大数运算

高精度整数：使用vector/数组存储每一位数字

下面介绍具体的实现

加法和减法

加法：

a/b + -
+ a+b a-(-b)
- -((-a)-b) -((-a)+(-b))

减法：

a/b(a-b） + -
+ a-b / -(b-a) a+(-b)
- -((-a)+b) -((-a)-(-b)) / (-b)-(-a)

从上表看出，实现加法和减法只需要实现：

  1. 两个正整数相加
  2. 一个较大的正整数减去一个较小的正整数

注意：减法后可能出现vector后面有多余的零的情况，需要去掉

乘法

以123*345=42435为例

存储在vector中为

BigNum.vec 0 1 2 3
num1 123 0 3 2 1
num2 345 0 5 4 3

结果存储在ans中

ans.vec 0 1 2 3 4 5
循环过程 0 15 12+10 9+8+5 6+4 3
循环结果 0 15 22 22 10 3
进位后 0 5（进1） 3（进2） 4（进2） 2（进1） 4

除法和取余

以7546除以23等于328为例

时间 b1 sub ans add
i=0 7546 2300 0 100
i=1 646 230 300 10
i=2 186 23 320 1
从第三次循环出来后 2 2 328 0

取余借助除法实现即可

完整代码

BigNum.h

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#ifndef TERMPJ_BIGNUM_H
#define TERMPJ_BIGNUM_H

#include <vector>
#include <iostream>

using std::vector;

class BigNum {
    friend BigNum absv(const BigNum& num);

    friend std::istream& operator>>(std::istream&, BigNum&);
    friend std::ostream& operator<<(std::ostream&, const BigNum&);

    friend BigNum operator-(const BigNum& num);

    friend BigNum operator+(const BigNum& num1, const BigNum& num2);
    friend BigNum operator-(const BigNum& num1, const BigNum& num2);
    friend BigNum operator+(const BigNum& num1, int num2);
    friend BigNum operator-(const BigNum& num1, int num2);
    friend BigNum operator*(const BigNum& num1, const BigNum& num2);
    friend BigNum operator*(const BigNum& num1, int num2);
    friend BigNum operator/(const BigNum& num1, const BigNum& num2);
    friend BigNum operator/(const BigNum& num1, int num2);
    friend BigNum operator%(const BigNum& num1, const BigNum& num2);
    friend BigNum operator%(const BigNum& num1, int num2);

    friend bool operator<(const BigNum& num1, const BigNum& num2);
    friend bool operator>(const BigNum& num1, const BigNum& num2);
    friend bool operator<=(const BigNum& num1, const BigNum& num2);
    friend bool operator>=(const BigNum& num1, const BigNum& num2);
    friend bool operator==(const BigNum& num1, const BigNum& num2);
    friend bool operator!=(const BigNum& num1, const BigNum& num2);

public:
    typedef vector<int>::size_type size_type;

    // 构造函数
    BigNum() { create(); }
    explicit BigNum(int num) { create(num); }
    BigNum(size_t n, int v);

    size_type size() const { return vec.size() - 1; }
    bool isNegative() const { return vec[0] != 0; }
    bool isZero() const { return vec.size() == 2 && vec[1] == 0; }

    BigNum& operator=(const BigNum& num);

    int& operator[](size_type i) {
        return vec[i];
    }

    void trim() {
        while (vec.size() > 2) {
            if (*(vec.end() - 1) == 0) {
                vec.erase(vec.end() - 1);
            } else return;
        }
    } // 减法后可能出现前面有多余的零的情况，需要去掉

private:
    void create();
    void create(int num);

protected:
    vector<int> vec;
};


#endif //TERMPJ_BIGNUM_H

BigNum.cpp

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#include "BigNum.h"
#include <vector>
#include <iostream>
#include <cctype>
#include <stdexcept>

using std::cout;
using std::endl;
using std::vector;
using std::ostream;
using std::istream;
using std::isspace;
using std::domain_error;

void BigNum::create() {
}

void BigNum::create(int num) {
    vec.push_back(num < 0 ? 1 : 0);
    int temp = this->isNegative() ? -num : num;
    while (temp != 0) {
        int digit = temp % 10;
        vec.push_back(digit);
        temp = temp / 10;
    }
}

