任意长度的两个大数的乘法

方法（一）：

关于大数乘法，可以使用数组来模拟小学三年级的乘法竖式计算过程，代码如下：

#include "iostream"
#include "string"
using namespace std;
int main(void)
{
    char str1[1000],str2[1000];
    int i,j,len1,len2,len;
    bool flag=false;
    cout<<"任意两个大数的乘法（用数组来模拟小学三年级的乘法竖式计算过程）："<<endl;
    cout<<"请输入被乘数：";
    gets(str1);
    cout<<"请输入乘数：";
    gets(str2);
    len1=strlen(str1);
    len2=strlen(str2);
    int *a=new int[len1];
    int *b=new int[len2];
    len=len1*len2+1;
    int *result= new int[len];
    for(i=0;i<len;i++)
        result[i]=0;
    for(i=0;i<len1;i++) //将字符串转换为整数，并倒置过来
        a[i]=str1[len1-1-i]-'0';
    for(i=0;i<len2;i++)
        b[i]=str2[len2-1-i]-'0';
    for(i=0;i<len1;i++)  //用数组来模拟小学三年级的乘法竖式计算过程
    {
        for(j=0;j<len2;j++)
            result[i+j]+=a[i]*b[j];
    }
    for(i=0;i<len;i++)   //处理进位的情况
    {
        if(result[i]>9)
        {
            result[i+1]+=result[i]/10;
            result[i]%=10;
        }
    }
    cout<<"两个大整数相乘的结果为：";
    for(i=len-1;i>=0;i--)
    {
        if(flag)
            cout<<result[i];
        else if(result[i]!=0)
        {
            cout<<result[i];
            flag=true;
        }
    }
    cout<<endl;
    delete []a;
    delete []b;
    delete []result;
    system("pause");
    return 0;
}

方法（二）：关于大数乘法，可以使用大整数乘法的分治方法：

设X和Y都是n位的整数，现在要计算它们的乘积XY。如果
**利用小学所学的方法，将每两个一位数都进行相乘，最后
**再相加，效率比较低下，乘法需要n^2次。分治的方法可以
**减少乘法的次数，设X被分成2部分A和B，即X=A*10^(n/2)+B
**Y也同样处理，即Y=C*10^(n/2)+D.
**那么，XY=(A*10^(n/2)+B)*(C*10^(n/2)+D)
               =AC*10^n+(AD+BC)*10^(n/2)+BD ------>(1)
**AD和BC可以利用AC和BD来表示，AD+BC=(A-B)*(D-C)+AC+BD --->(2)
**这样(1)的乘法次数由4次减少到3次。

**最后的运算效率会有所提高。

***以上出自   计算机算法设计与分析(王晓东) *******/

代码如下：

#include "iostream"
#include "list"
#include "string"
using namespace std;

/*******大整数减法*********/
list<char> long_sub(list<char> clist1, list<char> clist2);
/*******大整数加法*********/
list<char> long_add(list<char> clist1, list<char> clist2)
{
    list<char> rs; //存放最后的结果
    //如果一个正，一个负，做两数相减
    if ( *(clist1.begin()) == '-' && *(clist2.begin()) != '-' )
    {
        clist1.erase(clist1.begin()); //去掉符号位
        rs = long_sub(clist2, clist1);
        return rs;
    }
    //如果一个负，一个正，做两数相减
    if ( *(clist1.begin()) != '-' && *(clist2.begin()) == '-' )
    {
        clist2.erase(clist2.begin()); //去掉符号位
        rs = long_sub(clist1, clist2);
        return rs;
    }
    if ( *(clist1.begin()) == '-' && *(clist2.begin()) == '-' )
    {
        clist1.erase(clist1.begin());
        clist2.erase(clist2.begin());
        rs = long_add(clist1,clist2);
        rs.push_front('-');
        return rs;
    }
    if ( *(clist1.begin()) != '-' && *(clist2.begin()) != '-' )
    {
        //首先保证两数位数相同(填充0)
        int   length1 = clist1.size();
        int   length2 = clist2.size();
        if( length1 < length2 )
        {
            for(int i=0; i<(length2 -length1); ++i)
            {
                clist1.push_front('0');
            }
        }
        else if ( length1 > length2 )
        {
            for(int i=0; i<(length1 -length2); ++i)
            {
                clist2.push_front('0');
            }
        }

