Fixing numerics for Finite Differences

src/Equations/eq_1_ag_unit_test.hpp

+/*
+ * eq_1_ag_unit_test.hpp
+ *
+ * This file model the Eq1 in 2D of Active Gel
+ *
+ * R. Ramaswamy, G. Bourantas, F. Jülicher, and I. F. Sbalzarini.
+ * A hybrid particle-mesh method for incompressible active polar
+ * viscous gels. J. Comput. Phys., 291:334–361, 2015.
+ *
+ *  Created on: Sep 18, 2015
+ *      Author: Pietro Incardona
+ */
+
+#ifndef OPENFPM_NUMERICS_SRC_EQUATIONS_EQ_1_AG_UNIT_TEST_HPP_
+#define OPENFPM_NUMERICS_SRC_EQUATIONS_EQ_1_AG_UNIT_TEST_HPP_
+
+
+////////////////////////////////////// Left part  eq 1 //////////////////////////////
+
+// Creating square parentesis term left part equation 1
+
+// Derivation and multiplication of t1 ... t6
+
+namespace eq1
+{
+	typedef mul<etad2,D<x,t1>> t1_l;
+	typedef mul<eta,D<x,t2>> t2_l;
+	typedef mul<etad2,D<x,t3>> t3_l;
+
+	typedef mul<eta,D<y,t4>> t4_l;
+	typedef mul<eta2,D<y,t5>> t5_l;
+	typedef mul<eta,D<y,t6>> t6_l;
+
+	typedef mul<eta,Lap<v>> meLv;
+	typedef D<x,P> DxP;
+
+	// Assemble the left part
+	typedef add<meLV,minus<DxP>,t1_l,t2_l,t3_l,t4_l,t5_l,t6_l> left;
+
+	////////////////////////////////////////////////////////////////////////////////////
+
+	////////////////////////////////////// Right part  eq 1 /////////////////////////////
+
+	typedef div<2,D<x,hp>> t1_r;
+	typedef mul<zeta,D<x,mul<del_nu,px2>>> t2_r;
+	typedef mul<zeta,D<y,mul<del_nu,px_py>>> t3_r;
+	typedef mul<zeta,D<x,mul<del_nu,dev<2,px2_p_py2>>>> t4_r;
+	typedef mul<nud2,D<x,mul<-2,hp,px_py>> t5_r;
+	typedef mul<etad2,D<y,mul<hp,px2_m_py2>>> t6_r;
+	typedef D<x,sigma[x][x]> t7_r;
+	typedef D<y,sigma[x][y]> t8_r;
+
+	typedef add<t1_r,t2_r,t3_r,t4_r,t5_r,t6_r,t7_r,t8_r,g_ext> eq1r;
+
+	typedef Eq<eq1l,eq1r> equation;
+}
+
+/////////////////////////////////////////////////////////////////////////////////////
+
+#endif /* OPENFPM_NUMERICS_SRC_EQUATIONS_EQ_1_AG_UNIT_TEST_HPP_ */

src/Equations/eq_2_ag_unit_test.hpp

+/*
+ * eq_2_ag_unit_test.hpp
+ *
+ *  Created on: Sep 18, 2015
+ *      Author: i-bird
+ */
+
+#ifndef OPENFPM_NUMERICS_SRC_EQUATIONS_EQ_2_AG_UNIT_TEST_HPP_
+#define OPENFPM_NUMERICS_SRC_EQUATIONS_EQ_2_AG_UNIT_TEST_HPP_
+
+// Left part equation 2
+
+namespace eq2
+{
+	typedef mul<etad2,D<y,t1>> t1_l;
+	typedef mul<eta,D<y,t2>> t2_l;
+	typedef mul<etad2,D<y,t3>> t3_l;
+
+	typedef mul<eta,D<x,t4>> t4_l;
+	typedef mul<eta2,D<x,t5>> t5_l;
+	typedef mul<eta,D<x,t6>> t6_l;
+
+	typedef mul<eta,Lap<v>> meLv;
+	typedef D<y,P> DyP;
+
+	// left part equation 2
+
+	typedef add<meLv,DyP,Dt1,Dt2,Dt3,Dt4,Dt6> left;
+
+	typedef div<-2,D<x,hp>> t1_r;
+	typedef mul<zeta,D<y,mul<del_nu,py2>>> t2_r;
+	typedef mul<zeta,D<x,mul<del_nu,px_py>>> t3_r;
+	typedef mul<zeta,D<y,mul<del_nu,dev<2,px2_p_py2>>>> t4_r;
+	typedef mul<zeta,D<y,mul<del_nu,dev<2,px2_p_py2>>>> t5_r;
+	typedef mul<nud2,D<x,mul<hp,px2_m_py2>>> t6_r;
+	typedef D<x,sigma[x][y]> t7_r;
+	typedef D<y,sigma[y][x]> t8_r;
+	typedef mul<nud2,D<y,mul<gm,lambda,del_nu,px2_m_py2>> t9_r
+
+	typedef add<t1_r,t2_r,t3_r,t4_r,t5_r,t6_r,t7_r,t8_r,t9_r> right;
+}
+
+
+
+#endif /* OPENFPM_NUMERICS_SRC_EQUATIONS_EQ_2_AG_UNIT_TEST_HPP_ */

