inmost_dense.h 29.6 KB
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#ifndef INMOST_DENSE_INCLUDED
#define INMOST_DENSE_INCLUDED
#include "inmost_common.h"
#if defined(USE_AUTODIFF)
#include "inmost_expression.h"
#endif
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#include <iomanip>
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// Matrix with n columns and m rows
//   __m__
//  |     |
// n|     |
//  |_____|
//
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// todo:
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// 1. expression templates for operations
//    (???) how to for multiplication?
// 2. (ok) template matrix type for AD variables
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// 3. template container type for data storage.
// 4. option for wrapper container around provided data storage. (to perform matrix operations with existing data)
// 5. class Subset for fortran-like access to matrix.
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namespace INMOST
{
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	template<class A, class B> struct Promote;
	template<> struct Promote<INMOST_DATA_REAL_TYPE, INMOST_DATA_REAL_TYPE> {typedef INMOST_DATA_REAL_TYPE type;};
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#if defined(USE_AUTODIFF)
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	template<> struct Promote<INMOST_DATA_REAL_TYPE, variable>  {typedef variable type;};
	template<> struct Promote<variable, INMOST_DATA_REAL_TYPE>  {typedef variable type;};
	template<> struct Promote<variable, variable> {typedef variable type;};
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	template<> struct Promote<INMOST_DATA_REAL_TYPE, hessian_variable>  {typedef hessian_variable type;};
	template<> struct Promote<hessian_variable, INMOST_DATA_REAL_TYPE>  {typedef hessian_variable type;};
	template<> struct Promote<variable, hessian_variable>  {typedef hessian_variable type;};
	template<> struct Promote<hessian_variable, variable>  {typedef hessian_variable type;};
	template<> struct Promote<hessian_variable, hessian_variable> {typedef hessian_variable type;};
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#else
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	__INLINE INMOST_DATA_REAL_TYPE get_value(INMOST_DATA_REAL_TYPE x) {return x;}
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#endif
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	template<typename Var>
	class Matrix
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	{
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	public:
		typedef unsigned enumerator;
	protected:
		array<Var> space;
		enumerator n, m;
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		static Var sign_func(const Var & a, const Var & b) {return (b >= 0.0 ? fabs(a) : -fabs(a));}
		static INMOST_DATA_REAL_TYPE max_func(INMOST_DATA_REAL_TYPE x, INMOST_DATA_REAL_TYPE y) { return x > y ? x : y; }
		static Var pythag(const Var & a, const Var & b)
		{
			Var at = fabs(a), bt = fabs(b), ct, result;
			if (at > bt)       { ct = bt / at; result = at * sqrt(1.0 + ct * ct); }
			else if (bt > 0.0) { ct = at / bt; result = bt * sqrt(1.0 + ct * ct); }
			else result = 0.0;
			return result;
		}
	public:
		bool CheckNans()
		{
			for(enumerator k = 0; k < n*m; ++k)
				if( check_nans(space[k]) ) return true;
			return false;
		}
		void RemoveRow(enumerator row)
		{
			for(enumerator k = row+1; k < n; ++k)
			{
				for(enumerator l = 0; l < m; ++l)
					(*this)(k-1,l) = (*this)(k,l);
			}
			space.resize((n-1)*m);
			--n;
		}
		void RemoveRows(enumerator first, enumerator last)
		{
			enumerator shift = last-first;
			for(enumerator k = last+1; k < n; ++k)
			{
				for(enumerator l = 0; l < m; ++l)
					(*this)(k-shift-1,l) = (*this)(k,l);
			}
			space.resize((n-shift)*m);
			n-=shift;
		}
		void RemoveColumn(enumerator col)
		{
			Matrix<Var> tmp(n,m-1);
			for(enumerator k = 0; k < n; ++k)
			{
				for(enumerator l = 0; l < col; ++l)
					tmp(k,l) = (*this)(k,l);
				for(enumerator l = col+1; l < m; ++l)
					tmp(k,l-1) = (*this)(k,l);
			}
			this->Swap(tmp);
		}
		void RemoveColumns(enumerator first, enumerator last)
		{
			enumerator shift = last-first;
			Matrix<Var> tmp(n,m-shift);
			for(enumerator k = 0; k < n; ++k)
			{
				for(enumerator l = 0; l < first; ++l)
					tmp(k,l) = (*this)(k,l);
				for(enumerator l = last+1; l < m; ++l)
					tmp(k,l-shift-1) = (*this)(k,l);
			}
			this->Swap(tmp);
		}
		void RemoveSubset(enumerator firstrow, enumerator lastrow, enumerator firstcol, enumerator lastcol)
		{
			enumerator shiftrow = lastrow-firstrow;
			enumerator shiftcol = lastcol-firstcol;
			Matrix<Var> tmp(n-shiftrow, m-shiftcol);
			for(enumerator k = 0; k < firstrow; ++k)
			{
				for(enumerator l = 0; l < firstcol; ++l)
					tmp(k,l) = (*this)(k,l);
				for(enumerator l = lastcol+1; l < m; ++l)
					tmp(k,l-shiftcol-1) = (*this)(k,l);
			}
			for(enumerator k = lastrow+1; k < n; ++k)
			{
				for(enumerator l = 0; l < firstcol; ++l)
					tmp(k-shiftrow-1,l) = (*this)(k,l);
				for(enumerator l = lastcol+1; l < m; ++l)
					tmp(k-shiftrow-1,l-shiftcol-1) = (*this)(k,l);
			}
			this->Swap(tmp);
		}
		void Swap(Matrix & b)
		{
			space.swap(b.space);
			std::swap(n,b.n);
			std::swap(m,b.m);
		}
		/// Singular value decomposition.
