Updating documentation.

src/CMakeLists.txt

@@ -63,7 +63,7 @@ if ( CUDA_ON_BACKEND STREQUAL "HIP" AND HIP_FOUND )
 #                                                BoundaryConditions/tests/method_of_images_cylinder_unit_test.cpp
 #               level_set/closest_point/closest_point_unit_tests.cpp
                level_set/redistancing_Sussman/tests/redistancingSussman_unit_test.cpp
-               level_set/redistancing_Sussman/tests/convergence_test.cpp
+#               level_set/redistancing_Sussman/tests/convergence_test.cpp
                 )
 
 
@@ -92,15 +92,18 @@ else()
 #						FiniteDifference/eq_unit_test_3d.cpp
 #						FiniteDifference/eq_unit_test.cpp
 #						FiniteDifference/tests/Eno_Weno_unit_test.cpp
-#						FiniteDifference/tests/Upwind_gradient_unit_test.cpp
-#						FiniteDifference/tests/FD_simple_unit_test.cpp
+						FiniteDifference/tests/Upwind_gradient_unit_test.cpp
+						FiniteDifference/tests/FD_simple_unit_test.cpp
 #						Operators/Vector/vector_dist_operators_unit_tests.cpp
 #						Operators/Vector/vector_dist_operators_apply_kernel_unit_tests.cpp
 #						../../src/lib/pdata.cpp
 #						BoundaryConditions/tests/method_of_images_cylinder_unit_test.cpp
-#		level_set/closest_point/closest_point_unit_tests.cpp
-		level_set/redistancing_Sussman/tests/redistancingSussman_unit_test.cpp
-		level_set/redistancing_Sussman/tests/convergence_test.cpp
+#		    level_set/closest_point/closest_point_unit_tests.cpp
+			level_set/redistancing_Sussman/tests/redistancingSussman_unit_test.cpp
+			level_set/redistancing_Sussman/tests/redistancingSussman_fast_unit_test.cpp
+			level_set/redistancing_Sussman/tests/convergence_test.cpp
+			level_set/redistancing_Sussman/tests/help_functions_unit_test.cpp
+			level_set/redistancing_Sussman/tests/narrowBand_unit_test.cpp
 		)
 
 	set_property(TARGET numerics PROPERTY CUDA_ARCHITECTURES OFF)

src/FiniteDifference/Upwind_gradient.hpp

@@ -33,7 +33,7 @@
  * @param dplus: Gradient approximated using RHS neighbors.
  * @param dminus: Gradient approximated using LHS neighbors.
  * @param sign: Sign of the velocity with which the wave front is moving.
- * @return Scalar double upwind gradient approximation in the dimension given.
+ * @return Scalar upwind gradient approximation in the dimension given.
  */
 template <typename field_type>
 static field_type upwinding(field_type dplus, field_type dminus, int sign)

