Fast power and Matrix fast power

The idea of a fast power:

is still the algorithm related to the 2-point method: (Many O (LOGN) algorithms are two-way ah ... ）

But the fast power has a previous problem, that is, the data type must meet the binding law

for the general solution: a^8 = A * A * a * a * a * a * a * a * a total of 7 multiplication operations, the average decomposition: a^82 so we just need 4 times multiplication; we can also decompose it: A^6222 This reduces the number of multiplication operations to 3 times. 

 int  Qpow (int  A, int   b) { if  (b = = 0 ) return  1  ;  int  x = 1      while   (n) { if  (N&1 ) x *= A;        A  *= A;    n  >>= 1  ;  return   x;}  

The idea of a matrix fast power:

Define a matrix class that overloads the operator to satisfy the binding law:

　　

classmatrix{ Public:    intN; int**m; Matrix (intn =2) {m=New int*[n];  for(inti =0; I < n; i++) {M[i]=New int[n]; } N=N;    Clear (); }    voidClear () {///empties the matrix to a 0 matrix         for(inti =0; i < N; i++) {memset (m[i],0,sizeof(int) *N); }    }    voidUnit () {///To set a matrix as a cell matrixClear ();  for(inti =0; i < N; i++) {M[i][i]=1; }} Matrixoperator= (Matrix &se) {////Assignment Matrix (SE.        N);  for(inti =0; I < SE. N i++){             for(intj =0; J < SE. N J + +) {M[i][j]=Se.m[i][j]; }        }        return* This; } Matrixoperator* (Matrix &se) {///matrix multiplication; matrix rslt (SE.        N);  for(inti =0; I < SE. N i++){             for(intj =0; J < SE. N J + +){                 for(intK =0; K < SE. N k++) {Rslt.m[i][j]+ = m[i][k] *Se.m[k][j]; }            }        }        returnrslt; }};

int N) {    Matrix rslt (A.N);    Rslt.unit ();     if 0 return rslt;      while (n) {        if(n&1) rslt *= A;        A*=A;        n>>=1;    }     return rslt;}

　　　　

Fast power and Matrix fast power