数值分析 算法C语言程序
数值分析插值算法源程序
#include<stdio.h>#include<math.h>float f(float x) //计算ex的值{return (exp(x));}float g(float x) //计算根号x的值{return (pow(x,0.5));}void linerity () //线性插值{float px,x;float x0,x1;printf("请输入x0，x1的值\n");scanf("%f,%f",&x0,&x1);printf("请输入x的值: ");scanf("%f",&x);px=(x-x1)/(x0-x1)*f(x0)+(x-x0)/(x1-x0)*f(x1);printf("f(%f)=%f \n",x,px);}void second () //二次插值{float x0,x1,x2,x,px;x0=0;x1=0.5;x2=2;printf("请输入x的值:");scanf("%f",&x);px=((x-x1)*(x-x2))/((x0-x1)*(x0-x2))*f(x0)+((x-x0)*(x-x2))/((x1-x0)*(x1-x2))*f(x1)+((x-x0)* (x-x1))/((x2-x0)*(x2-x1))*f(x2);printf("f(%f)=%f\n",x,px);}void Hermite () //Hermite插值{int i,k,n=2;int flag1=0;printf("Hermite插值多项式H5(x)=");for(i=0;i<=n;i++){int flag=0;flag1++;if(flag1==1){printf("y%d[1-2(x-x%d)*(",i,i);}else{printf("+y%d[1-2(x-x%d)*(",i,i);}for(k=0;k<=n;k++){if(k!=i){flag++;if(flag==1){printf("(1/x%d-x%d)",i,k);}else{printf("+(1/x%d-x%d)",i,k);}}}printf(")]*（");for(k=0;k<=n;k++){if(i!=k){printf("[(x-x%d)/(x%d-x%d)]2",i,k,i);}}printf("）");}printf("\n");}void sectionl () //分段线性插值{float x[5]={2.0,2.1,2.2,2.3,2.4};float y;printf("请输入y：");scanf("%f",&y);if(y>=2.0&&y<2.1){float px;px=((y-x[1])/(x[0]-x[1]))*g (x[0])+((y-x[0])/(x[1]-x[0]))*g (x[1]);printf("f(%f)=%f\n",y,px);}else if(y>=2.1&&y<2.2){float px;px=((y-x[2])/(x[1]-x[2]))*g (x[1])+((y-x[1])/(x[2]-x[1]))*g (x[2]);printf("f(%f)=%f\n",y,px);}else if(y>=2.2&&y<2.3){float px;px=((y-x[3])/(x[2]-x[3]))*g (x[2])+((y-x[2])/(x[3]-x[2]))*g (x[3]);printf("f(%f)=%f\n",y,px);}else if(y>=2.3&&y<2.4){float px;px=((y-x[4])/(x[3]-x[4]))*g (x[3])+((y-x[3])/(x[4]-x[3]))*g (x[4]);printf("f(%f)=%f\n",y,px);}else if(y>2.4) printf("**********ERROR!******************\n"); }void sectionp (){int i;float a[5]={2.0,2.1,2.2,2.3,2.4};float x,y;printf("input the data: x?\n");scanf("%f",&x);if(x<a[1]){i=1;goto loop;}if(x>a[4]){i=4;goto loop;}i=1;loop1:i++;if(x>a[i])goto loop1;if(fabs(x-a[i-1])<=fabs(x-a[i]))i=i-1;loop:y=g(a[i-1])*(x-a[i])*(x-a[i+1])/((a[i-1]-a[i])*(a[i-1]-a[i+1]));y=y+g(a[i])*(x-a[i-1])*(x-a[i+1])/((a[i]-a[i-1])*(a[i]-a[i+1]));y=y+g(a[i+1])*(x-a[i-1])*(x-a[i])/((a[i+1]-a[i-1])*(a[i+1]-a[i]));printf("f(%f)=%f\n",x,y);}int main(){char flag1='y';while(flag1=='y'){int flag=0;printf("*******[1]:线性插值***************\n");printf("*******[2]:二次插值***************\n");printf("*******[3]:Hermite插值************\n");printf("*******[4]:分段线性插值***********\n");printf("*******[5]:分段抛物线插值*********\n");printf("请输入：");scanf("%d",&flag);switch(flag){case 1:linerity ();break;case 2:second ();break;case 3:Hermite ();break;case 4:sectionl ();break;case 5:sectionp ();break;default:printf("error!!\n");}printf("是否继续？y/n \n");getchar();scanf("%c",&flag1);}return 0;}。
数值分析常微分数值解的求法C语言
本科生课程设计报告实习课程数值分析学院名称管理科学学院专业名称信息与计算科学学生姓名学生学号指导教师实验地点实验成绩二〇一六年六月二〇一六年六摘要常微分方程数值解法是计算数学的一个分支.是解常微分方程各类定解问题的数值方法.现有的解析方法只能用于求解一些特殊类型的定解问题，实用上许多很有价值的常微分方程的解不能用初等函数来表示，常常需要求其数值解.所谓数值解，是指在求解区间内一系列离散点处给出真解的近似值.这就促成了数值方法的产生与发展.?关键词：数值解法；常微分1. 实验内容和要求常微分方程初值问题 有精确解2()cos(2)x y x x e x -=+。
要求：分别取步长h = 0.1，0.01，0.001，采用改进的Euler 方法、4阶经典龙格-库塔R －K 方法和4阶Adams 预测-校正方法计算初值问题。
以表格形式列出10个等距节点上的计算值和精确值，并比较他们的计算精确度。
其中多步法需要的初值由经典R-K 法提供。
运算时，取足以表示计算精度的有效位。
2. 算法说明2.1函数及变量说明表1 函数及变量定义1、欧拉方法：1()()(,())i i i i y x y x hf x y x +=+ （i=0，1，2，3，......n-1）(0)y a= (其中a 为初值)2、改进欧拉方法：~1~111()()(,())()()[(,())(,())]2(0)i i i i i i i i i i y x y x hf x y x hy x y x f x y x f x y x y a ++++=+=++=（i=0，1，2......n-1） (其中a 为初值)3、经典K-R 方法： 11213243()6(,)(,)22(,)22(,)i i i i i i i i i i h y y K f x y h hK f x y K h h K f x y K K f x h y hK +⎧=+⎪⎪=⎪⎪⎪=++⎨⎪⎪=++⎪⎪=++⎪⎩4、4阶adams 预测-校正方法 预测： 校正：Adsms 内插外插公式联合使用称为Adams 预测-校正系统，利用外插公式计算预测，用内插公式进行校正。
C语言数据分析统计方法和数据可视化
C语言数据分析统计方法和数据可视化随着信息时代的不断发展，大量的数据被产生和积累。
为了从这些海量数据中提取有价值的信息，数据分析和统计方法成为了必不可少的工具。