src/level_set/redistancing_Sussman/HelpFunctionsForGrid.hpp

@@ -21,7 +21,7 @@
 
 /**@brief Computes the time step for the iterative re-distancing fulfilling CFL condition.
  *
- * @tparam grid_type Inferred type of the input grid.
+ * @tparam grid_type Template type of the input grid.
  * @param grid Input OpenFPM grid.
  * @return Time step.
  */
@@ -40,7 +40,7 @@ typename grid_type::stype get_time_step_CFL(grid_type &grid)
 /**@brief Computes the time step size fulfilling CFL condition according to https://www.cfd-online
  * .com/Wiki/Courant–Friedrichs–Lewy_condition for arbitrary dimensionality.
  *
- * @tparam grid_type Inferred type of the input grid.
+ * @tparam grid_type Template type of the input grid.
  * @param grid Input OpenFPM grid.
  * @param u Array of size grid_type::dims containing the velocity in each dimension.
  * @param Cmax Courant number.
@@ -59,7 +59,7 @@ typename grid_type::stype get_time_step_CFL(grid_type & grid, typename grid_type
 /**@brief Computes the time step size fulfilling CFL condition according to https://www.cfd-online
  * .com/Wiki/Courant–Friedrichs–Lewy_condition for arbitrary dimensionality.
  *
- * @tparam grid_type Inferred type of the input grid.
+ * @tparam grid_type Template type of the input grid.
  * @param grid Input OpenFPM grid.
  * @param u Velocity of propagating wave if isotropic for each direction.
  * @param Cmax Courant number.
@@ -80,8 +80,8 @@ typename grid_type::stype get_time_step_CFL(grid_type & grid, typename grid_type
  * init_value.
  *
  * @tparam Prop Index of property that should be initialized with the value from \p init_value.
- * @tparam grid_type Inferred type of input OpenFPM grid.
- * @tparam T Inferred type of the variable containing the initialization value \p init_value.
+ * @tparam grid_type Template type of input OpenFPM grid.
+ * @tparam T Template type of the variable containing the initialization value \p init_value.
  * @param grid OpenFPM grid whose property \p Prop should be initialized, including its ghost layer.
  * @param init_value Variable that contains the value that should be copied to \p Prop.
  */
@@ -101,8 +101,8 @@ void init_grid_and_ghost(grid_type & grid, T init_value)
  *
  * @tparam attr_sc Index of property that contains the value that should be copied.
  * @tparam attr_ds Index of property to which the value should be copied to.
- * @tparam grid_source_type Inferred type of the source grid \p grid_sc.
- * @tparam grid_dest_type Inferred type of the destination grid \p grid_ds.
+ * @tparam grid_source_type Template type of the source grid \p grid_sc.
+ * @tparam grid_dest_type Template type of the destination grid \p grid_ds.
  * @param grid_sc OpenFPM grid from which value is copied (source).
  * @param grid_ds OpenFPM grid to which value is copied to (destination).
  * @param include_ghost Bool variable that defines if the copying should include the ghost layer. False, if not; true,
@@ -144,9 +144,9 @@ void copy_gridTogrid(const grid_source_type & grid_sc, grid_dest_type & grid_ds,
 /**@brief Determines the biggest spacing of a grid which is potentially anisotropic when comparing x, y (and z)
  * coordinate axis.
  *
- * @tparam grid_type Inferred type of input grid.
+ * @tparam grid_type Template type of input grid.
  * @param grid Input OpenFPM grid.
- * @return Double variable that contains the value of the biggest spacing.
+ * @return Biggest spacing of same type as grid-space.
  */
 template <typename grid_type>
 typename grid_type::stype get_biggest_spacing(grid_type & grid)
@@ -162,9 +162,9 @@ typename grid_type::stype get_biggest_spacing(grid_type & grid)
 /**@brief Determines the smallest spacing of a grid which is potentially anisotropic when comparing x, y (and z)
  * coordinate axis.
  *
- * @tparam grid_type Inferred type of input grid.
+ * @tparam grid_type Template type of input grid.
  * @param grid Input OpenFPM grid.
- * @return Double variable that contains the value of the smallest spacing.
+ * @return Smallest spacing of same type as grid-space.
  */
 template <typename grid_type>
 typename grid_type::stype get_smallest_spacing(grid_type & grid)
@@ -183,10 +183,10 @@ typename grid_type::stype get_smallest_spacing(grid_type & grid)
  *
  * @tparam Prop1 Index of the first property.
  * @tparam Prop2 Index of the second property.
- * @tparam grid_type Inferred type of the input grid.
+ * @tparam grid_type Template type of the input grid.
  * @param grid Input OpenFPM grid.
- * @return Double variable that contains the sum over all grid nodes of the difference between the value stored at \p
- *         Prop1 and the value stored at \p Prop2.
+ * @return Sum over all grid nodes of the difference between the value stored at \p Prop1 and the value stored at \p
+ * Prop2.
  */
 template <size_t Prop1, size_t Prop2, typename grid_type>
 auto average_difference(grid_type & grid)
@@ -206,9 +206,9 @@ auto average_difference(grid_type & grid)
 /**@brief Determines the maximum value stored on a given grid at a given property.
  *
  * @tparam Prop Index of property for which maximum should be found.
- * @tparam grid_type Inferred type of the input grid.
+ * @tparam grid_type Template type of the input grid.
  * @param grid Input OpenFPM grid.
- * @return Double variable that contains the maximum value of property \p Prop in \p grid.
+ * @return Maximum value of property \p Prop in \p grid.
  */
 template <size_t Prop, typename grid_type>
 auto get_max_val(grid_type & grid)
@@ -231,9 +231,9 @@ auto get_max_val(grid_type & grid)
 /**@brief Determines the minimum value stored on a given grid at a given property.
  *
  * @tparam Prop Index of property for which minimum should be found.
- * @tparam grid_type Inferred type of the input grid.
+ * @tparam grid_type Template type of the input grid.
  * @param grid Input OpenFPM grid.
- * @return Double variable that contains the minimum value of property \p Prop in \p grid.
+ * @return Minimum value of property \p Prop in \p grid.
  */
 template <size_t Prop, typename grid_type>
 auto get_min_val(grid_type & grid)
@@ -257,7 +257,7 @@ auto get_min_val(grid_type & grid)
  *
  * @tparam Prop_in Index of property that contains the value whose sign should be determined.
  * @tparam Prop_out Index of property where the sign should be written to.
- * @tparam grid_type Inferred type of the input grid.
+ * @tparam grid_type Template type of the input grid.
  * @param grid Input OpenFPM grid.
  */
 template <size_t Prop_in, size_t Prop_out, typename grid_type>

