MatrixPowers-矩阵连乘

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本文介绍了一个使用C++实现的矩阵快速幂算法，该算法能够高效地计算矩阵的高次幂，并支持模运算。适用于需要频繁进行矩阵乘法及求幂操作的应用场景，如在数论和组合数学中。

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#include <iostream>

using namespace std;

#define N 105

#define FF(i,n) for(i=0;i<n;i++)

int ans[N][N];

int init[N][N];

int buf[N][N];

void matrixMul(int a[][N],int b[][N],int n,int mod){

int i,j,k;

FF(i,n)FF(j,n) buf[i][j]=0;

FF(i,n)FF(k,n) if(a[i][k])FF(j,n) if(b[k][j])

buf[i][j]=(buf[i][j]+a[i][k]*b[k][j])%mod;

FF(i,n)FF(j,n) a[i][j]=buf[i][j];

}

void matrixMul(int n,int m,int mod){

int i,j;

FF(i,n)FF(j,n) ans[i][j]=(i==j);

for(;m;m>>=1){

if(m&1) matrixMul(ans,init,n,mod);

matrixMul(init,init,n,mod);

}

int main()

{

int i,j,n,m,mod;

while (scanf("%d%d%d",&n,&mod,&m)) {

if (!n&&!m&&!mod)

break;

FF(i, n) FF(j, n) scanf("%d",&init[i][j]);

matrixMul(n,m,mod);

printf("\n");

FF(i,n) {

FF(j,n) {

if (j)

printf(" ");

printf("%d",ans[i][j]);

}

printf("\n");

}

return 0;

}

Problem description

Your task is to write a program to raise an integer matrix to a given power using modulo arithmetic.There is no back story here; at least not one that can be told. The application is too confidential(spying and military intelligence and all that) to be described in public. For example, to raise the 2 by 2 matrix

to the power 2 using modulo 17 arithmetic

Input

The input consists of a number of problems. Each problem starts with a line holding three numbers (N, M, and P) separated by single spaces. 1 <= N <= 100 is the size (N by N) of the matrix to be processed. 2 <= M <= 32000 is the modulo base and 1 <= P <= 32000 is the power to which the matrix must be raised. Following this line are N lines, each holding the n integer values of successive rows of the matrix as a series of positive integers i: 0 <= i < M separated by single spaces. Input is terminated by a line with three zeros.

Output

Output for each problem consists of a blank line, followed by the N rows of the result matrix. Each row is output on a single line as a sequence of integer values separated by single spaces. It doesn´t matter that lines may be quite long-no-one will be allowed to read them anyway.

Sample Input

2 17 2

1 2

3 4

0 0 0

Sample Output

7 10

15 5