src/Equations/eq_base_unit_test.hpp

+/*
+ * eq_base_unit_test.hpp
+ *
+ *  Created on: Sep 18, 2015
+ *      Author: i-bird
+ */
+
+#ifndef OPENFPM_NUMERICS_SRC_EQUATIONS_EQ_BASE_UNIT_TEST_HPP_
+#define OPENFPM_NUMERICS_SRC_EQUATIONS_EQ_BASE_UNIT_TEST_HPP_
+
+// quadratic and cubic term in p
+
+
+typedef mul<p[x],p[x]> px2;
+typedef mul<p[y],p[y]> py2;
+typedef mul<p[x],p[y]> px_py;
+typedef add<px2,py2> px2_p_py2;
+typedef add<px2,py2> px2_s_py2;
+typedef mul<px2,px> px3;
+typedef mul<py2,py> py3;
+typedef mul<gm,nu> gm_nu;
+
+// quadratic term in p
+
+typedef mul<px3,py> px3_m_py;
+typedef mul<px2,py2> px2_m_py2;
+typedef mul<px,py3> px_m_py3;
+
+typedef mul<px2,px2_s_py2> px2_r;
+typedef mul<px_py,px2_s_py2> px_py_r;
+typedef mul<py2,px2_s_py2> py2_r;
+
+// Division of the quadratic term by px2_s_py2
+
+typedef div<px3_m_py,px2_p_py2> p_c1;
+typedef div<px2_m_py2,px2_p_py2> p_c2;
+typedef div<px_m_py3,px2_p_py2> p_c3;
+
+typedef div<px2_r,px2_p_py2> p_c4;
+typedef div<px_py_r,px2_p_py2> p_c5;
+typedef div<py2_r,px2_p_py2> p_c6;
+
+// Create u_alpha_beta equation
+
+template <unsigned int i, unsigned int j> using u = div<2,add<D<i,v[j]>,D<j,v[i]>>>;
+
+// terms
+
+typedef mul<gm,p_c1,u[x][x]> t1;
+typedef mul<gm,p_c2,u[x][y]> t2;
+typedef mul<gm,p_c3,u[y][y]> t3;
+
+typedef mul<gm,p_c4,u[x][x]> t4;
+typedef mul<gm,p_c5,u[x][y]> t5;
+typedef mul<gm,p_c6,u[y][y]> t6;
+
+#endif /* OPENFPM_NUMERICS_SRC_EQUATIONS_EQ_BASE_UNIT_TEST_HPP_ */