src/level_set/redistancing_Sussman/NarrowBand.hpp

@@ -36,7 +36,7 @@
 /**@brief Class for getting the narrow band around the interface
  * @file NarrowBand.hpp
  * @class NarrowBand
- * @tparam grid_in_type Inferred type of input grid that stores the signed distance function Phi_SDF.
+ * @tparam grid_in_type Template type of input grid that stores the signed distance function Phi_SDF.
  */
 template <typename grid_in_type, typename phi_type>
 class NarrowBand
@@ -115,8 +115,8 @@ public:
 	 *
 	 * @tparam Phi_SDF_grid Index of property storing the signed distance function in the input grid.
 	 * @tparam Phi_SDF_vd Index of property that should store the SDF in the narrow band particle vector.
-	 * @tparam vector_type Inferred type of the particle vector.
-	 * @tparam grid_type Inferred type of the grid storing the SDF.
+	 * @tparam vector_type Template type of the particle vector.
+	 * @tparam grid_type Template type of the grid storing the SDF.
 	 *
 	 * @param grid Grid of arb. dims. storing the SDF (result of redistancing).
 	 * @param vd Empty vector with same spatial scaling (box) as the grid.
@@ -149,8 +149,8 @@ public:
 	 * @tparam Phi_SDF_grid Index of property storing the signed distance function in the input grid.
 	 * @tparam Phi_SDF_vd Index of property that should store the SDF in the narrow band particle vector.
 	 * @tparam Phi_grad Index of property that should store the gradient of phi in the narrow band particle vector.
-	 * @tparam vector_type Inferred type of the particle vector.
-	 * @tparam grid_type Inferred type of the grid storing the SDF.
+	 * @tparam vector_type Template type of the particle vector.
+	 * @tparam grid_type Template type of the grid storing the SDF.
 	 *
 	 * @param grid Grid of arb. dims. storing the SDF (result of redistancing).
 	 * @param vd Empty vector with same spatial scaling (box) as the grid.
@@ -186,8 +186,8 @@ public:
 	 * @tparam Phi_SDF_vd Index of property that should store the SDF in the narrow band particle vector.
 	 * @tparam Phi_grad Index of property that should store the gradient of phi in the narrow band particle vector.
 	 * @tparam Phi_magnOfGrad Index of property that should store the gradient magnitude of Phi.
-	 * @tparam vector_type Inferred type of the particle vector.
-	 * @tparam grid_type Inferred type of the grid storing the SDF.
+	 * @tparam vector_type Template type of the particle vector.
+	 * @tparam grid_type Template type of the grid storing the SDF.
 	 *
 	 * @param grid Grid of arb. dims. storing the SDF (result of redistancing).
 	 * @param vd Empty vector with same spatial scaling (box) as the grid.
@@ -225,8 +225,8 @@ public:
 	 * be a scalar).
 	 * @tparam Prop1_vd Index of particle property, where grid property should be copied to (prop. value must be a
 	 * scalar).
-	 * @tparam vector_type Inferred type of the particle vector.
-	 * @tparam grid_type Inferred type of the grid storing the SDF.
+	 * @tparam vector_type Template type of the particle vector.
+	 * @tparam grid_type Template type of the grid storing the SDF.
 	 *
 	 * @param grid Grid of arb. dims. storing the SDF (result of redistancing) and any arbitrary other (scalar)
 	 * property.
@@ -263,8 +263,8 @@ public:
 	 * @tparam Index1Vd Index of 1st scalar property on particle to which value from grid should be copied to.
 	 * @tparam Index2Vd Index of 2nd scalar property on particle to which value from grid should be copied to.
 	 * @tparam Index3Vd Index of 3rd scalar property on particle to which value from grid should be copied to.
-	 * @tparam vector_type Inferred type of the particle vector.
-	 * @tparam grid_type Inferred type of the grid storing the SDF.
+	 * @tparam vector_type Template type of the particle vector.
+	 * @tparam grid_type Template type of the grid storing the SDF.
 	 *
 	 * @param grid Grid of arb. dims. storing the SDF (result of redistancing) and any arbitrary other (scalar)
 	 * property.
@@ -360,10 +360,12 @@ private:
 	}
 	/**@brief Checks if a value for Phi_SDF lays within the narrow band.
 	 *
+	 * @tparam T Template type of parameter Phi.
 	 * @param Phi Value of the signed distance function Phi_SDF.
 	 * @return True, if point lays within the narrow band; False if not.
 	 */
-	bool within_narrow_band(double Phi)
+	template<typename T>
+	bool within_narrow_band(T Phi)
 	{
 		return (Phi >= b_low && Phi <= b_up);
 	}

