7 files changed, 158 insertions, 3 deletions

diff --git a/content/math/linearRecurrence.cpp b/content/math/linearRecurrence.cpp
new file mode 100644
index 0000000..c15c25c
--- /dev/null
+++ b/content/math/linearRecurrence.cpp
@@ -0,0 +1,30 @@
+// constexpr ll mod = 998244353;
+// vector<ll> mul(const vector<ll> &a, const vector<ll> &b){
+// 	vector<ll> c(sz(a) + sz(b) - 1);
+// 	for(int i = 0; i < sz(a); i++){
+// 		for(int j = 0; j < sz(b); j++){
+// 			c[i+j] += a[i]*b[j] % mod;
+// 		}
+// 	}
+// 	for(ll &x : c) x %= mod;
+// 	return c;
+// }
+
+ll kthTerm(const vector<ll>& f, const vector<ll>& c, ll k){
+	int n = sz(c);
+	vector<ll> q(n+1, 1);
+	for(int i = 1; i <= n; i++) q[i] = (mod-c[i-1])%mod;
+	vector<ll> p = mul(f, q);
+	p.resize(n);
+	p.push_back(0);
+	do{
+		vector<ll> q2 = q;
+		for(int i = 1; i <= n; i += 2) q2[i] = (mod - q2[i]) % mod;
+		vector<ll> x = mul(p, q2), y = mul(q, q2);
+		for(int i = 0; i <= n; i++){
+			p[i] = i == n ? 0 : x[2*i + (k&1)];
+			q[i] = y[2*i];
+		}
+	}while(k /= 2);
+	return p[0];
+}
\ No newline at end of file
diff --git a/content/math/linearRecurence.cpp b/content/math/linearRecurrenceOld.cpp
index 2501e64..2501e64 100644
--- a/content/math/linearRecurence.cpp
+++ b/content/math/linearRecurrenceOld.cpp
diff --git a/content/math/math.tex b/content/math/math.tex
index dd88a5b..fb66110 100644
--- a/content/math/math.tex
+++ b/content/math/math.tex
@@ -136,9 +136,10 @@ sich alle Lösungen von $x^2-ny^2=c$ berechnen durch:
 	Sei $f(n)=c_{0}f(n-1)+c_{1}f(n-2)+\dots + c_{n-1}f(0)$ eine lineare Rekurrenz.

 	

 	\begin{methods}

-		\method{kthTerm}{Berechnet $k$-ten Term einer Rekurrenz $n$-ter Ordnung}{\log(k)\cdot n^2}

+		\method{kthTerm}{Berechnet $k$-ten Term einer Rekurrenz $n$-ter Ordnung}{\log(k)\cdot \text{mul}(n)}

 	\end{methods}

-	\sourcecode{math/linearRecurence.cpp}

+	Die Polynom-Multiplikation kann auch mit NTT gemacht werden!

+	\sourcecode{math/linearRecurrence.cpp}

 	Alternativ kann der \mbox{$k$-te} Term in \runtime{n^3\log(k)} berechnet werden:

 	$$\renewcommand\arraystretch{1.5}

 	\setlength\arraycolsep{3pt}

diff --git a/tcr.pdf b/tcr.pdf
index 9ed6eae..1bf4c26 100644
--- a/tcr.pdf
+++ b/tcr.pdf
diff --git a/test/math/linearRecurence.cpp b/test/math/linearRecurenceOld.cpp
index a5290e5..dab2256 100644
--- a/test/math/linearRecurence.cpp
+++ b/test/math/linearRecurenceOld.cpp
@@ -1,5 +1,5 @@
 #include "../util.h"
-#include <math/linearRecurence.cpp>
+#include <math/linearRecurrenceOld.cpp>
 