src/Equations/eq_unit_tests.hpp

+/*
+ * eq_unit_tests.hpp
+ *
+ *  Created on: Sep 18, 2015
+ *      Author: i-bird
+ */
+
+#ifndef OPENFPM_NUMERICS_SRC_EQUATIONS_EQ_UNIT_TESTS_HPP_
+#define OPENFPM_NUMERICS_SRC_EQUATIONS_EQ_UNIT_TESTS_HPP_
+
+#include "FiniteDifference/FDScheme.hpp"
+
+BOOST_AUTO_TEST_SUITE( eq_test )
+
+struct Sys_p
+{
+	static const unsigned int dims = 3;
+	typedef float stype;
+};
+
+// define coordinates id
+
+constexpr unsigned int x = 0;
+constexpr unsigned int y = 1;
+
+// define constants
+
+constexpr unsigned int eta = 0;
+constexpr unsigned int zeta = 1;
+constexpr unsigned int gamma = 2;
+constexpr unsigned int nu = 3;
+constexpr unsigned int lambda = 4;
+constexpr unsigned int dmu = 5;
+constexpr unsigned int sigma[2][2] = {{6,7},{8,9}};
+constexpr unsigned int p[] = {10,11};
+constexpr unsigned int del_nu = 12;
+constexpr unsigned int g_ext = 13;
+
+// define variables fields
+
+// velocity
+constexpr unsigned int V[2] = {0,1};
+
+// pressure
+constexpr unsigned int P = 2;
+
+// model Eq1 active Gel
+
+//// First level expression
+
+
+
+/*
+ *
+ * p_x^2
+ *
+ */
+//typedef mul<p[x],p[x],Sys_p> px2;
+
+
+/*
+ *
+ * p_y^2
+ *
+ */
+//typedef mul<p[y],p[y],Sys_p> py2;
+
+
+/*
+ * p_x * p_y
+ *
+ */
+//typedef mul<p[x],p[y],Sys_p> px_py;
+
+
+
+BOOST_AUTO_TEST_CASE( eq_test_use)
+{
+	// create an equation object
+
+
+}
+
+BOOST_AUTO_TEST_SUITE_END()
+
+#endif /* OPENFPM_NUMERICS_SRC_EQUATIONS_EQ_UNIT_TESTS_HPP_ */