src/level_set/redistancing_Sussman/RedistancingSussman.hpp

@@ -156,7 +156,7 @@ struct DistFromSol
 /**@brief Class for reinitializing a level-set function into a signed distance function using Sussman redistancing.
  * @file RedistancingSussman.hpp
  * @class RedistancingSussman
- * @tparam grid_in_type Inferred type of input grid, which stores the initial level-set function Phi_0.
+ * @tparam grid_in_type Template type of input grid, which stores the initial level-set function Phi_0.
  */
 template <typename grid_in_type, typename phi_type>
 class RedistancingSussman

src/level_set/redistancing_Sussman/tests/TestHelpers.hpp

@@ -12,13 +12,13 @@ bool AlmostEqualRelativeAndAbs(T A, T B,
 {
 	// Check if the numbers are really close -- needed
 	// when comparing numbers near zero.
-	float diff = fabs(A - B);
+	T diff = fabs(A - B);
 	if (diff <= maxDiff)
 		return true;
 	
 	A = fabs(A);
 	B = fabs(B);
-	float largest = (B > A) ? B : A;
+	T largest = (B > A) ? B : A;
 	
 	if (diff <= largest * maxRelDiff)
 		return true;

src/level_set/redistancing_Sussman/tests/analytical_SDF/AnalyticalSDF.hpp

@@ -12,7 +12,7 @@
  * @file AnalyticalSDF.hpp
  *
  * @brief Header file containing functions that compute the analytical solution of the signed distance function (SDF)
- *        for a 3D sphere and a 2D circle of defined radius on a grid.
+ *        for a 3D sphere and a 2D disk of defined radius on a grid.
  *
  * @author Justina Stark
  * @date October 2020
@@ -21,6 +21,9 @@
 #define ANALYTICAL_SDF_HPP
 
