Matrix Fast Power--(recursive expression)

Matrix Fast Power

First know the Matrix

Matrix is a set of complex numbers or real numbers arranged in a rectangular array ;

matrix multiplication:

Definition: Set AAs a matrix of MXP, BFor the PXN matrix, then the matrix of the MXN is called CAs a matrix AAnd BThe product of the C=AXB, where the J column element of line I of Matrix C can be represented as:


knowing that matrix multiplication, such as Fibonacci, is a recursive type,F (n) =f (n-1) +f (n-2), because matrix multiplication, so
set matrix A to                 Matrix B is                  then A*b is f (n) = f (n-1) *1+f (n-2) * * because we need matrices for the a*b matrix to be                               so the B matrix is inverted to the B-matrix.，so the matrix of F (n) is the initial matrix * (b^ (n-2)) n>=3;The resulting matrix a[0][0] is the value of f (n), where the time complexity is optimized in the power exponent that part of the fast power, saving time. so the key is to construct the matrix from a recursive type For example:
The structure of this matrix is
You can write this matrix, you understand; Here's a simple question, practice practiced hand.

Give a simple exercise: Click the Open link

#include <cstdio> #include <cstring> #include <cctype> #include <cmath> #include <set># Include <map> #include <list> #include <queue> #include <deque> #include <stack> #include <string> #include <bitset> #include <vector> #include <iostream> #include <algorithm># Include <stdlib.h>using namespace std;typedef long long ll;const int inf=2e9+1e8;const int Mod=1000007;const int MAX _size=1005;const int mm=3; LL f1,f2,a,b,c,n;struct mat{    LL maze[mm][mm];    void Set_empty ()     {    &NBSP ;   memset (maze,0,sizeof (Maze));   }}; Mat unit_mat={    1,0,0,    0,1,0,    0,0,1};  //  Define a unit matrix, multiplying any matrix by the unit matrix whose value equals itself; Mat operator * (Mat A,mat b)  //overloaded operator  *//  defines multiplication of two matrices, based on matrix multiply The definition of the law to write {    Mat c;    c.set_empty ();  //be sure to set to empty. Qing Zero Group; otherwise, a number will be added to the garbage value     LL i,j,k;    for (i=0;i<mm; i++)     {        for (j=0; j<mm; j + +)         {      &NB Sp     for (k=0; k<mm; k++)             {            & nbsp   C.MAZE[I][J]+=A.MAZE[I][K] * b.maze[k][j];                C.MAZE[I][J]%= M od;           }       }   }    return c;} Mat operator^ (Mat a,ll N)  //  optimization time is here,//  similar to the general int number of the fast power evaluation, the idea is the same. Don't understand can Baidu fast power. {    Mat c=unit_mat;    while (N)     {        if (n&1) c=a*c;  &N Bsp     a=a*a;        n>>=1;   }    return c;} void Solve () {    Mat a,b;    b.set_empty ();    A.set_empty ();    B.maze[0][0]=b, b.maze[1][0]=a,b.maze[2][0]=c;    B.MAZE[0][1]=B.Maze[2][2]=1;    b=b^ (n-2);    a.maze[0][0]=f2,a.maze[0][1]=f1,a.maze[0][2]=1;    LL ans =0;    Mat c=a*b;    printf ("%lld\n", (c.maze[0][0]+mod)%mod);} int main () {    int times;    scanf ("%d", &times);    while (times--)     {         scanf ("%lld%lld%lld%lld%lld%lld", &f1,&f2,&a,&b,&c,&n);  & nbsp     if (n = = 1)             printf ("%lld\n", (f1+mod)%mod);      &NBSP ; else if (n = = 2)             printf ("%lld\n", (f2+mod)%mod);        else so Lve ();  //its upper special sentence because the formula is defined in the field  n>=3;   }    return 0;}

Matrix Fast Power--(recursive expression)