BigNum::BigNum(size_t n, int v) {
    vec.push_back(0);
    for (BigNum::size_type i = 0; i != n; i++) {
        vec.push_back(v);
    }
}

ostream& operator<<(ostream& os, const BigNum& num)
{
    if (num.size() == 0) {
        os << "0";
        return os;
    }
    if ( num.isNegative() ) {
        os << "-";
    }
    for (BigNum::size_type i = num.size(); i != 0; i--) {
        os << num.vec[i];
    }
    return os;
}

istream& operator>>(istream& is, BigNum& num)
{
    num.vec.clear();
    vector<int> temp;

    char v;
    while (is.get(v) && isspace(v))
        ;

    if (is) {
        do	temp.push_back(v);
        while (is.get(v) && !isspace(v));

        if (is)
            is.unget();
    }
    num.vec.push_back(temp[0] == '-' ? 1 : 0);
    if (num.vec[0] == 0) {
        for (int i = temp.size() - 1; i >= 0; i--) {
            num.vec.push_back(temp[i] - '0');
        }
    } else {
        for (int i = temp.size() - 1; i >= 1; i--) {
            num.vec.push_back(temp[i] - '0');
        }
    }
    return is;
}

bool operator<(const BigNum& num1, const BigNum& num2) {
    if ( num1.isNegative() ) {
        if ( !num2.isNegative() ) // -x < y
            return true;
        if ( num1.size() > num2.size() ) // -xx < -y
            return true;
        if ( num1.size() < num2.size() ) // -x > -yy
            return false;

        for (BigNum::size_type i = num1.size(); i >= 1; i--) {
            if ( num1.vec[i] < num2.vec[i] )
                return false;
            if ( num1.vec[i] > num2.vec[i] )
                return true;
        }

        return false;
    }  else {
        if ( num2.isNegative() ) // x > -y
            return false;
        if ( num1.size() > num2.size() ) // xx > y
            return false;
        if ( num1.size() < num2.size() ) // x < yy
            return true;

        for (BigNum::size_type i = num1.size(); i >= 1; i--) {
            if ( num1.vec[i] < num2.vec[i] )
                return true;
            if ( num1.vec[i] > num2.vec[i] )
                return false;
        }

        return false;
    }
}

bool operator>(const BigNum& num1, const BigNum& num2) {
    return num2 < num1;
}

bool operator<=(const BigNum& num1, const BigNum& num2) {
    return !(num1 > num2);
}

bool operator>=(const BigNum& num1, const BigNum& num2) {
    return !(num1 < num2);
}

bool operator==(const BigNum& num1, const BigNum& num2) {
    return (num1 <= num2) && (num1 >= num2);
}

bool operator!=(const BigNum& num1, const BigNum& num2) {
    return !(num1 == num2);
}

BigNum operator-(const BigNum& num) {
    BigNum negative;
    negative.vec.push_back(num.isNegative() ? 0 : 1);
    for (BigNum::size_type i = 1; i <= num.size(); i++) {
        negative.vec.push_back(num.vec[i]);
    }
    return negative;
}

BigNum & BigNum::operator=(const BigNum &num){
    this->vec = *new vector<int>();
    for (BigNum::size_type i = 0; i <= num.size(); i++) {
        this->vec.push_back(num.vec[i]);
    }
    return *this;
}

BigNum absv(const BigNum& num) {
    BigNum abs;
    abs.vec.push_back(0);
    for (BigNum::size_type i = 1; i <= num.size(); i++) {
        abs.vec.push_back(num.vec[i]);
    }
    return abs;
}

BigNum operator+(const BigNum& num1, const BigNum& num2) {
    BigNum b1 = absv(num1) > absv(num2) ? num1 : num2;
    BigNum b2 = absv(num1) > absv(num2) ? num2 : num1;
    BigNum ans;
    int carry = 0;
    if (!b1.isNegative() && !b2.isNegative()) {
        ans.vec.push_back(0);
        for (BigNum::size_type i = 1; i <= b1.size(); i++) {
            carry += b1[i];
            if(i <= b2.size()) carry += b2[i];
            ans.vec.push_back(carry % 10);
            carry /= 10;
        }
        if (carry != 0) ans.vec.push_back(carry);
        return ans;
    } else if (b1.isNegative() && !b2.isNegative()) {
        return -( (-b1) - b2 );
    } else if (!b1.isNegative() && b2.isNegative()) {
        return b1 - (-b2);
    } else if (b1.isNegative() && b2.isNegative()) {
        return -((-b1) + (-b2));
    }