在编程领域中，C语言作为一种高效且功能强大的语言，提供了各种数据分析和统计的方法，同时结合数据可视化技术，可以帮助程序员更好地理解和解释数据。
一、基本数据分析统计方法C语言提供了许多基础的数据分析和统计方法，如求和、平均值、中位数、极值等。
通过使用循环和条件语句，可以编写简洁高效的代码来实现这些统计功能。
以下是一些常见的基本统计方法的示例代码：1. 求和：```cdouble calculateSum(double array[], int length) {double sum = 0;for (int i = 0; i < length; i++) {sum += array[i];}return sum;}```2. 平均值：```cdouble calculateAverage(double array[], int length) { double sum = calculateSum(array, length);double average = sum / length;return average;}```3. 中位数：```cdouble calculateMedian(double array[], int length) { if (length % 2 == 0) {return (array[length/2 - 1] + array[length/2]) / 2.0; } else {return array[length/2];}}```二、高级数据分析统计方法除了基本的统计方法外，C语言还支持更加高级的数据分析和统计方法，如方差、标准差、相关性等。
这些方法能更全面地描述数据的分布和关系。
1. 方差和标准差：方差度量了数据集合的离散程度，标准差是方差的平方根。
数值计算C语言常用小程序
数值计算C语言常用小程序C语言是一种流行的编程语言，广泛应用于计算机科学和软件开发领域。
在数值计算方面，C语言提供了一些常用的函数和技巧，可以帮助我们进行各种数值计算任务。
本文将介绍一些C语言常用的数值计算小程序。
1.求平均数```c#include <stdio.h>int maiint num;int sum = 0;int count = 0;float average;printf("请输入数字: ");while(num != -1)scanf("%d", &num);if(num != -1)sum += num;count += 1;}}average = (float)sum / count;printf("平均值为: %.2f\n", average);return 0;```这个程序会要求用户输入一系列数字，直到输入-1为止。
然后计算这些数字的平均值并输出。
2.求阶乘```c#include <stdio.h>int factorial(int n)if(n == 0)return 1;} elsereturn n * factorial(n-1);}int maiint n;int result;printf("请输入一个正整数: ");scanf("%d", &n);result = factorial(n);printf("%d的阶乘为: %d\n", n, result);return 0;```这个程序会要求用户输入一个正整数，然后使用递归的方式计算该整数的阶乘，并输出结果。
3.求平方根```c#include <stdio.h>#include <math.h>int maidouble num;printf("请输入一个数字: ");scanf("%lf", &num);if(num < 0)printf("无法计算负数的平方根\n");} elsedouble result = sqrt(num);printf("该数字的平方根为: %.2lf\n", result);}return 0;```这个程序会要求用户输入一个数字，然后计算该数字的平方根并输出。
C++编写数值分析程序
高斯消去法；#include <iostream>using namespace std;template <class T>T** Allocation2D(int m, int n){T **a;a = new T*[m];for (int i=0; i<m; i++){a[i] = new T[n];}return a;}int main(){int i,j,k;int n;float** a;cout<<"输入系数矩阵的N值，N：";cin>>n;a = Allocation2D<float>(n, n+1);cout<<endl<<"输入增广矩阵的各值：/n";for(i=0; i<n; i++){for(j=0; j<n+1; j++){cin>>a[i][j];}}for(k=0; k<n-1; k++){for(i=k+1; i<n; i++){for(j=k+1; j<n+1; j++){a[i][j] = a[i][j] - a[i][k] * a[k][j] / a[k][k];}}}float temp;a[n-1][n] = a[n-1][n] / a[n-1][n-1];for(k=n-2; k>=0; k--){temp = 0;for(j=k+1; j<n; j++){temp = temp + a[k][j] * a[j][n];}a[k][n] = (a[k][n] - temp) / a[k][k];}for(i=0; i<n; i++){cout<<"x"<<i<<": "<<a[i][n]<<endl;}return 0;}拉格朗日插值法#include<iostream>#include<math.h>#define N 3using namespace std;double fun(double *x,double *y, int n,double p);void main(){int i;double a[N],p;double b[N];cout<<"please input xiangliang a= "<<endl;for(i=0;i<N;i++){cin>>a[i];}cout<<"please input xiangliang b= "<<endl;for(i=0;i<N;i++){cin>>b[i];}cout<<"please input LagelangrichazhiJieDian p= "<<endl;cin>>p;cout<<"The Answer= "<<fun(a,b,N,p)<<endl;}double fun(double x[],double y[], int n,double p){double z=0,s=1.0;int k,i=0;double L[N];k=0;while(k<n){if(k==0){for(i=1;i<n;i++){s=s*(p-x[i])/(x[0]-x[i]);}L[0]=s*y[0];k=k+1;}else{s=1.0;for(i=0;i<=k-1;i++){s=s*((p-x[i])/(x[k]-x[i]));}for(i=k+1;i<n;i++){s=s*((p-x[i])/(x[k]-x[i]));}L[k]=s*y[k];k++;}}for(i=0;i<n;i++){z=z+L[i];}return z;}Newton插值法；#include <iostream.h>#include <math.h>void main(){char L;do{double M[100][100];double x[100],y[100];double X=1,xx=0,w=1,N=0,P,R=1;int n;cout<<"请输入所求均差阶数：";cin>>n;for(int i=0;i<=n;i++){cout<<"请输入x"<<i<<"的值："<<endl;cin>>x[i];cout<<"请输入y"<<i<<"的值："<<endl;cin>>y[i];M[i][0]=x[i];M[i][1]=y[i];}for( int j=2;j<=n+1;j++){for( i=1;i<=n;i++){M[i][j]=(M[i][j-1]-M[i-1][j-1])/(M[i][0]-M[i-j+1][0]);}}for(i=1;i<=n;i++){cout<<"其"<<i<<"阶均差为："<<M[i][i+1]<<endl;}cout<<"请输入x的值：x=";cin>>xx;for(i=0;i<n;i++){X*=xx-x[i];N+=M[i+1][i+2]*X;P=M[0][1]+N; cout<<"其函数值：y="<<P<<endl;cout<<endl<<"如还想算其它插值请按'y'否则按'n'"<<endl;cin>>L;}while(L=='y'); }高斯列主元消去法；#include<iostream>#include<cmath>using