 struct RandomRecurence {
 	vector<ll> f, c, cache;
diff --git a/test/math/linearRecurrence.cpp b/test/math/linearRecurrence.cpp
new file mode 100644
index 0000000..50e98a0
--- /dev/null
+++ b/test/math/linearRecurrence.cpp
@@ -0,0 +1,57 @@
+#include "../util.h"
+#include <math/modPowIterativ.cpp>
+#include <math/transforms/ntt.cpp>
+#include <math/transforms/multiplyNTT.cpp>
+#include <math/linearRecurrence.cpp>
+
+struct RandomRecurence {
+	vector<ll> f, c, cache;
+	RandomRecurence(int n) : f(Random::integers<ll>(n, 0, mod)), c(Random::integers<ll>(n, 0, mod)), cache(f) {}
+
+	ll operator()(ll k){
+		while (sz(cache) <= k) {
+			ll cur = 0;
+			for (ll i = 0; i < sz(c); i++) {
+				cur += (c[i] * cache[sz(cache) - i - 1]) % mod;
+			}
+			cur %= mod;
+			cache.push_back(cur);
+		}
+		return cache[k];
+	}
+};
+
+void stress_test() {
+	int queries = 0;
+	for (int i = 0; i < 1'000; i++) {
+		int n = Random::integer<int>(1, 10);
+		RandomRecurence f(n);
+		for (int j = 0; j < 100; j++) {
+			ll k = Random::integer<ll>(0, 1000);
+
+			ll got = kthTerm(f.f, f.c, k);
+			ll expected = f(k);
+
+			if (got != expected) cerr << "got: " << got << ", expected: " << expected << FAIL;
+			queries++;
+		}
+	}
+	cerr << "tested random queries: " << queries << endl;
+}
+
+constexpr int N = 100'000;
+void performance_test() {
+	timer t;
+	RandomRecurence f(N);
+	t.start();
+	hash_t hash = kthTerm(f.f, f.c, 1e18);
+	t.stop();
+	if (t.time > 8000) cerr << "too slow: " << t.time << FAIL;
+	cerr << "tested performance: " << t.time << "ms (hash: " << hash << ")" << endl;
+}
+
+int main() {
+	stress_test();
+	performance_test();
+}
+
diff --git a/test/math/linearRecurrenceSlowMul.cpp b/test/math/linearRecurrenceSlowMul.cpp
new file mode 100644
index 0000000..205e584
--- /dev/null
+++ b/test/math/linearRecurrenceSlowMul.cpp
@@ -0,0 +1,67 @@
+#include "../util.h"
+
+constexpr ll mod = 998244353;
+vector<ll> mul(const vector<ll> &a, const vector<ll> &b){
+	vector<ll> c(sz(a) + sz(b) - 1);
+	for(int i = 0; i < sz(a); i++){
+		for(int j = 0; j < sz(b); j++){
+			c[i+j] += a[i]*b[j] % mod;
+		}
+	}
+	for(ll &x : c) x %= mod;
+	return c;
+}
+
+#include <math/linearRecurrence.cpp>
+
+struct RandomRecurence {
+	vector<ll> f, c, cache;
+	RandomRecurence(int n) : f(Random::integers<ll>(n, 0, mod)), c(Random::integers<ll>(n, 0, mod)), cache(f) {}
+
+	ll operator()(ll k){
+		while (sz(cache) <= k) {
+			ll cur = 0;
+			for (ll i = 0; i < sz(c); i++) {
+				cur += (c[i] * cache[sz(cache) - i - 1]) % mod;
+			}
+			cur %= mod;
+			cache.push_back(cur);
+		}
+		return cache[k];
+	}
+};
+
+void stress_test() {
+	int queries = 0;
+	for (int i = 0; i < 10'000; i++) {
+		int n = Random::integer<int>(1, 10);
+		RandomRecurence f(n);
+		for (int j = 0; j < 100; j++) {
+			ll k = Random::integer<ll>(0, 1000);
+
+			ll got = kthTerm(f.f, f.c, k);
+			ll expected = f(k);
+
+			if (got != expected) cerr << "got: " << got << ", expected: " << expected << FAIL;
+			queries++;
+		}
+	}
+	cerr << "tested random queries: " << queries << endl;
+}
+
+constexpr int N = 1'000;
+void performance_test() {
+	timer t;
+	RandomRecurence f(N);
+	t.start();
+	hash_t hash = kthTerm(f.f, f.c, 1e18);
+	t.stop();
+	if (t.time > 500) cerr << "too slow: " << t.time << FAIL;
+	cerr << "tested performance: " << t.time << "ms (hash: " << hash << ")" << endl;
+}
+
+int main() {
+	stress_test();
+	performance_test();
+}
+