Starting implementing persistence, but there is a serious problem with the third derivative right now.

2 files changed, 22 insertions, 9 deletions

diff --git a/Makefile b/Makefile
index 6b77cca..25e23dd 100644
--- a/Makefile
+++ b/Makefile
@@ -1,6 +1,6 @@
 all: walk correlations integrator
 
-CC := clang++ -std=c++17 -Wno-mathematical-notation-identifier-extension -O3 -march=native -mtune=native
+CC := clang++ -std=c++17 -Wno-mathematical-notation-identifier-extension -O3 -march=native -mtune=native -fopenmp
 
 walk: walk.cpp
 	$(CC) walk.cpp -o walk
diff --git a/integrator.cpp b/integrator.cpp
index ea5c477..7d0ad1d 100644
--- a/integrator.cpp
+++ b/integrator.cpp
@@ -16,7 +16,7 @@ public:
   }
 };
 
-unsigned p = 3;
+unsigned p = 2;
 
 Real f(Real q) {
   return 0.5 * pow(q, p);
@@ -30,25 +30,38 @@ Real ddf(Real q) {
   return 0.5 * p * (p - 1) * pow(q, p - 2);
 }
 
-Real integrate(const std::vector<Point>& Cₜ) {
+Real integrate(const std::vector<Point>& Cₜ, double τ₀) {
   Real I = 0;
+#pragma omp parallel for reduction(+:I)
   for (unsigned i = 0; i < Cₜ.size() - 1; i++) {
     unsigned j = Cₜ.size() - 1 - i;
     Real Δτ = Cₜ[i + 1].τ() - Cₜ[i].τ();
     Real C = (Cₜ[j].C() + Cₜ[j - 1].C()) / 2;
     Real dC = (Cₜ[i + 1].C() - Cₜ[i].C()) / Δτ;
-    I += Δτ * df(C) * dC;
+
+    Real dddC = 0;
+    if (i > 4 && i < Cₜ.size() && Cₜ.size() > 4) {
+      dddC += (Cₜ[j].C() - 3 * Cₜ[j+1].C() + 3 * Cₜ[j+2].C() - Cₜ[j+3].C()) / pow(Δτ, 3);
+      dddC += (Cₜ[j+1].C() - 3 * Cₜ[j+2].C() + 3 * Cₜ[j+3].C() - Cₜ[j+4].C()) / pow(Δτ, 3);
+      dddC /= 2;
+    }
+    I += Δτ * df(C) * (dC - pow(τ₀, 2) * dddC);
   }
   return I;
 }
 
-Real energy(const std::vector<Point>& Ct) {
+Real energy(const std::vector<Point>& Ct, Real τ₀) {
   Real I = 0;
   for (unsigned i = 0; i < Ct.size() - 1; i++) {
     Real Δτ = Ct[i + 1].τ() - Ct[i].τ();
     Real C = (Ct[i].C() + Ct[i + 1].C()) / 2;
     Real dC = (Ct[i + 1].C() - Ct[i].C()) / Δτ;
-    I += Δτ * df(C) * dC;
+
+    Real dddC = 0;
+    if (i > 1 && i < Ct.size() - 2 && Ct.size() > 3) {
+      dddC = (Ct[i+1].C() - 3 * Ct[i].C() + 3 * Ct[i-1].C() - Ct[i-2].C()) / pow(Δτ, 3);
+    }
+    I += Δτ * df(C) * (dC - pow(τ₀, 2) * dddC);
   }
   return I;
 }
@@ -80,7 +93,7 @@ int main(int argc, char* argv[]) {
     }
   }
 
-  Real z = 0.5 + 0.0 * pow(y, 2);
+  Real z = (sqrt(1 + 2 * τ₀) - 1) / (2 * τ₀);
 
   std::vector<Point> Cₜ;
   Cₜ.reserve(τₘₐₓ / Δτ + 1);
@@ -88,12 +101,12 @@ int main(int argc, char* argv[]) {
   Cₜ.push_back({0, 1});
 
   while (Cₜ.back().τ() < τₘₐₓ) {
-    Real dC = -z * Cₜ.back().C() - 2 * pow(y, 2) * integrate(Cₜ);
+    Real dC = -z * Cₜ.back().C() - 2 * pow(y, 2) * integrate(Cₜ, τ₀);
     Cₜ.push_back({Cₜ.back().τ() + Δτ, Cₜ.back().C() + Δτ * dC});
     std::cout << Cₜ.back().τ() << " " << Cₜ.back().C() << std::endl;
   }
 
-  std::cerr << - 2 * y * energy(Cₜ) << std::endl;
+  std::cerr << - 2 * y * energy(Cₜ, τ₀) << std::endl;
 
   return 0;
 }