namespace std;int main(){double max(double a, double b,double c=0);int i,j,t;double temp[3];double a[3][3]={0.012,0.01,0.167,1,0.8334,5.91,3200,1200,4.2}; double b[3]={0.6781,12.1,981};double x[3];for (i=0;i<=2;i++){if(a[i][0]=max(a[0][0] ,a[1][0],a[2][0]))t=i;for(i=0;i<=2;i++)temp[i]=a[t][i];a[t][i]=a[0][i];a[0][i]=temp[i];}double l21=a[1][0]/a[0][0];a[1][1]=a[1][1]-a[0][1]*l21;a[1][2]=a[1][2]-a[0][2]*l21;b[1]=b[1]-b[0]*l21;double l31=a[2][0]/a[0][0];a[2][1]=a[2][1]-a[0][1]*l31;a[2][2]=a[2][2]-a[0][2]*l31;b[2]=b[2]-b[0]*l31;for (i=0;i<=2;i++){if(a[i][1]==max(a[1][0] ,a[2][0]))t=i;for(i=0;i<=2;i++)temp[i]=a[t][i];a[t][i]=a[0][i];a[0][i]=temp[i];}double l32=a[2][1]/a[1][1];a[2][2]=a[2][2]-a[1][2]*l32;b[2]=b[2]-b[1]*l32;x[2]=b[2]/a[2][2];x[1]=(b[1]-a[1][2]*x[2])/a[1][1];x[0]=(b[0]-a[0][1]*x[1]-a[0][2]*x[2])/a[0][0];cout<<"x1="<<x[0]<<" "<<"x2="<<x[1]<<" "<<"x3="<<x[2]<<endl; return 0;}double max(double a,double b,double c){if(a>b){if(b>c) return a;elseif(a>c) return a; else return c; }elseif(b>c) return b; else return c;}迭代法；#include<iostream>using namespace std;double f(double);int main(){double a,i=0,a1,pr;cout<<"please input the start number:"; cin>>a;cout<<"please input precision:";cin>>pr;do{a1=a;a=f(a);i++;if(i>1000000){cout<<"Iterative is defeated！";return 0;} }while((a-a1)>pr||(a-a1)<-pr);cout<<"Iterative Numbers is "<<i<<endl; cout<<"The answer is "<<a<<endl; double f(double x){return pow((x+1.0),(1.0/3));}数值积分（梯形和辛普生公式）#include<stdio.h>double f(double x){return(x/(4+x*x));}double S(double a,double b){double u,h,c;h=(b-a)/6;c=(a+b)/2;u=f(a)+f(b)+4*f(c);return (h*u);}double F(double x){return(x/(4+x*x));}double T(double a,double b){double u,h;h=(b-a)/2;u=F(a)+F(b);return (h*u);}void main(){int c; //循环控制量int i=1; //求解次数printf("----数值积分求解-----\n");printf("1、利用辛普生公式；\n");printf("2、利用梯形公式；\n");printf("0、退出。
数值分析程序
拉格朗日#include<iostream>#include <cmath>using namespace std;double func (double m);int main(){const int j=100;int nn=0;double x=0.0,y=0.0,temp=0;cout<<"请输入要插节点个数"<<endl;cin>>nn;cout<<"请输入x的值:"<<endl;cin>>x;double a[j],b[j];for(int i=0;i<nn;++i){cout<<"请输入x的第"<<i+1<<"个值:";cin>>a[i];temp=a[i];b[i]=func(temp);}for(int k=0;k<nn;++k){double t=1.0;for(int n=0;n<nn;++n){if(k==n) ++n;if(n==nn) break;t=t*(x-a[n])/(a[k]-a[n]);}y=y+t*b[k];}cout<<"结果为:"<<y<<endl;return 0;}double func (double m){return sqrt(m) ;}四阶龙格库塔#include<iostream>using namespace std; double g(double,double);double rungekutta(double x0,double y0,double h,double N);int main(){double x0,y0,h;int N;cout<<"Please input the beginning point x0, y0, step length h, calculating times N."<<endl;cin>>x0>>y0>>h>>N;cout<<rungekutta(x0,y0,h,N);}double rungekutta(double x0,double y0,double h,double N){int i;double x1,y1,K1,K2,K3,K4;for(i=0;i!=N;i++){x1=x0+h;K1=g(x0,y0);K2=g(x0+h/2,y0+h/2*K1);K3=g(x0+h/2,y0+h/2*K2);K4=g(x1,y0+h*K3);y1=y0+h/6*(K1+2*K2+2*K3+K4);x0=x1;y0=y1;}return y1;}//注：以下的函数需要根据实际情况进行调整double g(double x,double y){return y-2*x/y;}快速弦截法#include<iostream>using namespace std;//下列函数的参数表按实际函数的不同定义一般只需要xdouble h(double a,double b,double c,double x);int main(){double x,x1,x2,eps,n,a,b,c;int k=1;//此处求解的方程为a*x^2+b*x+c=0，根据情况改变cout<<"Solving Equation a*x^2+b*x+c=0, input a,b,c,x0,x1,iteration times N and precision Eps.\n";//根据函数本有的变量进行输入一般只需要输入第一个起始点x 第二个起始点x1 次数n和精度epscin>>a>>b>>c>>x>>x1>>n>>eps;while(k<=n){if(h(a,b,c,x1)-h(a,b,c,x)==0){cout<<"Iteration Failed. Try other starting points.\n";return 0;}x2=x1-(x1-x)*h(a,b,c,x1)/(h(a,b,c,x1)-h(a ,b,c,x));if(x2-x1<eps&&x1-x2<eps){cout<<"The solution is x="<<x2<<endl;return 0;}x=x1;x1=x2;++k;}cout<<"Iteration Failed. Cannot be precise enough.