}

BigNum operator-(const BigNum& num1, const BigNum& num2) {
    if (num1 == num2)
    {
        return *new BigNum(0);
    }
    BigNum ans;
    BigNum b1 = num1;
    BigNum b2 = num2;
    int carry = 0;
    if (!num1.isNegative() && !num2.isNegative()) {
        if (absv(num1) > absv(num2)) {
            ans.vec.push_back(0);
            for (BigNum::size_type i = 1; i <= b1.size(); i++) {
                if (i <= b2.size()) {
                    if (b1[i] + carry < b2[i]) {
                        ans.vec.push_back(10 + b1[i] - b2[i] + carry);
                        carry = -1;
                    } else {
                        ans.vec.push_back(b1[i] - b2[i] + carry);
                        carry = 0;
                    }
                } else {
                    if (b1[i] + carry < 0) {
                        ans.vec.push_back(10 + b1[i] + carry);
                        carry = -1;
                    } else {
                        ans.vec.push_back(b1[i] + carry);
                        carry = 0;
                    }
                }
            }
        }
        else {
            return -(num2 - num1);
        }
    } else if (num1.isNegative() && !num2.isNegative()) {
        return -( (-num1) + num2 );
    } else if (!num1.isNegative() && num2.isNegative()) {
        return num1 + (-num2);
    } else if (num1.isNegative() && num2.isNegative()) {
       if (absv(num1) > absv(num2)) {
           return -( (-num1) - (-num2) );
       } else {
           return (-num2) - (-num1);
       }
    }

    ans.trim();
    return ans;
}

BigNum operator+(const BigNum& num1, int num2) {
    return num1 + *new BigNum(num2);
}

BigNum operator-(const BigNum& num1, int num2) {
    return num1 - *new BigNum(num2);
}

BigNum operator*(const BigNum &num1, const BigNum &num2) {
    if (num1.isZero() || num2.isZero()) {
        return *new BigNum(0);
    }
    BigNum ans;
    if (!num1.isNegative() && !num2.isNegative()) {
        ans.vec.push_back(0);
        for (BigNum::size_type i = 1; i <= num1.size(); i++) {
            for (BigNum::size_type j = 1; j <= num2.size(); j++) {
                if (ans.vec.size() <= i + j - 1) {
                    ans.vec.push_back(0);
                }
                ans.vec[i + j - 1] += num1.vec[i] * num2.vec[j];
            }
        }
        for (BigNum::size_type i = 1; i < ans.size(); i++) {
            ans.vec[i + 1] += ans.vec[i] / 10;
            ans.vec[i] = ans.vec[i] % 10;
        }
        if (ans.vec[ans.size()] >= 10) {
            ans.vec.push_back(ans.vec[ans.size()] / 10);
            ans.vec[ans.size()] = ans.vec[ans.size()] % 10;
        }
        return ans;
    } else if (!num1.isNegative() && num2.isNegative()) {
        return - (num1 * (-num2));
    } else if (num1.isNegative() && !num2.isNegative()) {
        return - ((-num1) * num2);
    } else if (num1.isNegative() && num2.isNegative()) {
        return (-num1) * (-num2);
    }
}

BigNum operator*(const BigNum &num1, int num2) {
    return num1 * *new BigNum(num2);
}

BigNum operator/(const BigNum &num1, const BigNum &num2) {
    if ( num2.isZero() ) {
        throw domain_error("dividend cannot be zero!");
    }
    if ( num1.isZero() ) {
        return *new BigNum(0);
    }
    if ( num1 < num2 ) {
        return num1;
    }
    BigNum ans = *new BigNum(0);
    BigNum b1 = num1;
    BigNum b2 = num2;
    if (!num1.isNegative() && !num2.isNegative()) {
        BigNum::size_type diff = b1.size() - b2.size();
        BigNum sub = b2;
        BigNum add = *new BigNum(1);
        for (BigNum::size_type i = 0; i < diff; i++) {
            sub = sub * 10;
            add = add * 10;
        }
        for (BigNum::size_type i = 0; i <= diff; i++) {
            while (b1 >= sub) {
                b1 = b1 - sub;
                ans = ans + add;
            }
            sub.vec.erase(sub.vec.begin() + 1);
            add.vec.erase(add.vec.begin() + 1);
        }
        return ans;
    } else if (!num1.isNegative() && num2.isNegative()) {
        return - (num1 / (-num2));
    } else if (num1.isNegative() && !num2.isNegative()) {
        return - ((-num1) / num2);
    } else if (num1.isNegative() && num2.isNegative()) {
        return (-num1) / (-num2);
    }

    return ans;
}

BigNum operator/(const BigNum &num1, int num2) {
    return num1 / *new BigNum(num2);
}

BigNum operator%(const BigNum &num1, const BigNum &num2) {
    return num1 - ( (num1 / num2) * num2 );
}

BigNum operator%(const BigNum &num1, int num2) {
    return num1 / *new BigNum(num2);
}
算法
C++

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