C++实现有分数结果的高斯消元法

1.程序源码

#include <iostream>

using namespace std;

#define MAX 100

//定义分数类
class Fractional
{
public:
    int getstd();
    int putstd();
    int iszero();
    friend Fractional operator+(const Fractional &a, const Fractional &b);
    friend Fractional operator-(const Fractional &a, const Fractional &b);
    friend Fractional operator*(const Fractional &a, const Fractional &b);
    friend Fractional operator/(const Fractional &a, const Fractional &b);

private:
    int a;
    int b;

    static int gcd(int c, int d);
    static int reduce(int *c, int *d);
};

int Fractional::iszero()
{
    int c = (a == 0);

    return c;
}

int Fractional::gcd(int c, int d)
{
    if(d == 0)
        return c;
    else
        return gcd(d, c%d);
}

int Fractional::reduce(int *c, int *d)
{
    int flag = 0;
    int x = 0;

    if(*d < 0)
    {
        *c = -*c;
        *d = -*d;
        flag = 1;
    }

    if(*c > 0)
    {
        x = gcd(*c, *d);

        if(x > 1)
        {
            *c /= x;
            *d /= x;
            flag = 1;
        }
    }
    else if(*c < 0)
    {
        x = gcd(-*c, *d);

        if(x > 1)
        {
            *c /= x;
            *d /= x;
            flag = 1;
        }
    }
    else
        *d = 1;

    return flag;
}

int Fractional::getstd()
{
    cin >> a;
    b = 1;

    return 0;
}

int Fractional::putstd()
{
    if(b == 1)
    {
        cout << a;
    }
    else
    {
        cout << a << "/" << b;
    }

    return 0;
}

Fractional operator+(const Fractional &a, const Fractional &b)
{
    Fractional c;
    int x;
    int y;
    int z;

    if(a.b != b.b)
    {
        x = Fractional::gcd(a.b, b.b);
        y = a.b / x;
        z = b.b / x;

        c.a = a.a * z + b.a * y;
        c.b = y * b.b;
    }
    else
    {
        c.a = a.a + b.a;
        c.b = a.b;
    }

    Fractional::reduce(&c.a, &c.b);

    return c;
}

Fractional operator-(const Fractional &a, const Fractional &b)
{
    Fractional c;
    int x;
    int y;
    int z;

    if(a.b != b.b)
    {
        x = Fractional::gcd(a.b, b.b);
        y = a.b / x;
        z = b.b / x;

        c.a = a.a * z - b.a * y;
        c.b = y * b.b;
    }
    else
    {
        c.a = a.a - b.a;
        c.b = a.b;
    }

    Fractional::reduce(&c.a, &c.b);

    return c;
}

Fractional operator*(const Fractional &a, const Fractional &b)
{
    Fractional c;

    c.a = a.a * b.a;
    c.b = a.b * b.b;

    Fractional::reduce(&c.a, &c.b);

    return c;
}

Fractional operator/(const Fractional &a, const Fractional &b)
{
    Fractional c;

    c.a = a.a * b.b;
    c.b = a.b * b.a;

    Fractional::reduce(&c.a, &c.b);

    return c;
}

template <class T>
int input_matrix(T **A, T *b)
{
    int n = 0;
    int i, j;

    cout << "系数矩阵A的阶数: ";
    cin >> n;

    for(i=0; i<n; i++)
    {
        cout << "系数矩阵第" << i+1 << "行元素" << endl;
        for(j=0; j<n; j++)
            A[i][j].getstd();
    }

    cout << "等式右端项b: ";

    for(i=0; i<n; i++)
        b[i].getstd();

    return n;
}

//代模板的高斯消元法
template <class T>
int gauss_sovle(T **A, T *b, T *X)
{
        T mik;
        T s;
    int n;
        int i, j, k;

        n = input_matrix(A, b);

        //消元
        for(k=0; k<n-1; k++)
        {
                if(A[k][k].iszero())
                        return -1;

                for(i=k+1; i<n; i++)
                {
                        mik = A[i][k] / A[k][k];

                        for(j=k; j<n; j++)
                        {
                                A[i][j] = A[i][j] - mik * A[k][j];
                        }

                        b[i] = b[i] - mik * b[k];
                }
        }

        cout << "消元后的矩阵A: " << endl;

        for(i=0; i<n; i++)
        {
                for(j=0; j<n; j++)
        {
            A[i][j].putstd();
            cout << " ";
        }

                cout << endl;
        }

        //回代
        X[n-1] = b[n-1] / A[n-1][n-1];

        for(k=n-2; k>=0; k--)
        {
                s = b[k];

                for(j=k+1; j<n; j++)
                {
                        s = s - A[k][j] * X[j];
                }

                X[k] = s / A[k][k];
        }

        cout << "结果X: ";

        for(i=0; i<n; i++)
    {
        X[i].putstd();
                cout << " ";
    }

        cout << endl;

    return 0;
}

//主函数
int main()
{
    Fractional *A[MAX];
    Fractional B[MAX][MAX];
    Fractional b[MAX];
    Fractional X[MAX];
    int i = 0;

    for(i=0; i<MAX; i++)
        A[i] = B[i];

    gauss_sovle(A, b, X);

    return 0;
}

2.编译源码

$ g++ -o gauss_frac gauss_frac.cpp

3.运行程序

./gauss_frac 
系数矩阵A的阶数: 3
系数矩阵第1行元素
1 3 4   
系数矩阵第2行元素
1 4 7
系数矩阵第3行元素
9 3 2
等式右端项b: 5 3 2
消元后的矩阵A: 
1 3 4 
0 1 3 
0 0 38 
结果X: -37/38 197/38 -91/38