = InitStruct();while (flag)cout << "方程未知量和解總個數(shù)： " << endl; cin >> Xnumbers;cout << &

8、quot;方程個數(shù)： " << endl;cin >> Fnumber;cout << "輸入方程系數(shù) ,輸入00結(jié)束（如輸入 2 1 5 或者2 1 5 3 4 1 00 "<< endl;cout << "00):" << endl;cin >> x;while (x != 00)for (i = 1; i <= Fnumber; i+)for (j = 1; j <= Xnumbers; j+)if (!Insert(A, x, i, j)ex

9、it(0); cin >> x;j = 1;printfunction(A);if (Xnumbers = Fnumber+1)flag = 0;elsecout << "方程可能無解 " << endl;A->m = 0;A->n = 0;/ 顯示交互 / void ShowMenu(void)int x;cout << "選擇求解方程根的方法： " << endl;cout << "1.列主元高斯消去法 " << endl;cout &

10、lt;< "2.三角分解法 " << endl;cout << "3.迭代法 " << endl;cin >> x;switch (x)case 1:ColumnGaussmethod(A);Tip(); break;case 2:LUmethod(A);Tip();break;case 3:ISODATAmethod(A);Tip();break;default:break;/ 打印輸出函數(shù) /void printfunction(datacoa *A)int i,j;cout << &

11、quot;矩陣=" << endl;for (i = 1; i <= A->m; i+)for (j = 1; j <= A->n; j+)cout <<setw(12)<< A->dataij;cout << endl;/ 打印結(jié)果 /void printresx(datacoa *A)int i;for (i = 1; i <= A->m; i+)cout << "X" << i << "=" << se

12、tw(12) << A->dataiXnumbers+1; cout << endl;/ 返回提示 /void Tip(void)int flag;cout << "輸入 000 退出 ,其余返回： " << endl; cin >> flag;if (flag = 000)Exflag = 0;3.3 核心算法部分算法部分頭文件 #ifndef _ALGORITHM_H #define _ALGORITHM_H#include "stdio.h"#include "stdlib

13、.h"#include "stdafx.h" int Judge(double x1, double x0, double e); void ColumnGaussmethod(datacoa *A); void LUmethod(datacoa *A);void ISODATAmethod(datacoa *A); #endif算法部分CPP文件#include "algorithm.h"#include "stdafx.h"#include "squencelist.h"#include "

14、equation.h"誤差判別 /#define max 100 /inline int Judge(double x1, double x0, double e) if (e = 0)return 0;if (x1 = 0)if (fabs(x0) < e)return 1;else return 0;if (x0 = 0)if (fabs(x1) < e)return 1;else return 0;if (fabs(x1 - x0) < e)return 1;else return 0;/ 列主元高斯消元法 / void ColumnGaussmethod(d

15、atacoa *A)cout << "/列主元高斯消元法 /" << endl;int i, j, i2, flagc, k, j2;float temp, res;for (i = 1; i < Fnumber; i+)flagc = i;for (i2 = i + 1; i2 <= Fnumber; i2+)if (fabs(A->datai2i)>(fabs(A->dataflagci) flagc = i2;if (flagc != i)for (k = i; k <= Xnumbers; k+)temp

16、= A->dataik;A->dataik = A->dataflagck; A->dataflagck = temp;for (i2 = i + 1; i2 <= Fnumber; i2+)temp = A->datai2i/A->dataii;for (j2 = i; j2 <= Xnumbers; j2+) A->datai2j2 = A->datai2j2 - temp*A->dataij2;for (i = Fnumber; i >= 1; i-)for (j = Fnumber; j >= i + 1;

17、j-) A->dataiXnumbers = A->dataiXnumbers A->dataij*A->datajXnumbers + 1;res = A->dataiXnumbers / A->dataii; Insert(A, res, i, Xnumbers+1);printresx(A);/ 直接三角分解法 / void LUmethod(datacoa *A)datacoa *L, *U;int i, j, k,q;float temp=0.0;L = InitStruct();U = InitStruct();for (i = 1; i <

18、;= Fnumber; i+)for (j = 1; j <= Xnumbers; j+)Insert(L, 0, i, j);Insert(U, 0, i, j);for (j = 1; j <= Xnumbers; j+)Insert(U, A->data1j, 1, j);for (i = 2; i <= Fnumber; i+)Insert(L, A->datai1 / A->data11, i, 1);for (k = 2; k <= Fnumber; k+)for (j = k; j <= Xnumbers; j+)for (q =

