高斯消去法求解方程组的实现(C++)

高斯消去法，其实就是利用初等变换把矩阵转换成上三角，便于求行列式，求逆，求线性方程组的解

只要动笔演算一下相关矩阵元素的索引，就能使得核心代码特别短，

初等变换如下

    for (int i = 0; i < column - 1; i++) {
        for (int j = 0; j < row - 1 - i; j++) {
            double coef = *(matrix + (row - 1 - j)*column + i) / *(matrix + i * column + i);
            for (int k = 0; k < column; k++) {
                *(matrix + (row - 1 - j)*column + k) -= (coef*(*(matrix + i * column + k)));
            }
        }
    }

方程求解代码如下：

    double* result = new double[column-1];
    for(int i=0;i<column-1;i++){
        double temp = 0;
        for(int j=0;j<i;j++){
            temp += *(matrix+(column-2-i)*column+column-1-i+j)*(*(result+column-1-i+j));
        }
        *(result+column-2-i) = (*(matrix+(column-2-i)*column+column-1) - temp) / *(matrix+(column-2-i)*column+column-2-i);

    }

完整代码给出如下：

#include <iostream>

void printMatrix(double *matrix, int row, int column) {
    for (int i = 0; i < row; i++) {
        for (int j = 0; j < column; j++) {
            printf("%6.2lf    ", *(matrix + i * column + j));
        }
        printf("\n");
    }
}

void printVector(double *matrix, int len) {
    for (int i = 0; i < len; i++) {
        printf("%6.2lf,\t", *(matrix + i));
    }
    printf("\n");
}

void gaussianElimination(double *matrix, int row, int column) {
    int i,j;
    for (i = 0; i < column - 1; i++) {
        for (j = 0; j < row - 1 - i; j++) {
            //printf("分母%lf，分子%lf\n", *(matrix + i * column + i), *(matrix + (row - 1 - j)*column + i));
            double coef = *(matrix + (row - 1 - j)*column + i) / *(matrix + i * column + i);
            //printf("系数：%lf，", coef);
            for (int k = 0; k < column; k++) {
                //printf("%lf -= %lf * %lf\n", *(matrix + (row - 1 - j)*column + k), coef, *(matrix + i * column + k));
                *(matrix + (row - 1 - j)*column + k) -= (coef*(*(matrix + i * column + k)));
                //printf("%lf\n", *(matrix + (row - 1 - j)*column + k));
            }
        }
        //printf("\n");
    }
    printMatrix(matrix, row, column);

    double* result = new double[column-1];
    for(i=0;i<column-1;i++){
        double temp = 0;
        for(j=0;j<i;j++){
            temp += *(matrix+(column-2-i)*column+column-1-i+j)*(*(result+column-1-i+j));
        }
        *(result+column-2-i) = (*(matrix+(column-2-i)*column+column-1) - temp) / *(matrix+(column-2-i)*column+column-2-i);

    }
    printf("求得方程的解为：");
     printVector(result, column-1);
    if(result){
        delete[] result;
    }
}

用主函数测试一下正确性

int main() {double matrix[] = {1,1,0,3,4,
        2,1,-1,1,1,
        3,-1,-1,2,-3,
        -1,2,3,-1,4};    
    int row = 4, column = 5;
    
    gaussianElimination(matrix, row, column);
return 0;
}

结果如下，显然正确

高斯消去法求解方程组的实现(C++)

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