Use triangular solves in default _solve

Makes the default `_solve` implementation of `LinearOperator` use
triangular solves if the linear operator is lower/upper triangular
Add test cases for triangular matrices

src/probnum/linops/_linear_operator.py

@@ -164,8 +164,7 @@ def _apply(self, x: np.ndarray, axis: int) -> np.ndarray:
         return np.moveaxis(self @ np.moveaxis(x, axis, -2), -2, axis)
 
     def __call__(self, x: np.ndarray, axis: Optional[int] = None) -> np.ndarray:
-        """Apply the linear operator to an input array along a specified
-        axis.
+        """Apply the linear operator to an input array along a specified axis.
 
         Parameters
         ----------
@@ -294,6 +293,10 @@ def _solve(self, B: np.ndarray) -> np.ndarray:
         """
         assert B.ndim == 2
 
+        if self.is_lower_triangular or self.is_upper_triangular:
+            return scipy.linalg.solve_triangular(
+                self.todense(), B, lower=self.is_lower_triangular, trans="N"
+            )
         if self.is_symmetric:
             if self.is_positive_definite is not False:
                 try:

tests/test_linops/test_linops_cases/linear_operator_cases.py

@@ -18,6 +18,12 @@
     np.array([[1.0, -2.0], [-2.0, 5.0]]),
     random_spd_matrix(np.random.default_rng(597), dim=10),
 ]
+lower_triangular_matrices = [
+    np.array([[5.0]]),
+    np.array([[1.0, 0.0], [2.0, 3.0]]),
+    np.array([[4.0, 0.0, 0.0], [48.0, 60.0, 0.0], [21.0, 39.0, 7.0]]),
+]
+upper_triangular_matrices = [mat.T for mat in lower_triangular_matrices]
 
 
 @pytest.mark.parametrize("matrix", matrices)
@@ -63,6 +69,28 @@ def case_matrix_spd(matrix: np.ndarray) -> Tuple[pn.linops.LinearOperator, np.nd
     return linop, matrix
 
 
+@pytest_cases.case(tags=("square", "lower-triangular"))
+@pytest_cases.parametrize("matrix", lower_triangular_matrices)
+def case_matrix_lower_triangular(
+    matrix: np.ndarray,
+) -> Tuple[pn.linops.LinearOperator, np.ndarray]:
+    linop = pn.linops.Matrix(matrix)
+    linop.is_lower_triangular = True
+
+    return linop, matrix
+
+
+@pytest_cases.case(tags=("square", "upper-triangular"))
+@pytest_cases.parametrize("matrix", upper_triangular_matrices)
+def case_matrix_upper_triangular(
+    matrix: np.ndarray,
+) -> Tuple[pn.linops.LinearOperator, np.ndarray]:
+    linop = pn.linops.Matrix(matrix)
+    linop.is_upper_triangular = True
+
+    return linop, matrix
+
+
 @pytest_cases.case(tags=("square", "symmetric"))
 def case_matrix_symmetric_indefinite() -> Tuple[pn.linops.LinearOperator, np.ndarray]:
     matrix = np.diag((2.1, 1.3, -0.5))