19、1; q < k; q+)temp = temp + L->datakq * U->dataqj;Insert(U, A->datakj - temp, k, j);temp = 0;for (i = k + 1; i <= Fnumber; i+)for (q = 1; q < k; q+)temp = temp + L->dataiq * U->dataqk;temp = A->dataik - temp;Insert(L, temp / U->datakk, i, k);temp = 0;Insert(U, U->data

20、FnumberXnumbers / U->dataFnumberXnumbers - 1, Fnumber, Xnumbers + 1);for (k = Fnumber - 1; k >= 1; k-)for (q = k + 1; q < Xnumbers; q+)temp = temp + U->datakq * U->dataqXnumbers + 1;temp = U->datakXnumbers - temp;Insert(U, temp / U->datakk, k, Xnumbers + 1); temp = 0; printresx(

21、U);/ 迭代法 /void ISODATAmethod(datacoa *A)int i=1, j, k=0;float x0 = 0, x1 = 1, temp = 0;for (i = 1; i <= Fnumber; i+)Insert(A, 0, i, Xnumbers + 1);while (1)for (i = 1; i <= Fnumber; i+)for (j = 1; j <= Fnumber; j+)if (j = i)continue;temp = temp + A->dataij * A->datajXnumbers + 1;temp =

22、 A->dataiXnumbers - temp;temp = temp / A->dataii;Insert(A, A->dataiXnumbers + 1, i, Xnumbers + 2);Insert(A, temp, i, Xnumbers + 1); temp = 0;k+;for (i = 1; i <= Fnumber; i+)temp = temp + fabs(A->dataiXnumbers + 1 - A->dataiXnumbers 2);if (temp < 0.000001) | k > max)break;temp

23、 = 0;DeleteLie(A, Xnumbers + 2);printfunction(A); cout << endl; printresx(A);cout << k << endl;3.4 數(shù)據(jù)結(jié)構部分數(shù)據(jù)結(jié)構頭文件 #ifndef _SQUENCELIST_H #define _SQUENCELIST_H#include "stdio.h"#include "stdlib.h"#include "stdafx.h" #include<iostream> using namesp

24、ace std;#define maxsize 1024*sequenlist*/ typedef float datatype;typedef structdatatype datamaxsizemaxsize; int m, n;datacoa;datacoa *InitStruct();int Length(datacoa*);int Insert(datacoa*, datatype, int ,int); void DeleteLie(datacoa*L, int j); void DeleteLine(datacoa*L, int i);#endif數(shù)據(jù)結(jié)構CPP文件#includ

25、e "stdafx.h"#include "squencelist.h"據(jù)結(jié)構部分/數(shù)/sequenlist/ datacoa *InitStruct()datacoa*L = (datacoa*)malloc(sizeof(datacoa);L->m = 0;L->n = 0;return L;/ datacoa*L = new datacoa;int Length(datacoa*L)return L->m*L->n;int Insert(datacoa*L, datatype x, int i,int j)int k;if

26、(L->m >= maxsize - 1) | (L->n >= maxsize - 1)cout << "表已滿 " << endl;return 0;for (k = L->n; k >= j; k-)L->dataik + 1 = L->dataik;L->dataij = x;if (i > L->m)L->m+;if (j > L->n)L->n+;return 1;void DeleteLie(datacoa*L,int j)int k,i;if (

27、j<1) | (j>L->n)cout << "非法刪除位置 " << endl;for (i = 1; i <= L->m; i+)for (k = j; k <= L->n; k+)L->dataij = L->dataij + 1; L->n-; void DeleteL in e(datacoa*L, int i)int k,j;if (i<1) II (i>L->m)cout << "非法刪除位置"<< endl;for (j = 1; j <= L->n; j+)for (k = i; k <= L->m; k+) L->dataij = L->datai+1j;L->m-;4運行結(jié)果4.1列主元高斯消去法運行結(jié)果5H匸:vi nd owssyste m32c md .e<e諭入方程系數(shù)輸入。醛束（如輸入2 1 S3 4 1或者Z 15 3 4L7?:L.93 :