\n";return 0;}//以下根据情况改变一般只需要未知数x double h(double a,double b,double c,double x)//Equation ax^2+bx+c=0{return (a*x*x+b*x+c);} 高斯-赛德尔（有点长）#include<iostream>using namespace std;int main(){int N,count,k,n;k=1;cout<<"Please input the number of equations.\n";cin>>N;cout<<"Please input max iretating number.\n";cin>>n;double **a=new double*[N];for(int i=0;i<N;++i){a[i]=new double [N];}double *b=new double[N];double *x=new double[N];double *y=new double[N];//上边的部分最好按自己的习惯定义数组此处给出的是可以随意调整方程阶数的解决方案double eps;cout<<"Please input the factors of x1,x2.....xn.\n";for(int i=0;i<N;++i){for(int j=0;j<N;++j){cin>>a[i][j];}x[i]=0;y[i]=x[i];}cout<<"Please input every result of the equations.\n";for(int i=0;i<N;++i){cin>>b[i];}cout<<"Please input the precision required.\n";cin>>eps;double sum;while(k<=n){for(int i=0;i<N;++i){sum=0.0;for(int j=0;j<N;++j){if(j==i)continue;sum+=a[i][j]*y[j];}y[i]=(b[i]-sum)/a[i][i];}count=0;for(int i=0;i<N;++i){if(x[i]-y[i]<=eps&&y[i]-x[i]<=eps){count++;}}cout<<"k="<<k-1;for(int i=0;i<N;++i){cout<<"y"<<i+1<<'='<<y[i]<<" ";}cout<<endl;if(count==N){cout<<"The result is:\n";for(int i=0;i<N;++i){cout<<"y"<<i+1<<'='<<y[i]<<" ";}return 0;}++k;for(int i=0;i<N;i++)x[i]=y[i];}cout<<"Iteration failed. Can not reach destination precision."<<endl;for(int i=0;i<N;++i){delete[N] a[i];}return 0;}高斯消去法#include<iostream>using namespace std;int main(){double **a;double *b;double *x,*y;int i,j,N;i=1;cout<<"Please input the number of the equations."<<endl;cin>>N;a=new double*[N+1];b=new double[N+1];x=new double[N+1];y=new double[N+1];for(i=1;i<=N;i++){a[i] = new double[N+1];}//以上按习惯定义数组cout<<"Please input the factors of x1,x2.....xn."<<endl;for(i=1;i<=N;i++)for(j=1;j<=N;j++){cin>>a[i][j];}cout<<"Please input the result of every equation."<<endl;for(i=1;i<=N;i++){cin>>b[i];}int k=1;int m;double sum,max,temp;i=1;do{max=fabs(a[k][k]);m=k;for(j=k;j<=N;j++){if(max<fabs(a[j][k])){max=a[j][k];m=j;}if(max==0){cout<<"No solution or Unlimited Solutions. "<<endl;return 0;}}for(int p=1;p<=N;p++){temp=a[m][p];a[m][p]=a[k][p];a[k][p]=temp;}temp=b[m];b[m]=b[k];b[k]=temp;for(j=k+1;j<=N;j++){a[k][j]=a[k][j]/a[k][k];}b[k]=b[k]/a[k][k];for(i=k+1;i<=N;i++){for(j=k+1;j<=N;j++){a[i][j]=a[i][j]-a[i][k]*a[k][j];}b[i]=b[i]-a[i][k]*b[k];}}while(k++!=N);for(i=N-1;i>=1;i--){sum=0;for(j=i+1;j<=N;j++){sum+=a[i][j]*b[j];}b[i]=b[i]-sum;}cout<<"The result is: "<<endl;for(i=1;i<=N;i++)cout<<'x'<<i<<'='<<b[i]<<" ";delete a;delete b;delete x;delete y;return 0;}高斯消去法#include<iostream>#include<cmath>const int n=3;using namespace std;int main(){int k=0;double a[n][n],b[n];cout<<"请输入各个系数:\n";for(int i=0;i<n;++i)for(int j=0;j<n;++j)cin>>a[i][j];cout<<"请输入各方程右边的值\n";for(int j=0;j<n;++j)cin>>b[j];do{double max=fabs(a[k][k]);for(int l=k;l<n;++l)if(max<fabs(a[l][k]) ) max=fabs(a[l][k]);for(int ll=k;ll<n;++ll)if(max==fabs(a[ll][k])){double temp;for(int i=k;i<n;++i){temp=a[k][i];a[k][i]=a[ll][i];a[ll][i]=temp;}double mm=b[k];b[k]=b[ll];b[ll]=mm;}for(int j=k+1;j<n;j++)a[k][j]=a[k][j]/a[k][k];b[k]=b[k]/a[k][k];for(int jj=k+1;jj<n;++jj){int i=jj;a[i][jj]=a[i][jj]-a[i][k]*a[k][jj];}for(int i=k+1;i<n;++i)b[i]=b[i]-a[i][k]*b[k];++k;}while(k!=n-1);for(int ii=n-2;ii>=0;--ii){double sum=0.0;for(int jjj=ii+1;jjj<n;++jjj)sum=sum+a[ii][jjj]*b[jjj];b[ii]=b[ii]-sum;}for(int mn=0;mn<n;++mn)cout<<"x["<<mn<<"]="<<b[mn]<<endl;return 0;}。
数值分析实验报告程序