src/FiniteDifference/Average.hpp

+/*
+ * Average.hpp
+ *
+ *  Created on: Nov 18, 2015
+ *      Author: Pietro Incardona
+ */
+
+#ifndef OPENFPM_NUMERICS_SRC_FINITEDIFFERENCE_AVERAGE_HPP_
+#define OPENFPM_NUMERICS_SRC_FINITEDIFFERENCE_AVERAGE_HPP_
+
+
+#define CENTRAL 0
+#define CENTRAL_B_ONE_SIDE 1
+#define FORWARD 2
+
+#include "util/mathutil.hpp"
+#include "Vector/map_vector.hpp"
+#include "Grid/comb.hpp"
+#include "FiniteDifference/util/common.hpp"
+#include "util/util_num.hpp"
+
+/*! \brief Average
+ *
+ * \tparam d on which dimension average
+ * \tparam Field which field average
+ * \tparam impl which implementation
+ *
+ */
+template<unsigned int d, typename Field, typename Sys_eqs, unsigned int impl=CENTRAL>
+class Avg
+{
+	/*! \brief Create the row of the Matrix
+	 *
+	 * \tparam ord
+	 *
+	 */
+	inline static void value(const grid_key_dx<Sys_eqs::dims> & pos, const grid_sm<Sys_eqs::dims,void> & gs, std::unordered_map<long int,typename Sys_eqs::stype > & cols, typename Sys_eqs::stype coeff)
+	{
+		std::cerr << "Error " << __FILE__ << ":" << __LINE__ << " only CENTRAL, FORWARD, BACKWARD derivative are defined";
+	}
+
+	/*! \brief Calculate the position where the derivative is calculated
+	 *
+	 * In case on non staggered case this function just return pos, in case of staggered,
+	 *  it calculate where the operator is calculated on a staggered grid
+	 *
+	 */
+	inline static grid_key_dx<Sys_eqs::dims> position(grid_key_dx<Sys_eqs::dims> & pos, const grid_sm<Sys_eqs::dims,void> & gs)
+	{
+		std::cerr << "Error " << __FILE__ << ":" << __LINE__ << " only CENTRAL, FORWARD, BACKWARD derivative are defined";
+	}
+};
+
+/*! \brief Average on direction i
+ *
+ *
+ */
+template<unsigned int d, typename arg, typename Sys_eqs>
+class Avg<d,arg,Sys_eqs,CENTRAL>
+{
+	public:
+
+	/*! \brief fill the row
+	 *
+	 *
+	 */
+	inline static void value(const typename stub_or_real<Sys_eqs,Sys_eqs::dims,typename Sys_eqs::stype,typename Sys_eqs::b_grid::decomposition>::type & g_map, grid_dist_key_dx<Sys_eqs::dims> & kmap , const grid_sm<Sys_eqs::dims,void> & gs, std::unordered_map<long int,typename Sys_eqs::stype > & cols, typename Sys_eqs::stype coeff)
+	{
+		// if the system is staggered the CENTRAL derivative is equivalent to a forward derivative
+		if (is_grid_staggered<Sys_eqs>::value() == true)
+		{
+			Avg<d,arg,Sys_eqs,FORWARD>::value(g_map,kmap,gs,cols,coeff);
+			return;
+		}
+
+		long int old_val = kmap.getKeyRef().get(d);
+		kmap.getKeyRef().set_d(d, kmap.getKeyRef().get(d) + 1);
+		arg::value(g_map,kmap,gs,cols,coeff);
+		kmap.getKeyRef().set_d(d,old_val);
+
+
+		old_val = kmap.getKeyRef().get(d);
+		kmap.getKeyRef().set_d(d, kmap.getKeyRef().get(d) - 1);
+		arg::value(g_map,kmap,gs,cols,coeff);
+		kmap.getKeyRef().set_d(d,old_val);
+	}
+
+
+	/*! \brief Calculate the position where the derivative is calculated
+	 *
+	 * In case on non staggered case this function just return pos, in case of staggered,
+	 *  it calculate where the operator is calculated on a staggered grid
+	 *
+	 *  \param pos from the position
+	 *  \param fld Field we are deriving, if not provided the function just return pos
+	 *  \param s_pos position of the properties in the staggered grid
+	 *
+	 */
+	inline static grid_key_dx<Sys_eqs::dims> position(grid_key_dx<Sys_eqs::dims> & pos, long int fld = -1, const openfpm::vector<comb<Sys_eqs::dims>> & s_pos = openfpm::vector<comb<Sys_eqs::dims>>())
+	{
+		if (is_grid_staggered<Sys_eqs>::value())
+		{
+			if (fld == -1)
+				return pos;
+
+			if (s_pos[fld][d] == 1)
+			{
+				grid_key_dx<Sys_eqs::dims> ret = pos;
+				ret.set_d(d,0);
+				return pos;
+			}
+			else
+			{
+				grid_key_dx<Sys_eqs::dims> ret = pos;
+				ret.set_d(d,1);
+				return pos;
+			}
+		}
+
+		return pos;
+	}
+};
+
+/*! \brief Average FORWARD on direction i
+ *
+ *g
+ */
+template<unsigned int d, typename arg, typename Sys_eqs>
+class Avg<d,arg,Sys_eqs,FORWARD>
+{
+	public:
+
+	/*! \brief fill the row
+	 *
+	 *
+	 */
+	inline static void value(const typename stub_or_real<Sys_eqs,Sys_eqs::dims,typename Sys_eqs::stype,typename Sys_eqs::b_grid::decomposition>::type & g_map, grid_dist_key_dx<Sys_eqs::dims> & kmap , const grid_sm<Sys_eqs::dims,void> & gs, std::unordered_map<long int,typename Sys_eqs::stype > & cols, typename Sys_eqs::stype coeff)
+	{
+
+		long int old_val = kmap.getKeyRef().get(d);
+		kmap.getKeyRef().set_d(d, kmap.getKeyRef().get(d) + 1);
+		arg::value(g_map,kmap,gs,cols,coeff);
+		kmap.getKeyRef().set_d(d,old_val);
+
+		// backward
+		arg::value(g_map,kmap,gs,cols,coeff);
+	}
+
+
+	/*! \brief Calculate the position where the derivative is calculated
+	 *
+	 * In case on non staggered case this function just return pos, in case of staggered,
+	 *  it calculate where the operator is calculated on a staggered grid
+	 *
+	 *  \param pos from the position
+	 *  \param fld Field we are deriving, if not provided the function just return pos
+	 *  \param s_pos position of the properties in the staggered grid
+	 *
+	 */
+	inline static grid_key_dx<Sys_eqs::dims> position(grid_key_dx<Sys_eqs::dims> & pos, long int fld = -1, const openfpm::vector<comb<Sys_eqs::dims>> & s_pos = openfpm::vector<comb<Sys_eqs::dims>>())
+	{
+		return pos;
+	}
+};
+
+
+#endif /* OPENFPM_NUMERICS_SRC_FINITEDIFFERENCE_AVERAGE_HPP_ */

src/FiniteDifference/Derivative.hpp

@@ -2,7 +2,7 @@
  * Derivative.hpp
  *
  *  Created on: Oct 5, 2015
- *      Author: i-bird
+ *      Author: Pietro Incardona
  */
 
