Multiplication of large numbers and its efficient algorithms

Test cases:

999 999

998001

999999999999 999999999999

999999999998000000000001

The following is an analysis of 999*999

6 5 4

6 5 4

36 30 24

30 25 20

24 20 16

-------------------------------

The result is clear.

However, you must note that the maximum ASC code is 128. Therefore, you must handle it when adding it.

The following is the most common code.

/* Name: hondely copyright: hondely Author: hondely Date: 05/11/11 14:34 Description: */# include <iostream> using namespace STD; void MUL (char * handle, char * CH2) {int len1 = strlen (substring), len2 = strlen (CH2); char CH3 [1000009]; int I, j, carry; for (I = 0; I <1000009; I ++) CH3 [I] = '\ 0'; // = 0for (I = 0; I <len1; I ++) {for (j = 0; j <len2; j ++) // The following ASC code may be greater than 128 {CH3 [I + J] = CH3 [I + J] + (shard [I]-'0 ') * (CH2 [J]-'0'); // here, CH3 is represented by the forward type. Note that if (CH3 [I + J]> 9 & (I + J)> 0) {CH3 [I + J-1] + = CH3 [I + J]/10; CH3 [I + J] = CH3 [I + J] % 10 ;}}} for (I = len1 + len2-1; I> 0; -- I) // prevents the case where the carry above is greater than 9 {If (CH3 [I]> 9) {CH3 [I-1] = CH3 [I-1] + CH3 [I]/10; CH3 [I] % = 10 ;}} if (CH3 [0]> 9) // If (CH [3]> 99) {cout <CH3 [0]/10; CH3 [0] = CH3 [0] % 10 ;} for (I = 0; I <len1 + len2-1; I ++) cout <char (CH3 [I] + 48); cout <Endl;} int main () {char sequence [10005], CH2 [10005]; cout <"Please input:"; while (CIN> sequence> CH2) {MUL (sequence, CH2 );} return 0 ;}

The following is the Fourier efficient algorithm. I don't know how to paste it.

#include <iostream>  #include <cmath>  #include <complex>  #include <cstring>  using namespace std;    const double PI = acos(-1);typedef complex<double> cp;  typedef long long int64;    const int N = 1 << 16;  int64 a[N], b[N], c[N << 1];    void bit_reverse_copy(cp a[], int n, cp b[])  {      int i, j, k, u, m;      for (u = 1, m = 0; u < n; u <<= 1, ++m);      for (i = 0; i < n; ++i)      {          j = i; k = 0;          for (u = 0; u < m; ++u, j >>= 1)              k = (k << 1) | (j & 1);          b[k] = a[i];      }  }    void FFT(cp _x[], int n, bool flag)  {      static cp x[N << 1];      bit_reverse_copy(_x, n, x);      int i, j, k, kk, p, m;      for (i = 1, m = 0; i < n; i <<= 1, ++m);      double alpha = 2 * PI;      if (flag) alpha = -alpha;      for (i = 0, k = 2; i < m; ++i, k <<= 1)      {          cp wn = cp(cos(alpha / k), sin(alpha / k));          for (j = 0; j < n; j += k)          {              cp w = 1, u, t;              kk = k >> 1;              for (p = 0; p < kk; ++p)              {                  t = w * x[j + p + kk];                  u = x[j + p];                  x[j + p] = u + t;                  x[j + p + kk] = u - t;                  w *= wn;              }          }      }      memcpy(_x, x, sizeof(cp) * n);  }    void polynomial_multiply(int64 a[], int na, int64 b[], int nb, int64 c[], int &nc)  {      int i, n;      i = max(na, nb) << 1;      for (n = 1; n < i; n <<= 1);      static cp x[N << 1], y[N << 1];      for (i = 0; i < na; ++i) x[i] = a[i];      for (; i < n; ++i) x[i] = 0;      FFT(x, n, 0);      for (i = 0; i < nb; ++i) y[i] = b[i];      for (; i < n; ++i) y[i] = 0;      FFT(y, n, 0);      for (i = 0; i < n; ++i) x[i] *= y[i];      FFT(x, n, 1);      for (i = 0; i < n; ++i)       {          c[i] =(int64)(x[i].real() / n + 0.5);    }      for (nc = na + nb - 1; nc > 1 && !c[nc - 1]; --nc);  }    const int LEN = 5, MOD = 100000;  void convert(char *s, int64 a[], int &n)  {      int len = strlen(s), i, j, k;      for (n = 0, i = len - LEN; i >= 0; i -= LEN)      {          for (j = k = 0; j < LEN; ++j)              k = k * 10 + (s[i + j] - '0');          a[n++] = k;      }      i += LEN;      if (i)      {          for (j = k = 0; j < i; ++j)              k = k * 10 + (s[j] - '0');          a[n++] = k;      }  }    void print(int64 a[], int n)  {      printf("%I64d", a[--n]);      while (n) printf("%05I64d", a[--n]);      puts("");  }    char buf[N + 10];    int main()  {      int na, nb, nc;            while (scanf("%s", buf) != EOF)      {          bool sign = false;          if (buf[0] == '-')          {              sign = !sign;               convert(buf + 1, a, na);          }          else convert(buf, a, na);          scanf("%s", buf);          if (buf[0] == '-')          {              sign = !sign;              convert(buf + 1, b, nb);          }          else convert(buf, b, nb);          polynomial_multiply(a, na, b, nb, c, nc);          int64 t1, t2;          t1 = 0;          for (int i = 0; i < nc; ++i)          {              t2 = t1 + c[i];              c[i] = t2 % MOD;              t1 = t2 / MOD;          }          for (; t1; t1 /= MOD) c[nc++] = t1 % MOD;          if (sign) putchar('-');          print(c, nc);      }      return 0;  }