 #include <iostream>
+#include "Vector/vector_dist.hpp"
+#include "Grid/grid_dist_id.hpp"
+
 /**@brief Computes the analytical signed distance function of a sphere for a given point in space.
  *
  * @details At the center of the sphere, \a &phi;_SDF_analytic = \a Radius. Moving outwards from the center on, the
@@ -30,13 +33,13 @@
  * @f[ \phi_{SDF}(x, y, z) = R - \sqrt{((x-a)^2 + (y-b)^2 + (z-c)^2)} @f]
  *
  * @tparam point_type Template type of OpenFPM Point<dimension, type>.
- * @tparam radius_type Template type of radius.
+ * @tparam space_type Template type of radius.
  * @param coords Point_type coordinates of point.
- * @param radius Radius_type radius of the sphere.
- * @param center_x Double x-coordinate of sphere center.
- * @param center_y Double y-coordinate of sphere center.
- * @param center_z Double z-coordinate of sphere center.
- * @return Double variable that contains the exact solution for the signed distance function of a given point in a
+ * @param radius Space_type radius of the sphere.
+ * @param center_x Space_type x-coordinate of sphere center.
+ * @param center_y Space_type y-coordinate of sphere center.
+ * @param center_z Space_type z-coordinate of sphere center.
+ * @return Space_type variable that contains the exact solution for the signed distance function of a given point in a
  *         sphere of given radius, where the SDF has positive values inside and negative values outside the sphere.
  */
 template <typename point_type, typename space_type>
@@ -61,13 +64,13 @@ space_type get_analytic_sdf_sphere(point_type coords, space_type radius,
  * @details Solves the exact SDF for each grid nodes and writes the solution to a given property.
  *
  * @tparam SDF_exact Index of property where the anaytical solution for the SDF should be written to.
- * @tparam grid_type Inferred type of the input grid.
- * @tparam radius_type Inferred type of radius the sphere is supposed to have.
+ * @tparam grid_type Template type of the input grid.
+ * @tparam space_type Template type of radius the sphere is supposed to have.
  * @param grid Input OpenFPM grid.
  * @param radius Radius of the filled sphere.
- * @param center_x Double x-coordinate of sphere center.
- * @param center_y Double y-coordinate of sphere center.
- * @param center_z Double z-coordinate of sphere center.
+ * @param center_x Space_type x-coordinate of sphere center.
+ * @param center_y Space_type y-coordinate of sphere center.
+ * @param center_z Space_type z-coordinate of sphere center.
 
  */
 template <size_t SDF_exact, typename grid_type, typename space_type>
@@ -100,13 +103,13 @@ void init_analytic_sdf_sphere(grid_type & grid, space_type radius, space_type ce
  * @f[ \phi_{SDF}(x, y, z) = R - \sqrt{((x-a)^2 + (y-b)^2 + (z-c)^2)} @f]
  *
  * @tparam point_type Template type of OpenFPM Point<dimension, type>.
- * @tparam radius_type Template type of radius.
+ * @tparam space_type Template type of radius.
  * @param coords Point_type coordinates of point.
- * @param radius Radius_type radius of the sphere.
- * @param center_x Double x-coordinate of sphere center.
- * @param center_y Double y-coordinate of sphere center.
- * @param center_z Double z-coordinate of sphere center.
- * @return Double variable that contains the exact solution for the signed distance function of a given point in a
+ * @param radius Space_type radius of the sphere.
+ * @param center_x Space_type x-coordinate of sphere center.
+ * @param center_y Space_type y-coordinate of sphere center.
+ * @param center_z Space_type z-coordinate of sphere center.
+ * @return Space_type variable that contains the exact solution for the signed distance function of a given point in a
  *         sphere of given radius, where the SDF has positive values inside and negative values outside the sphere.
  */
 template <typename point_type, typename space_type>
@@ -130,9 +133,8 @@ space_type get_analytic_sdf_circle(point_type coords, space_type radius,
  * @details Solves the exact SDF for each grid nodes and writes the solution to a given property.
  *
  * @tparam SDF_exact Index of property where the anaytical solution for the SDF should be written to.
- * @tparam grid_type Inferred type of the input grid.
- * @tparam radius_type Inferred type of radius the circle is supposed to have.
- * @tparam center_type Inferred type of the center coordinates.
+ * @tparam grid_type Template type of the input grid.
+ * @tparam space_type Template type of radius and center coordinates.
  * @param grid Input OpenFPM grid.
  * @param radius Radius of the filled circle.
  * @param center_x X-coordinate of the circle center.

src/level_set/redistancing_Sussman/tests/l_norms/LNorms.hpp

@@ -36,7 +36,7 @@
  * @tparam PropNumeric Index of grid property that contains the numerical value.
  * @tparam PropAnalytic Index of grid property that contains the analytical (exact) value.
  * @tparam Error Index of grid property where the computed error should be written to.
- * @tparam gridtype Inferred type of the input grid.
+ * @tparam gridtype Template type of the input grid.
  * @param grid Input OpenFPM grid. Can be of any dimension.
  */
 template <size_t PropNumeric, size_t PropAnalytic, size_t Error, typename gridtype>
@@ -68,7 +68,7 @@ void get_absolute_error(gridtype & grid)
  * @tparam PropNumeric Index of grid property that contains the numerical value.
  * @tparam PropAnalytic Index of grid property that contains the analytical (exact) value.
  * @tparam Error Index of grid property where the computed error should be written to.
- * @tparam gridtype Inferred type of the input grid.
+ * @tparam gridtype Template type of the input grid.
  * @param grid Input OpenFPM grid. Can be of any dimension.
  */
 template <size_t PropNumeric, size_t PropAnalytic, size_t Error, typename gridtype>
@@ -95,27 +95,26 @@ void get_relative_error(gridtype & grid)
 	}
 }
 
