数值分析在生活中应用举例及Matlab实现

一、最小二乘法,用MATLAB实现1. 数值实例下面给定地是乌鲁木齐最近1个月早晨7：00左右（新疆时间）地天气预报所得到地温度,按照数据找出任意次曲线拟合方程和它地图像.下面用MATLAB编程对上述数据进行最小二乘拟合.b5E2RGbCAP2008年10月26~11月26天数 1 2 3 4 5 6 7 8 9 10 温度9 10 11 12 13 14 13 12 11 9 天数11 12 13 14 15 16 17 18 19 20 温度10 11 12 13 14 12 11 10 9 8 天数21 22 23 24 25 26 27 28 29 30 温度7 8 9 11 9 7 6 5 3 1 下面用MATLAB编程对上述数据进行最小二乘拟合2、程序代码x=[1:1:30];y=[9,10,11,12,13,14,13,12,11,9,10,11,12,13,14,12,11,10,9,8,7,8,9,11,9,7,6,5,3,1];p1EanqFDPwa1=polyfit(x,y,3) %三次多项式拟合%a2= polyfit(x,y,9) %九次多项式拟合%a3= polyfit(x,y,15) %十五次多项式拟合%b1=polyval(a1,x)b2=polyval(a2,x)b3=polyval(a3,x)r1= sum((y-b1).^2) %三次多项式误差平方和%r2= sum((y-b2).^2) %九次次多项式误差平方和%r3= sum((y-b3).^2) %十五次多项式误差平方和%plot(x,y,'*') %用*画出x,y图像%hold onplot(x,b1, 'r') %用红色线画出x,b1图像%hold onplot(x,b2, 'g') %用绿色线画出x,b2图像%hold onplot(x,b3, 'b:o') %用蓝色o线画出x,b3图像%3、数值结果不同次数多项式拟合误差平方和为：r1=67.6659r2=20.1060r3=3.7952r1、r2、r3分别表示三次、九次、十五次多项式误差平方和.4、拟合曲线如下图二、线性方程组地求解（高斯-塞德尔迭代算法）1、实例：求解线性方程组（见书P233页）3612363311420238321321321xx xxx x x x x 记A x=b, 其中363320,,12361114238321bxAxx x 任取初始值Tx00,进行迭代.2、用C 语言实现,源程序如下：#include<stdio.h> #include<stdlib.h> struct Node {double value; // 矩阵非零元素值int index;// 元素地索引(序号或列标)};struct Matrix{struct Node *data; // 记录元素数据int total_elem; // 矩阵非零元素个数int total_ln; // 矩阵地总行数int total_col; // 矩阵地总列数};// 采用MSR法实现系数矩阵地存储void Create_Matrix( struct Matrix &matrix ){int index = 0, i = 0;double value = 0.0;printf( "请输入线性方程组系数矩阵地非零元素个数及总行数、总列数: " );scanf( "%d%d%d", &matrix.total_elem, &matrix.total_ln, &matrix.total_col );DXDiTa9E3dmatrix.data = (struct Node *)malloc( sizeof(struct Node)*(matrix.total_elem+1) );RTCrpUDGiTprintf( "请依次输入矩阵地主对角线元素及每行地非零非对角线元素(行优先)\n" );for( i = 0; i <= matrix.total_elem; i++ ){if( i == matrix.total_ln )value = 0;elsescanf( "%lf", &value );matrix.data[i].value = value;}printf( "\n系数矩阵地信息相关如下(Quote为添加项,表示序列号):" );printf( "\nValue:" );for( i = 0; i < matrix.total_elem + 1; i++ )printf( "%3.0lf", matrix.data[i].value );printf( "\nQuote:" );for( i = 0; i < matrix.total_elem + 1; i++ )printf( "%3d", i );printf( "\n\n" );for( i = 0; i <= matrix.total_elem; i++ ){if( i == 0 )printf( "请参照上表输入第每行第一个非零非主对角元素地序号\n " );if( i == matrix.total_ln )matrix.data[i].index = matrix.total_elem + 1;elseif( i == matrix.total_ln + 1 )printf( "请依次输入非零非对角元素地列标\n " );if( i != matrix.total_ln )scanf( "%d", &matrix.data[i].index );}}// 显示系数矩阵地相关信息void Display_Matrix( struct Matrix matrix ){int i = 