From 5715f5fbd77b86edbf8ff78f511fe88870e7b37e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 8 Nov 2021 15:45:57 +0100
Subject: Better method for finding saddles.

---
 dynamics.hpp | 50 ++++++++++++++++++++++++++++----------------------
 1 file changed, 28 insertions(+), 22 deletions(-)

diff --git a/dynamics.hpp b/dynamics.hpp
index 561714e..0597f35 100644
--- a/dynamics.hpp
+++ b/dynamics.hpp
@@ -4,6 +4,8 @@
 #include <eigen3/Eigen/LU>
 #include <random>
 
+#include <iostream>
+
 #include "p-spin.hpp"
 #include "stereographic.hpp"
 
@@ -45,35 +47,39 @@ std::tuple<Real, Vector<Scalar>> gradientDescent(const Tensor<Scalar, p>& J, con
 template <class Real, class Scalar, int p>
 Vector<Scalar> findSaddle(const Tensor<Scalar, p>& J, const Vector<Scalar>& z0, Real ε, Real δW = 2, Real γ0 = 1, Real δγ = 2) {
   Vector<Scalar> z = z0;
-  Vector<Scalar> ζ = euclideanToStereographic(z);
-
-  Real W;
-  std::tie(W, std::ignore) = WdW(J, z);
 
   Vector<Scalar> dH;
   Matrix<Scalar> ddH;
-  std::tie(std::ignore, dH, ddH) = stereographicHamGradHess(J, ζ, z);
+  std::tie(std::ignore, dH, ddH) = hamGradHess(J, z);
 
-  while (W > ε) {
-    // ddH is complex symmetric, which is (almost always) invertible, so a
-    // partial pivot LU decomposition can be used.
-    Vector<Scalar> dζ = ddH.partialPivLu().solve(dH);
-    Vector<Scalar> ζNew = ζ - dζ;
-    Vector<Scalar> zNew = stereographicToEuclidean(ζNew);
+  Scalar zz = z.transpose() * z;
+  Vector<Scalar> ż = zDot(z, dH) + z * (zz - (Real)z.size());
+  Matrix<Scalar> dż = dzDot(z, dH) + Matrix<Scalar>::Identity(z.size(), z.size()) * (zz - (Real)z.size()) + 2.0 * z * z.transpose();
+  Matrix<Scalar> dżc = dzDotConjugate(z, dH, ddH);
 
-    Real WNew;
-    std::tie(WNew, std::ignore) = WdW(J, zNew);
+  Vector<Scalar> b(2 * z.size());
+  Matrix<Scalar> M(2 * z.size(), 2 * z.size());
 
-    if (WNew < W) { // If Newton's step lowered the objective, accept it!
-      ζ = ζNew;
-      z = zNew;
-      W = WNew;
-    } else { // Otherwise, do gradient descent until W is a factor δW smaller.
-      std::tie(W, z) = gradientDescent(J, z, W / δW, γ0, δγ);
-      ζ = euclideanToStereographic(z);
-    }
+  b << ż.conjugate(), ż;
+  M << dż.conjugate(), dżc, dżc.conjugate(), dż;
+
+  while (ż.norm() > ε) {
+    Vector<Scalar> dz = M.partialPivLu().solve(b).tail(z.size());
+    dz -= z.conjugate().dot(dz) / z.squaredNorm() * z.conjugate();
+    z = normalize(z - dz);
+
+    std::cout << "error : " << z.transpose() * z << " "<< ż.norm() << " " << dz.norm() << std::endl;
+    getchar();
+
+    std::tie(std::ignore, dH, ddH) = hamGradHess(J, z);
+
+    zz = z.transpose() * z;
+    ż = zDot(z, dH) + z * (zz - (Real)z.size());
+    dż = dzDot(z, dH) + Matrix<Scalar>::Identity(z.size(), z.size()) * (zz - (Real)z.size()) + 2.0 * z * z.transpose();
+    dżc = dzDotConjugate(z, dH, ddH);
 
-    std::tie(std::ignore, dH, ddH) = stereographicHamGradHess(J, ζ, z);
+    b << ż.conjugate(), ż;
+    M << dż.conjugate(), dżc, dżc.conjugate(), dż;
   }
 
   return z;
-- 
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