一、Newton插值法#include<stdio.h>#define MAX_N 20typedef struct tagPOINT{double x;double y;}POINT;int main(){int n,i,j;POINT points[MAX_N+1];double diff[MAX_N+1]; double x,tmp,newton=0;printf("\nInput n value:");scanf("%d",&n);printf("Now input the (x_i,y_i),....,%d:\n",n);for(i=0;i<=n;i++)scanf("%lf%lf",&points[i].x,&points[i].y);printf("Now input the x value:");scanf("%lf",&x);for(i=0;i<=n;i++) diff[i]=points[i].y;for(i=0;i<=n;i++){for(j=n;j>i;j--){diff[j]=(diff[j]-diff[j-1])/(points[j].x-points[j-1-i].x); }}tmp=1;newton=diff[0];for(i=0;i<n;i++){tmp=tmp*(x-points[i].x);newton=newton+tmp*diff[i+1];}printf("newton(%f)=%f\n",x,newton);return 0;}二、Lagrange插值法double Lagrange(int n,double u,double x[],double y[]) {int i,j;double l,Ln=0;for(i=0;i<=n;i++){l=1.0;for(j=0;j<=n;j++)if (i!=j)l=l*(u-x[j])/(x(i)-x[j]);Ln=Ln+l*y[i];}return(Ln);}#include<stido.h>#define N 20V oid main(void){int i,n;double x[N],y[N],u.fu;double Lagrange(int n,double u,double x[],double y[]); printf(“请输入插值多项式的次数N及插入点U：\n”); scanf(“%d,%lf”,&n,&u);printf(“请输入n+1个插值节点x,y:\n”);for(i=0;i<=n;i++)scanf(“%lf,%lf”,&x[i],&y[i]);fu=Lagrange(n,u,x,y);printf(“插入点u的近似值为%lf\n”,fu);}三、用梯形公式#include<stdio.h>#include<math.h>double f(double x){return(sqrt(pow(x,4)+1));}double T(double a,double b){double u,h;h=(b-a)/2;u=f(a)+f(b);return (h*u);}void main(){ double a,b;printf("请输入积分区域：a,b\n"); scanf("%lf,%lf",&a,&b);double T(double ,double );printf("%lf,%lf",T(a,b));四、Simpson公式求积分#include<stdio.h>#include<math.h>double f(double x){return(sqrt(2-pow(sin(x),2))); }double S(double a,double b){double u,h,c;h=(b-a)/6;c=(a+b)/2;u=f(a)+f(b)+4*f(c);return (h*u);}void main(){ double a,b;printf("请输入积分区域：a,b\n");scanf("%lf,%lf",&a,&b);double S(double ,double );printf("%lf ",S(a,b))五、复化数值积分公式的自动控制误差算法复化Simpson自动控制误差#include<math.h>double AS(double (*f)(double x),double a,double b,double e) {int i;int n=2;double h,Jn,J2n,Sn,S2n;h=(b-a)/n;Jn=(*f)((a+b)/2);Sn=h/3*((*f)(a)+(*f)(b)+4*Jn);while(1){J2n=0;for(i=1;i<=n;i++)J2n=J2n+(*f)(a+(2*i-1)*h/2);S2n=Sn/2+h/6*(4*J2n-2*Jn);if(fabs(S2n-Sn)<15*e){return S2n;break;}else{ h=h/2;n=2*n;J2n=Jn;Sn=S2n;}}}复化梯形公式double comTx(double (*f)(double),double a,double b,int n) {int i; double h,Tn;h=(b-a)/n;Tn=(*f)(a)+(*f)(b);for(i=1;i<=n-1;i++)Tn=Tn+2*(*f)(a+i*h);Tn=(h/2)*Tn;return(Tn);}复化梯形自动控制误差double A T(double (*f)(double x),double a,double b,double e) {int i; int n=10;double h,Js,Tn,T2n;h=(b-a)/n;Tn=comTx(f,a,b,n);while(1){Js=0;for(i=1;i<=n;i++)Js=Js+(*f)(a+(2*i-1)*h/2);T2n=Tn/2+h/2*Js;if(fabs(T2n-Tn)<3*e)break;else{ h=h/2;n=2*n;Tn=T2n;}}return T2n;}六、Romberg公式计算积分#include "stdio.h"#include "math.h"#define MAX 20double f(double x)//要求的积分公式{ return(lnx);}double computeTn(double (*f)(double x),double a,double b, int n)//复化梯形公式{int i;double sum=0;double h=(b-a)/n;for(i=1;i<n;i++)sum+=f(a+i*h);sum+=(f(a)+f(b))/2;return (h*sum);}double Romberg(double (*f)(double x),double a,double b,double ep)//Romberg求积分{ int j,n=1,k;double R[3][MAX];R[2][1]=computeTn(f,a,b,n);printf("%lf\n",R[2][1]);for(k=2;k<MAX;k++){ for(j=1;j<k;j++)R[1][j]=R[2][j];n*=2;R[2][1]=computeTn(f,a,b,n);printf("%lf\t",R[2][1]);for(j=2;j<=k;j++){ R[2][j]=R[2][j-1]+(R[2][j-1]-R[1][j-1])/(pow(4,j-1)-1);printf("%lf\t",R[2][j]);}printf("\n");if(fabs(R[1][k-1]-R[2][k])<ep) return(R[2][k]); }k--;if(fabs(R[1][k-1]-R[2][k])>ep){printf("Return no solved...\n");return 0;}}七、用牛顿迭代法、弦截法求非线性方程的根///用牛顿迭代法的C源程序///#include<stdio.h>#include<math.h>#define m 20double f(double x){return((x*(x*(x-7.7)+19.2)-15.3));}double f1(double x){return(x*(3*x-15.4)+19.2);}main(){int k;double x,x0,e;printf("请输入x0和允许误差e的值:\n");scanf("%lf,%lf",&x0,&e);for(k=1;k<=m;k++){x=x0-f(x0)/f1(x0);if(fabs(x-x0)<e){printf("方程的近似根为%lf",x);return NULL;} x0=x;}printf("给定的迭代次数太小.停机");}///用弦截法的C源程序///#include<stdio.h>#include<math.h>#define m 20double f(double x){return((x*(x*(x-7.7)+19.2)-15.3));}main(){int k;double x,x0,x1,f0,f1,e;printf("请输入x0,x1和允许误差e的值:\n");scanf("%lf,%lf,%lf",&x0,&x1,&e);f0=f(x0);for(k=1;k<=m;k++){f1=f(x1);x=x1-(f1*(x1-x0)/(f1-f0));if(fabs(x-x1)<e){printf("方程的近似根为%lf",x);return NULL;} f0=f1;x0=x1;x1=x;}printf("给定的迭代次数太小.