        //整数加法，从低位加起，最低位的进位初始为0
        int   c=0; //低位借位初始为0
        int   low; //减完后本位的数值

        list<char>::iterator iter1 = clist1.end();
        --iter1;
        list<char>::iterator iter2 = clist2.end();
        --iter2;
        for(; iter1!=clist1.begin() && iter2!=clist2.begin();--iter1,--iter2)
        {
            int   num1 = *iter1 -'0';
            int   num2 = *iter2 -'0';
            low = (num1+num2+c)%10;
            c = (num1+num2+c)/10;
            rs.push_front(low+'0');
        }
        //双方最高位相加的处理
        int   num1 = *iter1 -'0';
        int   num2 = *iter2 -'0';
        low = (num1+num2+c)%10;
        c = (num1+num2+c)/10;
        rs.push_front(low+'0');
        if ( c != 0 )
        {
            rs.push_front(c+'0');
        }
        return rs;
    }
    return rs;
}

/*******大整数减法*********/
list<char> long_sub(list<char> clist1, list<char> clist2)
{
    list<char> rs; //存放最后的结果
    //如果一正一负相减，做相加
    if (*(clist1.begin()) != '-' && *(clist2.begin()) == '-' )
    {
        clist2.erase(clist2.begin()); //去掉符号位
        rs = long_add(clist1, clist2);
        return rs;
    }
    //如果一负一正相减，做相加(添符号)
    if ( *(clist1.begin()) == '-' && *(clist2.begin()) != '-' )
    {
        clist1.erase(clist1.begin()); //去掉符号位
        rs   = long_add(clist1, clist2);
        rs.push_front('-');
        return rs;
    }
    //如果两负相减，作相减
    if ( *(clist1.begin()) == '-' && *(clist2.begin()) == '-' )
    {
        clist1.erase(clist1.begin());
        clist2.erase(clist2.begin());   //去掉符号位
        rs = long_sub(clist2, clist1);
        return rs;
    }
    //如果两正相减，做相减
    if ( *(clist1.begin()) != '-' && *(clist2.begin()) != '-' )
    {
        int   sign = -1;   //代表两数相减结果的正负，如果最高位为'-'(ascii码为45)表示负，否则表示正
        //首先保证加数位数相同(填充0)
        int   length1 = clist1.size();
        int   length2 = clist2.size();
        if( length1 < length2 )
        {
            sign = '-';
            for(int i=0; i<(length2 -length1); ++i)
            {
                clist1.push_front('0');
            }
        }
        else if ( length1 > length2 )
        {
            for(int i=0; i<(length1 -length2); ++i)
            {
                clist2.push_front('0');
            }
        }
        else if ( *(clist1.begin()) < *(clist2.begin()) )
        {
            sign = '-';
        }
        //整数减法，从低位减起，最低位的借位初始为0
        int   c = 0; //低位借位初始为0
        int   low; //减完后本位的数值

