Merge pull request matthewsamuel95#685 from av4l4nc14/add_solving_linear_recurrence

Add solving linear recurrence

Author: matthewsamuel95

Math/Linear_Recurrences/Linear Recurrence.pdf

@@ -0,0 +1,113 @@
+#include <bits/stdc++.h>
+using namespace std;
+#define MOD 1000000000
+
+typedef long long ll;
+
+
+
+ll k;
+vector<ll> a,b,c;
+
+vector<vector<ll> > multiply(vector<vector<ll> > M1 , vector<vector<ll> > M2)
+{
+	vector<vector<ll> > M3(k+1,vector<ll>(k+1,0));
+	
+	for(ll i=1;i<=k;i++){
+		
+		for(ll j=1;j<=k;j++)
+		{
+			for(ll x=1;x<=k;x++)
+			{
+				M3[i][j]=(M3[i][j]+(M1[i][x]*M2[x][j])% MOD)%MOD;
+			}
+		}
+	}
+	return M3;
+	
+}
+
+vector<vector<ll> > power(vector<vector<ll> > M1, ll n)
+{
+	vector<vector<ll> > ANS(k+1,vector<ll>(k+1,0));
+	
+	for(ll i=0;i<=k;i++)
+	{
+		ANS[i][i]=1;
+	}
+	
+	while(n)
+	{
+		if(n&1)
+		ANS=multiply(ANS,M1);
+		M1=multiply(M1,M1);
+		n>>=1;
+		
+	}
+	return ANS;
+}
+
+
+ll compute(ll n){
+	if(n==0)
+	return 0;
+	if(n<=k)
+	return b[n-1];
+	
+	vector<ll> F1(k+1,0);
+	for(ll i=1;i<=k;i++)
+	F1[i]=b[i-1];
+	
+	vector<vector<ll> > T(k+1,vector<ll>(k+1,0));
+	
+	for(ll i=1;i<k;i++)
+	{
+		for(ll j=1;j<=k;j++)
+		{
+			if(j==i+1)
+			T[i][j]=1;
+			else
+			T[i][j]=0;
+		}
+	}
+	
+	for(ll i=1;i<=k;i++)
+	T[k][i]=c[k-i];
+	
+	T=power(T,n-1);
+	
+	ll res=0;
+	
+	for(ll i=1;i<=k;i++)
+	{
+		res= (res + (T[1][i]*F1[i]))%MOD;
+	}
+	
+	return res;
+	
+}
+int main() {
+	
+	ll x;
+	ll n;
+	cin>>k;
+	
+	for(ll i=0;i<k;i++)
+	{
+		cin>>x;
+		b.push_back(x);
+	}
+	
+	for(ll i=0;i<k;i++)
+	{
+		cin>>x;
+		c.push_back(x);
+	}
+	cin>>n;
+	cout<<compute(n)<<endl;
+	
+	b.clear();
+	c.clear();
+	
+	return 0;
+}