停机");}八、Gauss-Seidel迭代法解线性方程组的算法#include <math.h>#include <stdio.h>#define eps 0.000001#define max 100#define N 20double norm_inf(double x[],int n){ double norm;int i;norm=fabs(x[0]);for(i=1;i<n;i++){if(fabs(x[i])>norm)norm=fabs(x[i]);}return norm;}void seidel(double a[N][N],double g[N],int n){double b[N][N]={0},x0[N],x1[N],x1_x0[N],norm,temp; int i,j,k;for(i=0;i<n;i++){g[i]=g[i]/a[i][i];for(j=0;j<n;j++){ if(i==j)continue;b[i][j]=-a[i][j]/a[i][i];}}for(i=0;i<n;i++){x0[i]=0;x1[i]=1;x1_x0[i]=x1[i]-x0[i];}k=0;norm=norm_inf(x1_x0,n); while((norm>=eps)&&(k<max)) {for(i=0;i<n;i++)x0[i]=x1[i];for(i=0;i<n;i++){temp=0;for(j=0;j<=i-1;j++)temp=temp+b[i][j]*x1[j];for(j=i+1;j<n;j++)temp=temp+b[i][j]*x0[j];x1[i]=temp+g[i];x1_x0[i]=x1[i]-x0[i];}norm=norm_inf(x1_x0,n);k++;}for(i=0;i<n;i++)printf("x[%d]=%lf\n",i,x1[i]); printf("%d times iteration.\n",k); }int main(){double b[N][N]={{10,-2,-1},{-2,10,-1},{-1,-2,5}},f[N]={0,-21,-20}; printf("\n");seidel(b,f,3);return 0;}九、用列主元消元法#include<stdio.h>#include<math.h>#define n 3void print(double a[n][n+1]);void gauss(double a[n][n+1],double x[n]){ int i,j,k;double temp,s,l;for(i=0;i<n-1;i++){ k=i;for(j=i+1;j<n;j++){ if(fabs(a[j][i])>fabs(a[k][i]))k=j;}if(k!=i)for(j=i;j<=n;j++) {temp=a[i][j];a[i][j]=a[k][j];a[k][j]=temp;}for(j=i+1;j<n;j++){ l=1.0*a[j][i]/a[i][i];for(k=0;k<n+1;k++)a[j][k]=a[j][k]-a[i][k]*l; }print(a);printf("lf");}print(a);x[n-1]=a[n-1][n]/a[n-1][n-1]; for(i=n-2;i>=0;i--){ s=0.0;for(j=i;j<n;j++){ if(j==i)continue;s+=a[i][j]*x[j];}x[i]=(a[i][n]-s)/a[i][i];}}void print(double a[n][n+1]){int i,j;for(i=0;i<n;i++){ for(j=0;j<n+1;j++)printf("%lf",a[i][j]);printf("lf");}}十、Doolittle分解法#include "iostream.h"#include "stdio.h"#define max 20void main(){int i,k,n,j,p;double x,v,y;double a[max][max],r[max][max],l[max][max]; double b[max],Y[max],X[max];printf("Input n:"); scanf("%d",&n);printf("Input a[%d][%d]:\n",n,n);for(i=1;i<n+1;i++)for(j=1;j<n+1;j++){ scanf("%lf",&x); a[i][j]=x; }printf("b[i]:");for(i=1;i<n+1;i++){ scanf("%lf",&y); b[i]=y; }for(k=1;k<n+1;k++){ for(i=k;i<n+1;i++){ v=0;for(p=1;p<k;p++)v=l[k][p]*r[p][i]+v;r[k][i]=a[k][i]-v; printf("r[%d][%d]=%f\n",k,i,r[k][i]); } for(i=k+1;i<n+1;i++){ v=0;for(p=1;p<k;p++)v=v+l[i][p]*r[p][k];l[i][k]=(a[i][k]-v)/r[k][k]; printf("l[%d][%d]=%f\n",i,k,l[i][k]);} for(i=1;i<n+1;i++){ v=0;for(k=1;k<i;k++)v=v+l[i][k]*Y[k];Y[i]=b[i]-v;}for(i=n;i>0;i--){ v=0;for(k=i+1;k<n+1;k++)v=v+r[i][k]*X[k];X[i]=(Y[i]-v)/r[i][i];}printf("\n");for(i=1;i<n+1;i++)printf("x[%d]=%f\n",i,X[i]);printf("\n");return ;}。
数值分析C程序
b[k*n+i] = t;
}
}
// 为了使a的第k行的第k元素为1,a和b的第k行都除以a(k,k)
t = 1.0 / a[k*n+k];
for(i=k; i<n; i++) a[k*n+i] *= t;
for(i=0; i<n; i++) b[k*n+i] *= t;
for(i=0; i<n; i++)
{
if(i != k && a[i*n+k] != 0.0)
{
t = a[i*n+k];
for(j=k+1; j<n; j++) a[i*n+j] -= a[k*n+j] * t;
for(j=0; j<n; j++) b[i*n+j] -= b[k*n+j] * t;
// 带有列主元的约当消元法
// 功能： 求解线性方程组 Ax = b
// 参数： A － 指向n*n系数矩阵的指针
// b － 常数向量的指针
// n － 方程组的维数
// 返回值：0 － 如果成功。线性方程组的解保存在 b 中
// 1 － 求解失败
// ----------------------------------------------------------------------------
if(j != k) // 交换方程的第 j 行和 k 行
{
for(i=k; i<n; i++)
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一、拉格朗日插值#include<stdio.h>#include<stdlib.h>#include<math.h>void Lagrange(float s){double x[5]={0.2,0.4,0.6,0.8,1.0},y[5]={0.98,0.92,0.81,0.64,0.38},f,L=0;int i,j;for (i=0;i<5;i++){ f=1; for (j=0;j<5;j++) if(j!