A lot of progress.

src/kernel/constant.rs

@@ -33,7 +33,15 @@ impl Kernel for Constant {
         array![1.0]
     }
 
+    fn feedback(&self) -> Array2<f64> {
+        array![[0.0]]
+    }
+
     fn transition(&self, t0: f64, t1: f64) -> Array2<f64> {
-        todo!();
+        array![[1.0]]
+    }
+
+    fn noise_cov(&self, t0: f64, t1: f64) -> Array2<f64> {
+        array![[0.0]]
     }
 }

src/kernel/exponential.rs

@@ -34,7 +34,15 @@ impl Kernel for Exponential {
         array![1.0]
     }
 
+    fn feedback(&self) -> Array2<f64> {
+        (-1.0 / self.l_scale) * array![[1.0]]
+    }
+
     fn transition(&self, t0: f64, t1: f64) -> Array2<f64> {
-        todo!();
+        -(t1 - t0) / self.l_scale * array![[1.0]]
+    }
+
+    fn noise_cov(&self, t0: f64, t1: f64) -> Array2<f64> {
+        self.var * (1.0 - (-2.0 * (t1 - t0) / self.l_scale).exp()) * array![[1.0]]
     }
 }

src/kernel/matern52.rs

@@ -45,7 +45,56 @@ impl Kernel for Matern52 {
         array![1.0, 0.0, 0.0]
     }
 
+    fn feedback(&self) -> Array2<f64> {
+        let a = self.lambda;
+
+        array![
+            [0.0, 1.0, 0.0],
+            [0.0, 0.0, 1.0],
+            [-a.powi(3), -3.0 * a.powi(2), -3.0 * a],
+        ]
+    }
+
     fn transition(&self, t0: f64, t1: f64) -> Array2<f64> {
-        todo!();
+        let d = t1 - t0;
+        let a = self.lambda;
+        let da = d * a;
+
+        let ba = array![
+            [(da * da) / 2.0 + da + 1.0, d * (da + 1.0), d * d / 2.0],
+            [
+                -(da * da * a) / 2.0,
+                -da * da + da + 1.0,
+                -(d / 2.0) * (da - 2.0)
+            ],
+            [
+                (da * a * a / 2.0) * (da - 2.0),
+                (da * a) * (da - 3.0),
+                (da * da - 4.0 * da + 2.0) / 2.0
+            ]
+        ];
+
+        (-da).exp() * ba
+    }
+
+    fn noise_cov(&self, t0: f64, t1: f64) -> Array2<f64> {
+        let d = t1 - t0;
+        let a = self.lambda;
+        let da = d * a;
+
+        let c = (-2.0 * da).exp();
+
+        let x11 = -(1.0 / 3.0)
+            * (c * (2.0 * da.powi(4) + 4.0 * da.powi(3) + 6.0 * da * da + 6.0 * da + 3.0) - 3.0);
+        let x12 = c * (2.0 / 3.0) * a * da.powi(4);
+        let x13 = -(a * a / 3.0)
+            * (c * (2.0 * da.powi(4) - 4.0 * da.powi(3) - 2.0 * da * da - 2.0 * da - 1.0) + 1.0);
+        let x22 = -(a * a / 3.0)
+            * (c * (2.0 * da.powi(4) - 4.0 * da.powi(3) + 2.0 * da * da + 2.0 * da + 1.0) - 1.0);
+        let x23 = c * (2.0 / 3.0) * da * da * a.powi(3) * (da - 2.0).powi(2);
+        let x33 = -(a.powi(4) / 3.0)
+            * (c * (2.0 * da.powi(4) - 12.0 * da.powi(3) + 22.0 * da * da - 10.0 * da + 3.0) - 3.0);
+
+        self.var * array![[x11, x12, x13], [x12, x22, x23], [x13, x23, x33]]
     }
 }