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最小二乘法 多项式拟合 C语言实现

标签：计算方法实验

拟合结果：

/*    本实验根据数组x[], y[]列出的一组数据，用最小二乘法求它的拟合曲线。    近似解析表达式为y = a0 + a1 * x + a2 * x^2 + a3 * x^3;*/#include 
   
    #include 
    
     #define maxn 12#define rank_ 3int main(){    double x[maxn] = {
  0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55};    double y[maxn] = {
  0, 1.27, 2.16, 2.86, 3.44, 3.87, 4.15, 4.37, 4.51, 4.58, 4.02, 4.64};    double atemp[2 * (rank_ + 1)] = {
  0}, b[rank_ + 1] = {
  0}, a[rank_ + 1][rank_ + 1];    int i, j, k;    for(i = 0; i < maxn; i++){  //        atemp[1] += x[i];        atemp[2] += pow(x[i], 2);        atemp[3] += pow(x[i], 3);        atemp[4] += pow(x[i], 4);        atemp[5] += pow(x[i], 5);        atemp[6] += pow(x[i], 6);        b[0] += y[i];        b[1] += x[i] * y[i];        b[2] += pow(x[i], 2) * y[i];        b[3] += pow(x[i], 3) * y[i];    }    atemp[0] = maxn;    /*    for(i = 0; i <= 2 * rank_; i++)  printf("atemp[%d] = %f\n", i, atemp[i]);    printf("\n");    for(i = 0; i <= rank_; i++)  printf("b[%d] = %f\n", i, b[i]);    printf("\n");    */    for(i = 0; i < rank_ + 1; i++){  //构建线性方程组系数矩阵，b[]不变        k = i;        for(j = 0; j < rank_ + 1; j++)  a[i][j] = atemp[k++];    }    /*    for(i = 0; i < rank_ + 1; i++){        for(j = 0; j < rank_ + 1; j++)  printf("a[%d][%d] = %-17f  ", i, j, a[i][j]);        printf("\n");    }    printf("\n");    */    //以下为高斯列主元消去法解线性方程组    for(k = 0; k < rank_ + 1 - 1; k++){  //n - 1列        int column = k;        double mainelement = a[k][k];        for(i = k; i < rank_ + 1; i++)  //找主元素            if(fabs(a[i][k]) > mainelement){                mainelement = fabs(a[i][k]);                column = i;            }        for(j = k; j < rank_ + 1; j++){  //交换两行            double atemp = a[k][j];            a[k][j] = a[column][j];            a[column][j] = atemp;        }        double btemp = b[k];        b[k] = b[column];        b[column] = btemp;        for(i = k + 1; i < rank_ + 1; i++){  //消元过程            double Mik = a[i][k] / a[k][k];            for(j = k; j < rank_ + 1; j++)  a[i][j] -= Mik * a[k][j];            b[i] -= Mik * b[k];        }    }    /*    for(i = 0; i < rank_ + 1; i++){  //经列主元高斯消去法得到的上三角阵(最后一列为常系数)        for(j = 0; j < rank_ + 1; j++)  printf("%20f", a[i][j]);        printf("%20f\n", b[i]);    }    printf("\n");    */    b[rank_ + 1 - 1] /= a[rank_ + 1 - 1][rank_ + 1 - 1];  //回代过程    for(i = rank_ + 1 - 2; i >= 0; i--){        double sum = 0;        for(j = i + 1; j < rank_ + 1; j++)  sum += a[i][j] * b[j];        b[i] = (b[i] - sum) / a[i][i];    }    //高斯列主元消去法结束    printf("P(x) = %f%+fx%+fx^2%+fx^3\n\n", b[0], b[1], b[2], b[3]);    /*    for(i = 0; i < maxn; i++){  //误差比较        double temp = b[0] + b[1] * x[i] + b[2] * x[i] * x[i] + b[3] * x[i] * x[i] * x[i];        printf("%f    %f    error: %f\n", y[i], temp, temp - y[i]);    }    */    return 0;}
    
   

实验结果：

//清新版/*    本实验根据数组x[], y[]列出的一组数据，用最小二乘法求它的拟合曲线。    近似解析表达式为y = a0 + a1 * x + a2 * x^2 + a3 * x^3;*/#include 
   
    #include 
    
     #define maxn 12#define rank_ 3int main(){    double x[maxn] = {
  0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55};    double y[maxn] = {
  0, 1.27, 2.16, 2.86, 3.44, 3.87, 4.15, 4.37, 4.51, 4.58, 4.02, 4.64};    double atemp[2 * (rank_ + 1)] = {
  0}, b[rank_ + 1] = {
  0}, a[rank_ + 1][rank_ + 1];    int i, j, k;    for(i = 0; i < maxn; i++){  //        atemp[1] += x[i];        atemp[2] += pow(x[i], 2);        atemp[3] += pow(x[i], 3);        atemp[4] += pow(x[i], 4);        atemp[5] += pow(x[i], 5);        atemp[6] += pow(x[i], 6);        b[0] += y[i];        b[1] += x[i] * y[i];        b[2] += pow(x[i], 2) * y[i];        b[3] += pow(x[i], 3) * y[i];    }    atemp[0] = maxn;    for(i = 0; i < rank_ + 1; i++){  //构建线性方程组系数矩阵，b[]不变        k = i;        for(j = 0; j < rank_ + 1; j++)  a[i][j] = atemp[k++];    }    //以下为高斯列主元消去法解线性方程组    for(k = 0; k < rank_ + 1 - 1; k++){  //n - 1列        int column = k;        double mainelement = a[k][k];        for(i = k; i < rank_ + 1; i++)  //找主元素            if(fabs(a[i][k]) > mainelement){                mainelement = fabs(a[i][k]);                column = i;            }        for(j = k; j < rank_ + 1; j++){  //交换两行            double atemp = a[k][j];            a[k][j] = a[column][j];            a[column][j] = atemp;        }        double btemp = b[k];        b[k] = b[column];        b[column] = btemp;        for(i = k + 1; i < rank_ + 1; i++){  //消元过程            double Mik = a[i][k] / a[k][k];            for(j = k; j < rank_ + 1; j++)  a[i][j] -= Mik * a[k][j];            b[i] -= Mik * b[k];        }    }    b[rank_ + 1 - 1] /= a[rank_ + 1 - 1][rank_ + 1 - 1];  //回代过程    for(i = rank_ + 1 - 2; i >= 0; i--){        double sum = 0;        for(j = i + 1; j < rank_ + 1; j++)  sum += a[i][j] * b[j];        b[i] = (b[i] - sum) / a[i][i];    }    //高斯列主元消去法结束    printf("P(x) = %f%+fx%+fx^2%+fx^3\n\n", b[0], b[1], b[2], b[3]);    return 0;}
    
   

实验结果：