0;printf( "系数矩阵采用MSR存储法地相关信息如下(Quote为添加项,表示序列号):" ); printf( "\nQuote: " );for( i = 0; i <= matrix.total_elem; i++ )printf( "%3d", i );printf( "\nvalue: " );for( i = 0; i <= matrix.total_elem; i++ )printf( "%3.0lf", matrix.data[i].value );printf( "\nIndex: " );for( i = 0; i <= matrix.total_elem; i++ )printf( "%3d", matrix.data[i].index );printf( "\n" );}// 初始化向量,使之分量全为0void Init_X( double *X, int n ){int i = 0;for( i = 0; i < n; i++ )X[i] = 0;}// 初始化右端向量void Init_b( double *b, int n ){int i = 0;Init_X( b, n );printf( "\n请输入%d个数(方程组地右端项): ", n );for( i = 0; i < n; i++ )scanf( "%lf", &b[i] );}// 初始化未知向量,得到初始值void Init_temp_X( double *temp, int n ){int i = 0;Init_X( temp, n );printf( "请输入X 向量地初始值: " );for( i = 0; i < n; i++ )scanf( "%lf", &temp[i] );}// 采用高斯-塞德尔GS迭代法求值int Calculate_X_GS( struct Matrix A, double *X, double *b )5PCzVD7HxA{int i = 0, j = 0, sum = 0;double temp = 0.0, temp_1 = 0.0, *temp_X = 0, judge = 0.0;jLBHrnAILgtemp_X = (double *)new double[A.total_ln];Init_temp_X( temp_X, A.total_ln );for( i = 0; i < A.total_ln; i++ )X[i] = temp_X[i];sum = 0;do{judge = 0.0;for( i = 0; i < A.total_ln; i++ )temp_X[i] = X[i];for( i = 0; i < A.total_ln; i++ ){temp = 0.0;temp_1 = 0.0;for( j = A.data[i].index; A.data[j].index < i && j <= A.total_elem; j++ )xHAQX74J0X temp_1 += A.data[j].value*X[A.data[j].index];for( ; j < A.data[i+1].index && j <= A.total_elem; j++ )LDAYtRyKfEtemp += A.data[j].value * temp_X[A.data[j].index];X[i] = (b[i] - temp_1- temp)/A.data[i].value;}for( i = 0; i < A.total_ln; i++ ){if( X[i]-temp_X[i] < 0 )judge = temp_X[i]-X[i]>judge ? temp_X[i]-X[i] : judge;Zzz6ZB2Ltk elsejudge = X[i]-temp_X[i]>judge ? X[i]-temp_X[i] : judge;dvzfvkwMI1 }sum++;printf( "\n\n第%d 次迭代地结果: ", sum );for( i = 0; i < A.total_ln; i++ )printf( "\nX[%d] = %g", i, X[i] );printf( "\njudge = %g", judge );}while( judge > 0.000001 );return sum;}void main( ){struct Matrix A;// 实现系数矩阵地存储及显示Create_Matrix( A );Display_Matrix( A );double *X = 0, *b = 0;X = (double *)new double[A.total_ln];b = (double *)new double[A.total_ln];Init_X( X, A.total_ln );Init_b( b, A.total_ln );int i = 0, sum = 0;sum = Calculate_X_GS( A, X, b );printf( "\n\n线性方程组地解为: " );for( i = 0; i < A.total_ln; i++ )printf( "\nX[%d] = %g", i, X[i] );printf( "\n总共迭代了%d 次\n\n", sum );}3、运行结果如下：版权申明本文部分内容，包括文字、图片、以及设计等在网上搜集整理.版权为个人所有This article includes some p arts, including text, pictures, and design. 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