        list<char>::iterator iter1 = clist1.end();
        --iter1;
        list<char>::iterator iter2 = clist2.end();
        --iter2;
        if (sign != '-' )
        {
            for(; iter1!=clist1.begin() && iter2!=clist2.begin();--iter1,--iter2)
            {
                int   num1 = *iter1 -'0';
                int   num2 = *iter2 -'0';
                int   c_new = 0; //向高位的借位
                if ( num1 < num2+c )
                {
                    c_new = 1;
                    num1 = num1+10;
                }
                low = (num1-num2-c)%10;
                c = c_new;
                rs.push_front(low+'0');
            }
            //双方最高位相减的处理
            int   num1 = *iter1 -'0';
            int   num2 = *iter2 -'0';
            low = (num1-num2-c)%10;
            if ( low != 0 )
            {
                rs.push_front(low+'0');
            }
        }
        else if ( sign == '-' )
        {
            for(; iter1!=clist1.begin() && iter2!=clist2.begin();--iter1,--iter2)
            {
                int   num1 = *iter1 -'0';
                int   num2 = *iter2 -'0';
                int   c_new = 0; //向高位的借位
                if ( num2 < num1+c )
                {
                    c_new = 1;
                    num2 = num2+10;
                }
                low = (num2-num1-c)%10;
                c = c_new;
                rs.push_front(low+'0');
            }
            //双方最高位相减的处理
            int   num1 = *iter1 -'0';
            int   num2 = *iter2 -'0';
            low = (num2-num1-c)%10;
            if ( low != 0 )
            {
                rs.push_front(low+'0');
            }
            rs.push_front('-'); //最高位的'-'作为负数的标志
        }
        return rs;
    }
    return rs;
}
/*******大整数乘法*********/
list<char> long_mul(list<char> clist1, list<char> clist2)
{
    list<char> rs;   //保存最后的结果
    //递归出口
    if(clist1.size() == 1 && clist2.size() == 1)   //两个数都成1位了
    {
        int num1 = *(clist1.begin())-'0';
        int num2 = *(clist2.begin())-'0';
        int   high = (num1*num2)/10;   //积的高位
        int   low   = (num1*num2)%10;   //积的低位
        if ( high != 0 )
        {
            rs.push_back(high+'0');
        }
        rs.push_back(low+'0');
        return rs;
    }
    if (clist1.size() == 1 && clist2.size() > 1)
    {
        int sign = -1; //最后结果的正负标志，'-'(ascii码为45)表示负，其他表示正
        char high_bit2 = *(clist2.begin()); //clist2的最高位
        if ( high_bit2 == '-' )
        {
            sign = '-'; //相乘结果为负
            clist2.erase(clist2.begin()); //去掉高位的'-'
        }
        int length2 = clist2.size(); //clist2的有效长度(去除'-'号后)
        if ( length2 > 1 )
        {
            //对clist2进行拆分
            list<char>   clist2_1;   //高位部分
            list<char>   clist2_2;   //低位部分
            int i;
            list<char>::iterator iter;
            for (i=0,iter=clist2.begin(); i<length2/2; ++i,++iter)
            {
                clist2_1.push_back(*iter);
            }
            for(; iter != clist2.end(); ++iter)
            {
                clist2_2.push_back(*iter);
            }
            list<char> rs1; //高位部分和clist1的积
            list<char> rs2; //低位部分和clist1的积
            rs1 = long_mul(clist1, clist2_1);
            rs2 = long_mul(clist1, clist2_2);
            //对高位积进行移位(末尾添0)
            for( i=0; i<clist2_2.size(); ++i )
            {
                rs1.push_back('0');
            }
            rs = long_add(rs1,rs2); //两部分积相加
            if( sign == '-' )
            {
                rs.push_front('-');
            }
        }
        else
        {
            rs = long_mul(clist1, clist2);
            if ( sign == '-' )
            {
                rs.push_front('-'); //结果添上'-'号
            }
        }
        return rs;
    }
    if (clist1.size() >1 && clist2.size() == 1)
    {
        int sign = -1; //最后结果的正负标志，'-'表示负，其他表示正
        char high_bit1 = *(clist1.begin()); //clist1的最高位
        if ( high_bit1 == '-' )
        {
            sign = '-'; //相乘结果为负
            clist1.erase(clist1.begin()); //去掉高位的'-'
        }
        int length1 = clist1.size(); //clist1的有效长度(去除'-'号后)
        if ( length1 > 1 )
        {
            //对clist1进行拆分
            list<char>   clist1_1;   //高位部分
            list<char>   clist1_2;   //低位部分
            int i;
            list<char>::iterator iter;
            for (i=0,iter=clist1.begin(); i<length1/2; ++i,++iter)
            {
                clist1_1.push_back(*iter);
            }
            for(; iter != clist1.end(); ++iter)
            {
                clist1_2.push_back(*iter);
            }
            list<char> rs1; //高位部分和clist2的积
            list<char> rs2; //低位部分和clist2的积