 #ifndef OPENFPM_NUMERICS_SRC_FINITEDIFFERENCE_DERIVATIVE_HPP_

src/FiniteDifference/FDScheme.hpp

@@ -84,8 +84,10 @@ class FDScheme
 	// Domain Grid informations
 	const grid_sm<Sys_eqs::dims,void> & gs;
 
+	typedef grid_dist_id<Sys_eqs::dims,typename Sys_eqs::stype,scalar<size_t>,typename Sys_eqs::b_grid::decomposition> g_map_type;
+
 	// mapping grid
-	grid_dist_id<Sys_eqs::dims,typename Sys_eqs::stype,scalar<size_t>,typename Sys_eqs::b_grid::decomposition> g_map;
+	g_map_type g_map;
 
 	// row of the matrix
 	size_t row;
@@ -102,6 +104,23 @@ class FDScheme
 	// processor
 	size_t s_pnt;
 
+	/* \brief calculate the mapping grid size with padding
+	 *
+	 * \param gs original grid size
+	 *
+	 * \return padded grid size
+	 *
+	 */
+	inline const std::vector<size_t> padding( const size_t (& sz)[Sys_eqs::dims], Padding<Sys_eqs::dims> & pd)
+	{
+		std::vector<size_t> g_sz_pad(Sys_eqs::dims);
+
+		for (size_t i = 0 ; i < Sys_eqs::dims ; i++)
+			g_sz_pad[i] = sz[i] + pd.getLow(i) + pd.getHigh(i);
+
+		return g_sz_pad;
+	}
+
 	/*! \brief Check if the Matrix is consistent
 	 *
 	 */
@@ -142,14 +161,14 @@ class FDScheme
 		}
 	}
 
-	/*! \brief Convert from integer ghost to continuos
+	/* \brief Convert discrete ghost into continous ghost
 	 *
-	 * \return the continuos version of the ghost
+	 * \param Ghost
 	 *
 	 */
-	Ghost<dim,Sys_eqs::stype> convert_into_cg()
+	inline Ghost<Sys_eqs::dims,typename Sys_eqs::stype> convert_dg_cg(const Ghost<Sys_eqs::dims,typename Sys_eqs::stype> & g)
 	{
-
+		return g_map_type::convert_ghost(g,g_map.getCellDecomposer());
 	}
 
 public:
@@ -160,18 +179,20 @@ public:
 	 * \param gs grid infos where Finite differences work
 	 *
 	 */
-	FDScheme(Padding<Sys_eqs::dims> & pd, const Box<Sys_eqs::dims,typename Sys_eqs::stype> & domain, const grid_sm<Sys_eqs::dims,void> & gs, typename Sys_eqs::b_grid::decomposition & dec, Vcluster & v_cl)
-	:pd(pd),gs(gs),g_map(dec.duplicate(Ghost<Sys_eqs::dims,long int>(1)),gs.getSize(),domain)
+	FDScheme(Padding<Sys_eqs::dims> & pd, const Box<Sys_eqs::dims,typename Sys_eqs::stype> & domain, const grid_sm<Sys_eqs::dims,void> & gs, const typename Sys_eqs::b_grid::decomposition & dec, Vcluster & v_cl)
+	:pd(pd),gs(gs),g_map(dec,padding(gs.getSize(),pd),domain,Ghost<Sys_eqs::dims,typename Sys_eqs::stype>(1)),row(0),row_b(0)
 	{
 		// Calculate the size of the local domain
 		size_t sz = g_map.getLocalDomainSize();
 
 		// Get the total size of the local grids on each processors
 		v_cl.allGather(sz,pnt);
+		v_cl.execute();
+		s_pnt = 0;
 
 		// calculate the starting point for this processor
 		for (size_t i = 0 ; i < v_cl.getProcessUnitID() ; i++)
-			s_pnt += pnt.get(0);
+			s_pnt += pnt.get(i);
 
