Matrix Exponentiation (#589)

* Matrix Exponentiation

* Update and rename Others/Matrix_Expo.cpp to others/matrix_exponentiation.cpp

* Update matrix_exponentiation.cpp

* clang-format -i -style="{BasedOnStyle: Google, IndentWidth: 4}" matrix_exponentiation.cpp

* clang-format -i -style="{IndentWidth: 4}" matrix_exponentiation.cpp

* Fix cpplint readability/braces issue

* using std::cin; using std::cout; using std::vector;

* added int_64 instead of long long

* Minor changes

* Update matrix_exponentiation.cpp

others/matrix_exponentiation.cpp

@@ -0,0 +1,119 @@
+/*
+Matrix Exponentiation.
+The problem can be solved with DP but constraints are high.
+ai = bi (for i <= k)
+ai = c1*ai-1 + c2*ai-2 + ... + ck*ai-k (for i > k)
+Taking the example of Fibonacci series, K=2
+b1 = 1, b2=1
+c1 = 1, c2=1
+a = 0 1 1 2 ....
+This way you can find the 10^18 fibonacci number%MOD.
+I have given a general way to use it. The program takes the input of B and C
+matrix.
+Steps for Matrix Expo
+1. Create vector F1 : which is the copy of B.
+2. Create transpose matrix (Learn more about it on the internet)
+3. Perform T^(n-1) [transpose matrix to the power n-1]
+4. Multiply with F to get the last matrix of size (1xk).
+The first element of this matrix is the required result.
+*/
+
+#include <bits/stdc++.h>
+using std::cin;
+using std::cout;
+using std::vector;
+
+#define ll int64_t
+#define endl '\n'
+#define pb push_back
+#define MOD 1000000007
+ll ab(ll x) { return x > 0LL ? x : -x; }
+ll k;
+vector<ll> a, b, c;
+
+// To multiply 2 matrix
+vector<vector<ll>> multiply(vector<vector<ll>> A, vector<vector<ll>> B) {
+    vector<vector<ll>> C(k + 1, vector<ll>(k + 1));
+    for (ll i = 1; i <= k; i++) {
+        for (ll j = 1; j <= k; j++) {
+            for (ll z = 1; z <= k; z++) {
+                C[i][j] = (C[i][j] + (A[i][z] * B[z][j]) % MOD) % MOD;
+            }
+        }
+    }
+    return C;
+}
+
+// computing power of a matrix
+vector<vector<ll>> power(vector<vector<ll>> A, ll p) {
+    if (p == 1)
+        return A;
+    if (p % 2 == 1) {
+        return multiply(A, power(A, p - 1));
+    } else {
+        vector<vector<ll>> X = power(A, p / 2);
+        return multiply(X, X);
+    }
+}
+
+// main function
+ll ans(ll n) {
+    if (n == 0)
+        return 0;
+    if (n <= k)
+        return b[n - 1];
+    // F1
+    vector<ll> F1(k + 1);
+    for (ll i = 1; i <= k; i++)
+        F1[i] = b[i - 1];
+
+    // Transpose matrix
+    vector<vector<ll>> T(k + 1, vector<ll>(k + 1));
+    for (ll i = 1; i <= k; i++) {
+        for (ll j = 1; j <= k; j++) {
+            if (i < k) {
+                if (j == i + 1)
+                    T[i][j] = 1;
+                else
+                    T[i][j] = 0;
+                continue;
+            }
+            T[i][j] = c[k - j];
+        }
+    }
+    // T^n-1
+    T = power(T, n - 1);
+
+    // T*F1
+    ll res = 0;
+    for (ll i = 1; i <= k; i++) {
+        res = (res + (T[1][i] * F1[i]) % MOD) % MOD;
+    }
+    return res;
+}
+
+// 1 1 2 3 5
+
+int main() {
+    cin.tie(0);
+    cout.tie(0);
+    ll t;
+    cin >> t;
+    ll i, j, x;
+    while (t--) {
+        cin >> k;
+        for (i = 0; i < k; i++) {
+            cin >> x;
+            b.pb(x);
+        }
+        for (i = 0; i < k; i++) {
+            cin >> x;
+            c.pb(x);
+        }
+        cin >> x;
+        cout << ans(x) << endl;
+        b.clear();
+        c.clear();
+    }
+    return 0;
+}