From e4c074476027154ac615fbe1bf2a7219f647e665 Mon Sep 17 00:00:00 2001
From: nuclearkevin <66632997+nuclearkevin@users.noreply.github.com>
Date: Mon, 13 Jul 2026 16:01:24 -0500
Subject: [PATCH 1/3] Fix the two-component estimator.

---
 src/eigenvalue.cpp | 25 +++++++++++++++++--------
 1 file changed, 17 insertions(+), 8 deletions(-)

diff --git a/src/eigenvalue.cpp b/src/eigenvalue.cpp
index bc8dcc9ea77..cbdc017a343 100644
--- a/src/eigenvalue.cpp
+++ b/src/eigenvalue.cpp
@@ -568,17 +568,26 @@ int openmc_get_keff(double* k_combined)
     // These equations are derived analogously to that done in the paper by
     // Urbatsch, but are simpler than for the three estimators case since the
     // block matrices of the three estimator equations reduces to scalars here
+    // See LA-12658-MS, Eqs. 36 and 40.
 
-    // Store the commonly used term
-    double f = kv[i] - kv[j];
-    double g = cov(i, i) + cov(j, j) - 2.0 * cov(i, j);
+    // We can use variance/covariances for the mean as the division by (n - 1)
+    // cancels.
+    const double f = cov(i, i) + cov(j, j) - 2.0 * cov(i, j);
+    const double w_1 = (cov(j, j) - cov(i, j)) / f;
+    const double w_2 = (cov(i, i) - cov(i, j)) / f;
 
-    // Calculate combined estimate of k-effective
-    k_combined[0] = kv[i] - (cov(i, i) - cov(i, j)) / g * f;
+    k_combined[0] = kv[i] * w_1 + kv[j] * w_2;
 
-    // Calculate standard deviation of combined estimate
-    k_combined[1] = (cov(i, i) * cov(j, j) - cov(i, j) * cov(i, j)) *
-                    (g + n * f * f) / (n * (n - 2) * g * g);
+    // We must use the sums of squares for the variance as the division by
+    // (n - 1) does not cancel.
+    const double s_ii = cov(i, i) * (n - 1);
+    const double s_jj = cov(j, j) * (n - 1);
+    const double s_ij = cov(i, j) * (n - 1);
+
+    const double g = s_ii - (s_ii - s_ij) * (s_ii - s_ij) / (f * (n - 1));
+    const double h = 1.0 / n + (kv[i] - kv[j]) * (kv[i] - kv[j]) / (f * (n - 1));
+
+    k_combined[1] = 1.0 / (n - 2) * g * h;
     k_combined[1] = std::sqrt(k_combined[1]);
   }
   return 0;

From dad594879d39843f6d7d7d048ec0d2a7aa072e5b Mon Sep 17 00:00:00 2001
From: nuclearkevin <66632997+nuclearkevin@users.noreply.github.com>
Date: Mon, 13 Jul 2026 16:06:57 -0500
Subject: [PATCH 2/3] Regold MG survival biasing test for the 2-component fix.

---
 src/eigenvalue.cpp                                          | 4 ++--
 tests/regression_tests/mg_survival_biasing/results_true.dat | 2 +-
 2 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/eigenvalue.cpp b/src/eigenvalue.cpp
index cbdc017a343..fe39dc01689 100644
--- a/src/eigenvalue.cpp
+++ b/src/eigenvalue.cpp
@@ -570,7 +570,7 @@ int openmc_get_keff(double* k_combined)
     // block matrices of the three estimator equations reduces to scalars here
     // See LA-12658-MS, Eqs. 36 and 40.
 
-    // We can use variance/covariances for the mean as the division by (n - 1)
+    // We can use the \Sigma matrix for the mean as the division by (n - 1)
     // cancels.
     const double f = cov(i, i) + cov(j, j) - 2.0 * cov(i, j);
     const double w_1 = (cov(j, j) - cov(i, j)) / f;
@@ -578,7 +578,7 @@ int openmc_get_keff(double* k_combined)
 
     k_combined[0] = kv[i] * w_1 + kv[j] * w_2;
 
-    // We must use the sums of squares for the variance as the division by
+    // We must use the S matrix for the variance as the division by
     // (n - 1) does not cancel.
     const double s_ii = cov(i, i) * (n - 1);
     const double s_jj = cov(j, j) * (n - 1);
diff --git a/tests/regression_tests/mg_survival_biasing/results_true.dat b/tests/regression_tests/mg_survival_biasing/results_true.dat
index 4b26978ada6..4f14ad167a1 100644
--- a/tests/regression_tests/mg_survival_biasing/results_true.dat
+++ b/tests/regression_tests/mg_survival_biasing/results_true.dat
@@ -1,2 +1,2 @@
 k-combined:
-9.889968E-01 9.144186E-03
+9.889968E-01 1.823442E-02

From 62e74daa0cf03d991cf228545ddea9d4ae2f9c11 Mon Sep 17 00:00:00 2001
From: nuclearkevin <66632997+nuclearkevin@users.noreply.github.com>
Date: Mon, 13 Jul 2026 16:09:00 -0500
Subject: [PATCH 3/3] Style changes.

---
 src/eigenvalue.cpp | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/src/eigenvalue.cpp b/src/eigenvalue.cpp
index fe39dc01689..bde357721c7 100644
--- a/src/eigenvalue.cpp
+++ b/src/eigenvalue.cpp
@@ -585,7 +585,8 @@ int openmc_get_keff(double* k_combined)
     const double s_ij = cov(i, j) * (n - 1);
 
     const double g = s_ii - (s_ii - s_ij) * (s_ii - s_ij) / (f * (n - 1));
-    const double h = 1.0 / n + (kv[i] - kv[j]) * (kv[i] - kv[j]) / (f * (n - 1));
+    const double h =
+      1.0 / n + (kv[i] - kv[j]) * (kv[i] - kv[j]) / (f * (n - 1));
 
     k_combined[1] = 1.0 / (n - 2) * g * h;
     k_combined[1] = std::sqrt(k_combined[1]);