 		// Calculate the starting point
 
@@ -204,8 +225,18 @@ public:
 	 *
 	 *
 	 */
-	template<typename T> void imposeA(const T & op , grid_dist_iterator_sub<Sys_eqs::dims,typename Sys_eqs::b_grid::d_grid> it)
+	template<typename T> void imposeA(const T & op , grid_dist_iterator_sub<Sys_eqs::dims,typename Sys_eqs::b_grid::d_grid> it_d)
 	{
+		// key one
+		grid_key_dx<Sys_eqs::dims> gk_one;
+		gk_one.one();
+
+		// add padding to start and stop
+		grid_key_dx<Sys_eqs::dims> start = it_d.getStart() + pd.getKP1();
+		grid_key_dx<Sys_eqs::dims> stop = it_d.getStop() + pd.getKP1();
+
+		auto it = g_map.getSubDomainIterator(start,stop);
+
 		std::unordered_map<long int,float> cols;
 
 		// iterate all the grid points

src/FiniteDifference/FDScheme_unit_tests.hpp

@@ -12,6 +12,7 @@
 #include "FiniteDifference/Laplacian.hpp"
 #include "Decomposition/CartDecomposition.hpp"
 #include "util/grid_dist_testing.hpp"
+#include "FiniteDifference/Average.hpp"
 
 constexpr unsigned int x = 0;
 constexpr unsigned int y = 1;
@@ -113,7 +114,7 @@ const bool syss_pp::boundary[] = {PERIODIC,PERIODIC};
 
 BOOST_AUTO_TEST_SUITE( fd_test )
 
-BOOST_AUTO_TEST_CASE( fd_test_use_non_periodic)
+BOOST_AUTO_TEST_CASE( fd_test_use_der_non_periodic)
 {
 	// grid size
 	size_t sz[2]={16,16};
@@ -239,8 +240,78 @@ BOOST_AUTO_TEST_CASE( fd_test_use_non_periodic)
 	BOOST_REQUIRE_EQUAL(cols_y[13*16+15],0.5);
 }
 
+BOOST_AUTO_TEST_CASE( fd_test_use_avg_non_periodic)
+{
+	// grid size
+	size_t sz[2]={16,16};
+
+	// grid_dist_testing
+	grid_dist_testing<2> g_map(sz);
+
+	// grid_sm
+	grid_sm<2,void> ginfo(sz);
+
+	// Create several keys
+	grid_dist_key_dx<2> key11(0,grid_key_dx<2>(1,1));
+	grid_dist_key_dx<2> key00(0,grid_key_dx<2>(0,0));
+	grid_dist_key_dx<2> key22(0,grid_key_dx<2>(2,2));
+	grid_dist_key_dx<2> key1515(0,grid_key_dx<2>(15,15));
+
+	// filled colums
+	std::unordered_map<long int,float> cols_x;
+	std::unordered_map<long int,float> cols_y;
+
+	Avg<x,Field<V,sys_nn>,sys_nn>::value(g_map,key11,ginfo,cols_x,1);
+	Avg<y,Field<V,sys_nn>,sys_nn>::value(g_map,key11,ginfo,cols_y,1);
+
+	BOOST_REQUIRE_EQUAL(cols_x.size(),2);
+	BOOST_REQUIRE_EQUAL(cols_y.size(),2);
+
+	BOOST_REQUIRE_EQUAL(cols_x[17+1],1);
+	BOOST_REQUIRE_EQUAL(cols_x[17-1],1);
+
+	BOOST_REQUIRE_EQUAL(cols_y[17+16],1);
+	BOOST_REQUIRE_EQUAL(cols_y[17-16],1);
+
+	// filled colums
+
+	std::unordered_map<long int,float> cols_xx;
+	std::unordered_map<long int,float> cols_xy;
+	std::unordered_map<long int,float> cols_yx;
+	std::unordered_map<long int,float> cols_yy;
+
+	// Composed derivative
+
+	Avg<x,Avg<x,Field<V,sys_nn>,sys_nn>,sys_nn>::value(g_map,key22,ginfo,cols_xx,1);
+	Avg<x,Avg<y,Field<V,sys_nn>,sys_nn>,sys_nn>::value(g_map,key22,ginfo,cols_xy,1);
+	Avg<y,Avg<x,Field<V,sys_nn>,sys_nn>,sys_nn>::value(g_map,key22,ginfo,cols_yx,1);
+	Avg<y,Avg<y,Field<V,sys_nn>,sys_nn><