		/// Reconstruct matrix: A = U*Sigma*V.Transpose().
		/// @param U Left unitary matrix, U^T U = I.
		/// @param Sigma Diagonal matrix with singular values.
		/// @param V Right unitary matrix, not transposed.
		/// @param order_singular_values
		bool SVD(Matrix & U, Matrix & Sigma, Matrix & V, bool order_singular_values = true)
		{
			int flag, i, its, j, jj, k, l, nm;
			Var c, f, h, s, x, y, z;
			Var g = 0.0, scale = 0.0;
			INMOST_DATA_REAL_TYPE anorm = 0.0;
			if (n < m)
			{
				bool success = Transpose().SVD(U,Sigma,V);
				if( success )
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				{
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					U.Swap(V);
					U = U.Transpose();
					V = V.Transpose();
					return true;
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				}
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				else return false;
			} //m <= n
			array<Var> _rv1(m);
			shell<Var> rv1(_rv1);
			U = (*this);
			Sigma.Resize(m,m);
			Sigma.Zero();
			V.Resize(m,m);
			
			std::swap(n,m); //this how original algorithm takes it
			// Householder reduction to bidiagonal form
			for (i = 0; i < (int)n; i++)
			{
				// left-hand reduction
				l = i + 1;
				rv1[i] = scale * g;
				g = s = scale = 0.0;
				if (i < (int)m)
				{
					for (k = i; k < (int)m; k++) scale += fabs(U(k,i));
					if (get_value(scale))
					{
						for (k = i; k < (int)m; k++)
						{
							U(k,i) /= scale;
							s += U(k,i) * U(k,i);
						}
						f = U(i,i);
						g = -sign_func(sqrt(s), f);
						h = f * g - s;
						U(i,i) = f - g;
						if (i != n - 1)
						{
							for (j = l; j < (int)n; j++)
							{
								for (s = 0.0, k = i; k < (int)m; k++) s += U(k,i) * U(k,j);
								f = s / h;
								for (k = i; k < (int)m; k++) U(k,j) += f * U(k,i);
							}
						}
						for (k = i; k < (int)m; k++) U(k,i) *= scale;
					}
				}
				Sigma(i,i) = scale * g;
				// right-hand reduction
				g = s = scale = 0.0;
				if (i < (int)m && i != n - 1)
				{
					for (k = l; k < (int)n; k++) scale += fabs(U(i,k));
					if (get_value(scale))
					{
						for (k = l; k < (int)n; k++)
						{
							U(i,k) = U(i,k)/scale;
							s += U(i,k) * U(i,k);
						}
						f = U(i,l);
						g = -sign_func(sqrt(s), f);
						h = f * g - s;
						U(i,l) = f - g;
						for (k = l; k < (int)n; k++) rv1[k] = U(i,k) / h;
						if (i != m - 1)
						{
							for (j = l; j < (int)m; j++)
							{
								for (s = 0.0, k = l; k < (int)n; k++) s += U(j,k) * U(i,k);
								for (k = l; k < (int)n; k++) U(j,k) += s * rv1[k];
							}
						}
						for (k = l; k < (int)n; k++) U(i,k) *= scale;
					}
				}
				anorm = max_func(anorm,fabs(get_value(Sigma(i,i))) + fabs(get_value(rv1[i])));
			}
			
			// accumulate the right-hand transformation
			for (i = n - 1; i >= 0; i--)
			{
				if (i < (int)(n - 1))
				{
					if (get_value(g))
					{
						for (j = l; j < (int)n; j++) V(j,i) = ((U(i,j) / U(i,l)) / g);
						// double division to avoid underflow
						for (j = l; j < (int)n; j++)
						{
							for (s = 0.0, k = l; k < (int)n; k++) s += U(i,k) * V(k,j);
							for (k = l; k < (int)n; k++) V(k,j) += s * V(k,i);
						}
					}