            rs1 = long_mul(clist1_1, clist2);
            rs2 = long_mul(clist1_2, clist2);
            //对高位积进行移位(末尾添0)
            for( i=0; i<clist1_2.size(); ++i )
            {
                rs1.push_back('0');
            }
            rs = long_add(rs1,rs2); //两部分积相加
            if( sign == '-' )
            {
                rs.push_front('-');
            }
        }
        else
        {
            rs = long_mul(clist1, clist2);
            if ( sign == '-' )
            {
                rs.push_front('-'); //结果添上'-'号
            }
        }
        return rs;
    }
    if (clist1.size() >1 && clist2.size() > 1 )
    {
        int sign = -1; //最后结果的正负标志，'-'表示负，其他表示正
        char   high_bit1 = *(clist1.begin());   //clist1的最高位
        char   high_bit2 = *(clist2.begin());   //clist2的最高位
        if ( high_bit1 == '-'   && high_bit2 != '-' )
        {
            sign = '-';   //相乘结果为负
            clist1.erase(clist1.begin());   //去掉高位的'-'
        }
        if ( high_bit1 != '-' && high_bit2 == '-' )
        {
            sign = '-';   //相乘结果为负
            clist2.erase(clist2.begin());   //去掉高位的'-'
        }
        if ( high_bit1 =='-' && high_bit2 == '-' )
        {
            clist1.erase(clist1.begin());
            clist2.erase(clist2.begin());    //去掉高位的'-'
        }
        int length1 = clist1.size(); //clist1的有效长度
        int length2 = clist2.size(); //clist2的有效长度
        if ( length1 == 1 || length2 == 1 )
        {
            rs = long_mul(clist1, clist2);
            if ( sign == '-' )
            {
                rs.push_front('-');
            }
        }
        else if ( length1 > 1 && length2 > 1 )
        {
            //对clist1和clist2分别进行划分
            list<char> clist1_1;   //clist1的高位部分
            list<char> clist1_2;   //clist1的低位部分
            list<char> clist2_1;   //clist2的高位部分
            list<char> clist2_2;   //clist2的低位部分
            int i;
            list<char>::iterator iter;
            for(i=0,iter=clist1.begin(); i<length1/2; ++i,++iter)
            {
                clist1_1.push_back(*iter);
            }
            for(; iter!=clist1.end(); ++iter)
            {
                clist1_2.push_back(*iter);
            }
            for(i=0,iter=clist2.begin(); i<length2/2; ++i,++iter)
            {
                clist2_1.push_back(*iter);
            }
            for(; iter!=clist2.end(); ++iter)
            {
                clist2_2.push_back(*iter);
            }
            list<char> rs_hh;   //两个高位相乘的结果
            list<char> rs_ll;   //两个低位相乘的结果
            rs_hh = long_mul(clist1_1, clist2_1);
            //高位相乘结果移位(末尾添0)
            for(i=0; i<clist1_2.size()+clist2_2.size(); ++i)
            {
                rs_hh.push_back('0');
            }
            rs_ll = long_mul(clist1_2, clist2_2);
            list<char> sub1_hl;   //clist1的高位和低位部分的差
            list<char> sub2_lh;   //clist2的低位和高位部分的差
            //两个高位分别移位
            for(i=0; i<clist1_2.size(); ++i)
            {
                clist1_1.push_back('0');
            }
            for(i=0; i<clist2_2.size(); ++i)
            {
                clist2_1.push_back('0');
            }
            sub1_hl = long_sub(clist1_1, clist1_2);
            sub2_lh = long_sub(clist2_2, clist2_1);
            list<char> rs_sub1_sub2; //两个差的乘积
            rs_sub1_sub2 = long_mul(sub1_hl, sub2_lh);
            //把几个乘积的结果加起来
            list<char> tmp1 = long_add(rs_hh, rs_ll);
            list<char> tmp2 = long_add(tmp1, rs_sub1_sub2);
            list<char> tmp3 = long_add(tmp2, rs_hh);
            rs = long_add(tmp3, rs_ll);
            if ( sign == '-' )
            {
                rs.push_front('-');
            }
        }
        return rs;
    }
    return rs;
}

int main(void)
{
    list<char> clist1;
    list<char> clist2;
    cout <<"请您输入2个乘数: "<<endl;
    cout <<"请您输入被乘数: ";
    int i;
    string   str1;
    cin >> str1;
    for(i=0; i<str1.size(); ++i)
    {
        if (str1[i] >= '0' && str1[i] <= '9' )
        {
            clist1.push_back(str1[i]);
        }
        else
        {
            cout <<"被乘数中的数字只能为0~9" <<endl;
            exit(1);
        }
    }
    cout <<"请您输入乘数: ";
    string str2;
    cin >> str2;
    for(i=0; i<str2.size(); ++i)
    {
        if ( str2[i] >= '0' && str2[i] <= '9' )
        {
            clist2.push_back(str2[i]);
        }
        else
        {
            cout <<"乘数中的数字只能为0~9" <<endl;
            exit(1);
        }
    }
    list<char> rs = long_mul(clist1,clist2);
    cout << "两个大整数相乘的积为: ";
    for(list<char>::iterator iter=rs.begin(); iter!=rs.end(); ++iter)
    {
        cout << *iter;
    }
    cout <<endl;
    system("pause");
    return 0;
}