-/**@brief Class for computing relative/absolute errors and l2/l_infinity norm for distributed grids and vectors
+/**@brief Class for computing the l2/l_infinity norm for distributed grids and vectors based on given errors.
  *
- * @tparam lnorm_type Desired return type for l-norm.
+ * @tparam lnorm_type Return type for l-norm.
  */
 template <typename lnorm_type>
 class LNorms
 {
 public:
-	LNorms(){};
+	LNorms() = default;
 	// Member variables
-	lnorm_type l2;
-	lnorm_type linf;
+	lnorm_type l2; // L2 norm
+	lnorm_type linf; // L_infinity norm
 	
 	// Member functions
-/**@brief Computes the L_2 and L_infinity norm on the basis of the precomputed error on a grid.
- *
- * @tparam Error Index of grid property that contains the error.
- * @tparam gridtype Inferred type of the input grid.
- * @param grid Input OpenFPM grid. Can be of any dimension.
- * @return Object of type L_norms that contains #L_norms::l2 and #L_norms::linf.
- */
+	/**@brief Computes the L_2 and L_infinity norm on the basis of the precomputed error on a grid.
+	 *
+	 * @tparam Error Index of grid property that contains the error.
+	 * @tparam gridtype Template type of the input grid.
+	 * @param grid Input OpenFPM grid. Can be of any dimension.
+	 */
 	template <size_t Error, typename gridtype>
 	void get_l_norms_grid(gridtype & grid)
 	{
@@ -140,13 +139,12 @@ public:
 		l2 = (lnorm_type) sqrt( sumErrorSq / (error_type)grid.size());
 		linf = (lnorm_type) maxError;
 	}
-/**@brief Computes the L_2 and L_infinity norm on the basis of the precomputed error on a particle vector.
- *
- * @tparam Error Index of grid property that contains the error.
- * @tparam vectortype Inferred type of the input particle vector.
- * @param vd Input particle vector.
- * @return Object of type L_norms that contains #L_norms::l2 and #L_norms::linf.
- */
+	/**@brief Computes the L_2 and L_infinity norm on the basis of the precomputed error on a particle vector.
+	 *
+	 * @tparam Error Index of grid property that contains the error.
+	 * @tparam vectortype Template type of the input particle vector.
+	 * @param vd Input particle vector.
+	 */
 	template <size_t Error, typename vectortype>
 	void get_l_norms_vector(vectortype & vd)
 	{
@@ -173,16 +171,18 @@ public:
 		l2 = (lnorm_type) sqrt( sumErrorSq / (error_type)count);
 		linf = (lnorm_type) maxError;
 	}
-/**@brief Writes the N (number of grid points on a square grid) and L-norms as strings to a csv-file.
- *
- * @param N Size_t variable that contains the grid size in number of grid points in one dimension for an NxN(xN) grid
- * @param l_norms Object of type L_norms that contains #L_norms::l2 and #L_norms::linf.
- * @param filename Std::string containing the name of the csv file (without the .csv) to which the l-norms should be
- *                 written to.
- * @param path_output Std::string containing the path where the output csv file should be saved.
- */
-	void write_to_file(const size_t N, const int precision,
-	                                 const std::string & filename, const std::string & path_output)
+	/**@brief Writes the N (number of grid points on a square grid) and L-norms as strings to a csv-file.
+	 *
+	 * @param N Size_t variable that contains the grid size in number of grid points in one dimension for an NxN(xN) grid
+	 * @param precision Precision in number of digits after the points for writing of the l-norms to file.
+	 * @param filename Std::string containing the name of the csv file (without the .csv) to which the l-norms should be
+	 *                 written to.
+	 * @param path_output Std::string containing the path where the output csv file should be saved.
+	 */
+	 void write_to_file(const size_t N,
+						const int precision,
+						const std::string & filename,
+						const std::string & path_output)
 	{
 		auto &v_cl = create_vcluster();
 		if (v_cl.rank() == 0)