From 8fa78c60e548f784514ed88867bc1f4dfa575c3c Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 13:13:38 +0200
Subject: [PATCH 1/2] Stabilize Merwe sigma-point scaling

---
 src/pyrecest/sampling/sigma_points.py | 39 +++++++++++++++++++--------
 1 file changed, 28 insertions(+), 11 deletions(-)

diff --git a/src/pyrecest/sampling/sigma_points.py b/src/pyrecest/sampling/sigma_points.py
index 673f04cff9..c0dff54492 100644
--- a/src/pyrecest/sampling/sigma_points.py
+++ b/src/pyrecest/sampling/sigma_points.py
@@ -90,6 +90,17 @@ def _validate_finite_scalar(value, name: str) -> float:
     return result
 
 
+def _merwe_scale(n: int, alpha: float, kappa: float) -> float:
+    """Return the Merwe scale without subtracting and re-adding ``n``."""
+
+    scale = alpha * alpha * (n + kappa)
+    if not math.isfinite(scale) or scale <= 0.0:
+        raise ValueError(
+            "alpha and kappa must produce a positive finite sigma-point scale"
+        )
+    return scale
+
+
 def _validate_sigma_inputs(x, P, n: int):
     if _has_complex_dtype(x):
         raise ValueError("x must contain real values")
@@ -138,22 +149,28 @@ def __init__(self, n: int, alpha: float, beta: float, kappa: float):
 
     def _compute_weights(self):
         n = self.n
-        lam = self.alpha**2 * (n + self.kappa) - n
-        scale = n + lam
+        scale = _merwe_scale(n, self.alpha, self.kappa)
+        lam = scale - n
+        mean_weight = lam / scale
+        covariance_weight = mean_weight + (1.0 - self.alpha**2 + self.beta)
+        side_weight = 0.5 / scale
+        if not (
+            math.isfinite(mean_weight)
+            and math.isfinite(covariance_weight)
+            and math.isfinite(side_weight)
+        ):
+            raise ValueError("alpha and kappa must produce finite sigma-point weights")
 
         self.Wm = concatenate(
             [
-                asarray([lam / scale], dtype=float64),
-                full(2 * n, 0.5 / scale, dtype=float64),
+                asarray([mean_weight], dtype=float64),
+                full(2 * n, side_weight, dtype=float64),
             ]
         )
         self.Wc = concatenate(
             [
-                asarray(
-                    [lam / scale + (1.0 - self.alpha**2 + self.beta)],
-                    dtype=float64,
-                ),
-                full(2 * n, 0.5 / scale, dtype=float64),
+                asarray([covariance_weight], dtype=float64),
+                full(2 * n, side_weight, dtype=float64),
             ]
         )
 
@@ -168,11 +185,11 @@ def sigma_points(self, x, P):
             State covariance, shape ``(n, n)``.
         """
         n = self.n
-        lam = self.alpha**2 * (n + self.kappa) - n
+        scale = _merwe_scale(n, self.alpha, self.kappa)
 
         x, P = _validate_sigma_inputs(x, P, n)
 
-        U = linalg.cholesky((n + lam) * P)  # lower-triangular
+        U = linalg.cholesky(scale * P)  # lower-triangular
 
         positive = [x + U[:, i] for i in range(n)]
         negative = [x - U[:, i] for i in range(n)]

From 3a54837bc1b35c71c5dea13536b8aae62eb7b104 Mon Sep 17 00:00:00 2001
From: Florian Pfaff <6773539+FlorianPfaff@users.noreply.github.com>
Date: Mon, 13 Jul 2026 13:13:46 +0200
Subject: [PATCH 2/2] Add Merwe scale regression

---
 tests/test_merwe_sigma_point_scale.py | 35 +++++++++++++++++++++++++++
 1 file changed, 35 insertions(+)
 create mode 100644 tests/test_merwe_sigma_point_scale.py

diff --git a/tests/test_merwe_sigma_point_scale.py b/tests/test_merwe_sigma_point_scale.py
new file mode 100644
index 0000000000..4a29caa5d5
--- /dev/null
+++ b/tests/test_merwe_sigma_point_scale.py
@@ -0,0 +1,35 @@
+import unittest
+
+import numpy as np
+import numpy.testing as npt
+from pyrecest.backend import __backend_name__, asarray, to_numpy
+from pyrecest.sampling import MerweScaledSigmaPoints
+
+
+@unittest.skipIf(
+    __backend_name__ == "pytorch",
+    reason="Sigma-point tests use NumPy assertions and the PyTorch backend is unsupported",
+)
+class TestMerweSigmaPointScale(unittest.TestCase):
+    def test_small_positive_alpha_preserves_nonzero_sigma_spread(self):
+        alpha = 1.0e-9
+        points = MerweScaledSigmaPoints(n=2, alpha=alpha, beta=2.0, kappa=0.0)
+
+        sigmas = to_numpy(
+            points.sigma_points(asarray(np.zeros(2)), asarray(np.eye(2)))
+        )
+        offsets = sigmas[1:] - sigmas[0]
+        expected_radius = np.sqrt(alpha**2 * 2.0)
+
+        npt.assert_allclose(
+            np.linalg.norm(offsets, axis=1),
+            expected_radius,
+            rtol=1.0e-6,
+            atol=0.0,
+        )
+        self.assertTrue(np.all(np.isfinite(to_numpy(points.Wm))))
+        self.assertTrue(np.all(np.isfinite(to_numpy(points.Wc))))
+
+
+if __name__ == "__main__":
+    unittest.main()