Matrix Exponentiaton, Linear Recurrence

Author: ivorynoise

SEQ-d.cpp

@@ -0,0 +1,101 @@
+#include<bits/stdc++.h>
+using namespace std;
+
+typedef long long int ll;
+typedef vector<vector<ll> > vii;
+typedef vector<ll> vi;
+const int m = (int)1e9;
+int k;
+vi b;
+vi c;
+
+//matrix mul
+vii mul(const vii& A, const vii& B) {
+	vii C(k, vi(k, 0));
+
+	for (int r = 0; r < k; ++r) {
+		for (int c = 0; c < k; ++c) {
+			for (int x = 0; x < k; ++x) {
+				C[r][c] += (A[r][x] * B[x][c]) % m;
+				C[r][c] %= m;
+			}
+		}
+	}
+	return C;
+}
+
+//matrix exponentiation
+vii pow(vii& A, int n) {
+	if (n <= 1) return A;
+
+	//if odd
+	if (n & 1) return mul(A, pow(A, n - 1));
+
+	vii ans = pow(A, n / 2);
+	return mul(ans, ans);
+}
+
+int sol(int n){
+	vii t(k, vi(k, 0));
+
+	if (n <= k) return b[n - 1];
+	
+	//setting transformation matrix
+	reverse(c.begin(), c.end());
+
+	for(int r = 0; r < k; ++r){
+		for(int col = 0; col < k; ++col){
+			//Make sure that T.Fn-1 = Fn
+			//no skipping should be there else formula needs to be modified
+
+			t[r][col] = 0;
+			if (col == r + 1) t[r][col] = 1;
+			//last row
+			if (r == k - 1) t[r][col] = c[col];
+
+		}
+	}
+
+	//setting f
+	vi f(k);
+	copy(b.begin(), b.end(), f.begin());
+
+	//compute T^n
+	int exp = n-1;
+	vii tn = pow(t, exp);
+
+	ll ans = 0;
+	// getting the first row
+	for(int i = 0; i < k; ++i){
+		ans += (tn[0][i] * f[i]) % m;
+		ans %= m;
+	}
+	return ans;
+
+}
+
+int main() {
+	int t;
+	cin >> t;
+	while (t--)
+	{	
+		cin >> k;
+		b.resize(k);
+		c.resize(k);
+
+		for (int i = 0; i < k; ++i) {
+			cin >> b[i];
+		}
+
+		for (int i = 0; i < k; ++i) {
+			cin >> c[i];
+		}
+
+		int n;
+		cin >> n;
+		cout << sol(n) << "\n";	//never use endl
+		b.clear();
+		c.clear();
+	}
+
+}

SEQ-h.cpp

@@ -0,0 +1,82 @@
+#include<iostream>
+using namespace std;
+#define mod 1000000000
+int size;
+
+void multiply(long long int a[][10],long long int b[][10]){
+    int i,j,k;
+    long long int ans[20][20]={};
+    for(i=0;i<size;i++){
+        for(j=0;j<size;j++){
+            for(k=0;k<size;k++){
+                ans[i][j]=(ans[i][j] + a[i][k]*b[k][j])%mod;
+                ans[i][j]=ans[i][j]%mod;
+            }
+        }
+    }
+    for(i=0;i<size;i++){
+        for(j=0;j<size;j++){
+         a[i][j]=ans[i][j];   
+        }
+    }
+    
+}
+
+void power(long long int second[][10],long long int n){
+    long long int current[20][10]={},i,j;
+    for(i=0;i<size;i++){
+        for(j=0;j<size;j++)
+            current[i][j]=second[i][j];
+    }
+    if(n<=1){
+        return;
+    }
+    if(n&1){
+         power(second,n-1);
+        multiply(second ,current);
+    }
+    else{
+        power(second,n/2);
+    multiply(second,second);
+    }
+}
+int main(){
+    int t;
+    cin>>t;
+    while(t--){
+        int i,j;cin>>size;
+        long long int n,b[20],c[20];
+        for(i=0;i<size;i++){
+            cin>>b[i];
+        }
+        for(i=0;i<size;i++){
+            cin>>c[i];
+        }
+        cin>>n;
+        if(n<=size){
+            cout<<b[size-n]<<"\n";
+        }else{
+            long long int second[20][10]={},ans=0;
+            for(i=0;i<size;i++){
+                second[0][i]=c[i];
+            }
+            for(i=1;i<size;i++){
+                second[i][i-1]=1;
+            }
+            long long int e=n-size;
+            power(second,e);
+            
+            for(i=0;i<size;i++){
+                ans=(ans+second[0][i]*b[size-i-1])%mod;
+                ans%=mod;
+            }
+            // for(i=0;i<size;i++){
+            //     for(j=0;j<size;j++)
+            //         cout<<second[i][j]<<" ";
+            //     cout<<"\n";
+            // }
+            cout<<ans<<"\n";
+            
+        }
+    }
+}