=i) f=(s-x[j])/(x[i]-x[j])*f; L+=f*y[i]; } printf("输出：%f\n",L);}void main(){float x;printf("输入插值点：");scanf("%f",&x);Lagrange(x);}二、牛顿插值#include<stdlib.h>#include<stdio.h>#include<math.h>int ND(float s){double x[5]={0.2,0.4,0.6,0.8,1.0},y[5]={0.98,0.92,0.81,0.64,0.38},p=0,g,f;int i,j,k;for (i=0;i<5;i++){for (j=4;j>i;j--){ f=x[j]-x[j-i-1];y[j]=(y[j]-y[j-1])/f;}g=y[i+1];for (k=0;k<=i;k++)g=g*(s-x[k]);p=p+g;}printf("输出插值点函数值：%f\n",p+y[0]);return 1;}void main(){float x;printf("输入插值点：");scanf("%f",&x);ND(x);}三、埃尔米特插值#include<stdio.h>#include<stdlib.h>#include<math.h>void Hermite(float s){double x[3]={0.25,1,2.25},y[3]={0.125,1,3.375},z[3]={0.75,1.5,2.25};double H=0.0,a,b,f,g;int i,j;for (i=0;i<3;i++){f=1.0;g=0.0;for (j=0;j<3&&j!=i;j++){f=f*(s-x[j])/(x[i]-x[j]);g=g+1/(x[i]-x[j]);}a=(1-2*(s-x[i])*g)*f*f;b=(s-x[i])*f*f;H=H+y[i]*a+z[i]*b;}printf("%f\n",H);}void main(){float x;printf("输入插值点：");scanf("%f",&x);Hermite(x);}四、三次样条插值#include <math.h>#include <stdio.h>#include <stdlib.h>void main(){int N=7,R=2,i,k;double p1,p2,p3,p4;double x[8]={0.5,0.7,0.9,1.1,1.3,1.5,1.7,1.9};double y[8]={0.4794,0.6442,0.7833,0.8912,0.9636,0.9975,0.9917,0.9463};double P0=-0.4794,Pn=-0.9463,u[3]={0.6,0.8,1.2},s[3];double h[7],a[8],c[7], g[8],af[8],ba[7],m[8];for(k=0;k <N;k++)h[k]=x[k+1]-x[k];for(k=1;k <N;k++) a[k]=h[k]/(h[k]+h[k-1]);for(k=1;k <N;k++) c[k]=1-a[k];for(k=1;k <N;k++) g[k]=3*(c[k]*(y[k+1]-y[k])/h[k]+a[k]*(y[k]-y[k-1])/h[k-1]);c[0]=a[N]=1;g[0]=3*(y[1]-y[0])/h[0]-P0*h[0]/2;g[N]=3*(y[N]-y[N-1])/h[N-1]+Pn*h[N-1]/2;ba[0]=c[0]/2; g[0]=g[0]/2;for(i=1;i <N;i++){ af[i]=2-a[i]*ba[i-1]; g[i]=(g[i]-a[i]*g[i-1])/af[i]; ba[i]=c[i]/af[i]; }af[N]=2-a[N]*ba[N-1];g[N]=(g[N]-a[N]*g[N-1])/af[N];m[N]=g[N];for(i=N-1;i>=0;i--) m[i]=g[i]-ba[i]*m[i+1];for(i=0;i <=R;i++){ k=0;while(u[i]> x[k+1])k++;p1=(h[k]+2*(u[i]-x[k])*pow((u[i]-x[k+1]),2)*y[k])/pow(h[k],3);p2=(h[k]-2*(u[i]-x[k+1])*pow((u[i]-x[k]),2)*y[k+1])/pow(h[k],3);p3=(u[i]-x[k])*pow((u[i]-x[k+1]),2)*m[k]/pow(h[k],2);p4=(u[i]-x[k+1])*pow((u[i]-x[k]),2)*m[k+1]/pow(h[k],2);s[i]=p1+p2+p3+p4;}printf( "\nx= ");for(i=0;i <=N;i++)printf( "%8.1f ",x[i]);printf( "\ny= ");for(i=0;i <=N;i++)printf( "%8.4f ",y[i]);printf( "\n\nu= ");for(i=0;i <=R;i++)printf( "%9.2f ",u[i]);printf( "\n插值点：s= ");for(i=0;i <=R;i++)printf( "%9.5f ",s[i]);printf("\n");}五、复合梯形公式#include<stdio.h>#include<stdlib.h>#include<math.h>double FTX(int n,float a,float b){double f=0,t,h,*x,*y;int i;x=(double*)malloc((n+1)*sizeof(double));y=(double*)malloc((n+1)*sizeof(double));h=(b-a)/n;for(i=0;i<n+1;i++){x[i]=a+i*h;y[i]=sin(x[i]);}for(i=1;i<n;i++) f=f+2*y[i];t=h/2*(y[0]+f+y[n]);printf("输出函数值：%f\n",t);return 1;}void main(){float a,b;int n;printf("输入区间上，下限：");scanf("%f %f",&a,&b);printf("输入等分区间数：");scanf("%d",&n);FTX(n,a,b);}六、复合辛普森求积公式#include<stdio.h>#include<stdlib.h>#include<math.h>double FSP(int n,float a,float b){double f1=0,f2=0,h,*x1,*y1,*x2,*y2;int i;x1=(double*)malloc((n+1)*sizeof(double));y1=(double*)malloc((n+1)*sizeof(double));x2=(double*)malloc(n*sizeof(double));y2=(double*)malloc(n*sizeof(double));h=(b-a)/n;for(i=0;i<n+1;i++){x1[i]=a+h*i;y1[i]=sin(x1[i])*x1[i];}for(i=0;i<n;i++){x2[i]=x1[i]+h/2;y2[i]=sin(x2[i])*x2[i];}for(i=1;i<n;i++) f1=f1+2*y1[i];for(i=0;i<n;i++) f2=f2+4*y2[i];printf("输出函数值：%f\n",h/6*(y1[0]+f1+f2+y1[n]));return 1;}void main(){float a,b;int n;printf("输入区间上，下限：");scanf("%f %f",&a,&b);printf("输入等分区间数：");scanf("%d",&n);FSP(n,a,b);}七、直接三角分解法#include<stdio.h>#include<stdlib.h>#include<math.h>void main(){doubleA[3][3]={0.25,0.2,0.166667,0.3333,0.25,0.2,0.5,1,2},x[3],y[3],b[3]={9,8,8},L[3][3],U[3][3],f1=0,f2=0;int i,j,k;for(i=0;i<3;i++)for(j=0;j<3;j++){U[i][j]=0;L[i][j]=0;}for(i=0;i<3;i++){U[0][i]=A[0][i];L[i][0]=A[i][0]/U[0][0];L[i][i]=1;}for(i=1;i<3;i++)for(j=i;j<3;j++){for(k=0;k<=i-1;k++){f1=f1+L[i][k]*U[k][j];f2+=L[j][k]*U[k][i];} U[i][j]=A[i][j]-f1;L[j][i]=(A[j][i]-f2)/U[i][i];f1=0;f2=0;}y[0]=b[0];for(i=1;i<3;i++){for(j=0;j<=i-1;j++) f1+=L[i][j]*y[j];y[i]=b[i]-f1;f1=0;}x[2]=y[2]/U[2][2];for(i=1;i>=0;i--){for(j=i+1;j<3;j++) f2+=U[i][j]*x[j];x[i]=(y[i]-f2)/U[i][i];f2=0;} printf("输出L矩阵：\n");for(i=0;i<3;i++){for(j=0;j<3;j++)printf("%f ",L[i][j]);printf("\n");}printf("输出U矩阵：\n");for(i=0;i<3;i++){for(j=0;j<3;j++)printf("%f ",U[i][j]);printf("\n");}printf("输出求解结果：\n");for(i=0;i<3;i++)printf("%f ",x[i]);printf("\n");}八、改进的平方法#include<stdio.h>#include<stdlib.h>#include<math.h>void main(){double A[3][3]={2,-1,1,-1,-2,3,1,3,1},x[3],y[3],b[3]={4,5,6},d[3],L[3][3],U[3][3],f1=0,f2=0; int i,j,k,n=3;for(i=0;i<n;i++)for(j=0;j<n;j++) {U[i][j]=0;L[i][j]=0;}d[0]=A[0][0];L[0][0]=1;for(i=1;i<n;i++){L[i][i]=1;for(j=0;j<=i-1;j++) {for(k=0;k<=j-1;k++) f1+=U[i][k]*L[j][k];U[i][j]=A[i][j]-f1;L[i][j]=U[i][j]/d[j];f2+=U[i][j]*L[i][j];f1=0;}d[i]=A[i][j]-f2;f2=0;}y[0]=b[0];for(i=1;i<n;i++) {for(j=0;j<=i-1;j++) f1+=L[i][j]*y[j];y[i]=b[i]-f1;f1=0;}x[n-1]=y[n-1]/d[n-1];for(i=n-2;i>=0;i--) {for(j=i+1;j<n;j++) f2+=L[j][i]*x[j];x[i]=y[i]/d[i]-f2;f2=0;}printf("输出L矩阵：\n");for(i=0;i<n;i++){for(j=0;j<n;j++) printf("%f ",L[i][j]); printf("\n");}printf("输出U矩阵：\n");for(i=0;i<n;i++){for(j=0;j<n;j++) printf("%f ",U[i][j]); printf("\n");}printf("输出求解结果：\n");for(i=0;i<n;i++) printf("%f ",x[i]);printf("\n");}九、追赶法#include<stdio.h>#include<stdlib.h>#include<math.h>void main(){double A[5][5]={2,-1,0,0,0,-1,2,-1,0,0,0,-1,2,-1,0,0,0,-1,2,-1,0,0,0,-1,2},f[5]={1,0,0,0,0}; double x[5],y[5],U[5][5];int i,j,n=5;for(i=0;i<n;i++)for(j=0;j<n;j++) U[i][j]=0;U[0][0]=U[n-1][n-1]=1;U[0][1]=A[0][1]/A[0][0];for(i=1;i<n-1;i++){ U[i][i]=1;U[i][i+1]=A[i][i+1]/(A[i][i]-A[i][i-1]*U[i-1][i]);}y[0]=f[0]/A[0][0];for(i=1;i<n;i++)y[i]=(f[i]-A[i][i-1]*y[i-1])/(A[i][i]-A[i][i-1]*U[i-1][i]);x[n-1]=y[n-1];for(i=n-2;i>=0;i--) x[i]=y[i]-U[i][i+1]*x[i+1];printf("输出U矩阵：\n");for(i=0;i<n;i++){ for(j=0;j<n;j++) printf("%f ",U[i][j]); printf("\n");}printf("输出求解结果：\n");for(i=0;i<n;i++) printf("%f ",x[i]);printf("\n");}十、雅可比迭代法#include<stdio.h>#include<stdlib.h>#include<math.h>void main(){double A[3][3]={5,2,1,-1,4,2,2,-3,10},x[50][3],b[3]={-12,20,3},f=0,L=1;int n=3,i,j,k=1;for (i=0;i<3;i++) x[0][i]=0;/*初始值*/while(L<0.003){for(i=0;i<n;i++){ for(j=0;j<n;j++)if(j!=i) f=f+A[i][j]*x[k-1][j];x[k][i]=(b[i]-f)/A[i][i];}L=x[k][0]-x[k-1][0];for(i=1;i<n;i++)if((x[k][i]-x[k-1][i])/(x[k][i-1]-x[k-1][i-1])>1||(x[k][i]-x[k-1][i])/(x[k][i-1]-x[k-1][i-1])<-1) L=x[k][i]-x[k-1][i];k++;}printf("输出每次迭代求得的X值：\n");for(i=1;i<k;i++){printf("第%d次迭代：",i);for(j=0;j<n;j++)printf("%f ",x[i][j]);printf("\n");}printf("\n输出迭代次数：%d\n",k-1);}十一、高斯—塞德尔迭代法#include<stdio.h>#include<stdlib.h>#include<math.h>void main(){double A[3][3]={5,2,1,-1,4,2,2,-3,10},x[50][3],b[3]={-12,20,3},f1=0,f2=0,L=1;int i,j,k=1,n=3;for (i=0;i<3;i++) x[0][i]=0;while(L>0.00001||L<-0.00001){for(i=0;i<n;i++){ for(j=0;j<n;j++){if(j<i) f1+=A[i][j]*x[k][j];else if(j>i) f2+=A[i][j]*x[k-1][j];}x[k][i]=(b[i]-f1-f2)/A[i][i];}L=x[k][0]-x[k-1][0];for(i=1;i<n;i++)if((x[k][i]-x[k-1][i])/(x[k][i-1]-x[k-1][i-1])>1||(x[k][i]-x[k-1][i])/(x[k][i-1]-x[k-1][i-1])<-1) L=x[k][i]-x[k-1][i];k++;}printf("输出迭代X矩阵：\n");for(i=1;i<k;i++){for(j=0;j<n;j++) printf("%f ",x[i][j]); printf("\n");}printf("\n输出迭代次数：%d\n",k-1);}十二、超松弛迭代法#include<stdio.h>#include<stdlib.h>#include<math.h>void main(){double A[3][3]={5,2,1,-1,4,2,2,-3,10},x[30][3],b[3]={-12,20,3},f1=0,f2=0,L=1,w=0.9;int i,j,k=1,n=3;for(i=0;i<3;i++) x[0][i]=0;while(L>0.0001||L<-0.0001){for(i=0;i<n;i++){ for(j=0;j<n;j++){if(j<i) f1+=A[i][j]*x[k][j];else if(j>i) f2+=A[i][j]*x[k-1][j];}x[k][i]=w*(b[i]-f1-f2)/A[i][i];}L=x[k][0]-x[k-1][0];for(i=1;i<n;i++)if((x[k][i]-x[k-1][i])/(x[k][i-1]-x[k-1][i-1])>1||(x[k][i]-x[k-1][i])/(x[k][i-1]-x[k-1][i-1])<-1) L=x[k][i]-x[k-1][i];k++;}printf("输出迭代X矩阵：\n");for(i=1;i<k;i++){for(j=0;j<n;j++) printf("%f ",x[i][j]); printf("\n");}printf("\n输出迭代次数：%d\n",k-1);}数值分析算法程序班级：计算08Q2班姓名：甄彦福。