					for (j = l; j < (int)n; j++) V(i,j) = V(j,i) = 0.0;
				}
				V(i,i) = 1.0;
				g = rv1[i];
				l = i;
			}
			
			// accumulate the left-hand transformation
			for (i = n - 1; i >= 0; i--)
			{
				l = i + 1;
				g = Sigma(i,i);
				if (i < (int)(n - 1))
					for (j = l; j < (int)n; j++)
						U(i,j) = 0.0;
				if (get_value(g))
				{
					g = 1.0 / g;
					if (i != n - 1)
					{
						for (j = l; j < (int)n; j++)
						{
							for (s = 0.0, k = l; k < (int)m; k++) s += (U(k,i) * U(k,j));
							f = (s / U(i,i)) * g;
							for (k = i; k < (int)m; k++) U(k,j) += f * U(k,i);
						}
					}
					for (j = i; j < (int)m; j++) U(j,i) = U(j,i)*g;
				}
				else for (j = i; j < (int)m; j++) U(j,i) = 0.0;
				U(i,i) += 1;
			}
			
			// diagonalize the bidiagonal form
			for (k = n - 1; k >= 0; k--)
			{// loop over singular values
				for (its = 0; its < 30; its++)
				{// loop over allowed iterations
					flag = 1;
					for (l = k; l >= 0; l--)
					{// test for splitting
						nm = l - 1;
						if (fabs(get_value(rv1[l])) + anorm == anorm)
						{
							flag = 0;
							break;
						}
						if (fabs(get_value(Sigma(nm,nm))) + anorm == anorm)
							break;
					}
					if (flag)
					{
						c = 0.0;
						s = 1.0;
						for (i = l; i <= k; i++)
						{
							f = s * rv1[i];
							if (fabs(get_value(f)) + anorm != anorm)
							{
								g = Sigma(i,i);
								h = pythag(f, g);
								Sigma(i,i) = h;
								h = 1.0 / h;
								c = g * h;
								s = (- f * h);
								for (j = 0; j < (int)m; j++)
								{
									y = U(j,nm);
									z = U(j,i);
									U(j,nm) = (y * c + z * s);
									U(j,i) = (z * c - y * s);
								}
							}
						}
					}
					z = Sigma(k,k);
					if (l == k)
					{// convergence
						if (z < 0.0)
						{// make singular value nonnegative
							Sigma(k,k) = -z;
							for (j = 0; j < (int)n; j++) V(j,k) = -V(j,k);
						}
						break;
					}
					if (its >= 30)
					{
						std::cout << "No convergence after " << its << " iterations" << std::endl;
						std::swap(n,m);
						return false;
					}
					// shift from bottom 2 x 2 minor
					x = Sigma(l,l);
					nm = k - 1;
					y = Sigma(nm,nm);
					g = rv1[nm];
					h = rv1[k];
					f = ((y - z) * (y + z) + (g - h) * (g + h)) / (2.0 * h * y);
					g = pythag(f, 1.0);
					f = ((x - z) * (x + z) + h * ((y / (f + sign_func(g, f))) - h)) / x;
					// next QR transformation
					c = s = 1.0;
					for (j = l; j <= nm; j++)
					{
						i = j + 1;
						g = rv1[i];
						y = Sigma(i,i);
						h = s * g;
						g = c * g;
						z = pythag(f, h);
						rv1[j] = z;
						c = f / z;
						s = h / z;
						f = x * c + g * s;
						g = g * c - x * s;
						h = y * s;
						y = y * c;
						for (jj = 0; jj < (int)n; jj++)
						{
							x = V(jj,j);
							z = V(jj,i);
							V(jj,j) = (x * c + z * s);
							V(jj,i) = (z * c - x * s);
						}
						z = pythag(f, h);
						Sigma(j,j) = z;
						if (z)
						{
							z = 1.0 / z;
							c = f * z;
							s = h * z;
						}
						f = (c * g) + (s * y);
						x = (c * y) - (s * g);
						for (jj = 0; jj < (int)m; jj++)
						{
							y = U(jj,j);
							z = U(jj,i);
							U(jj,j) = (y * c + z * s);
							U(jj,i) = (z * c - y * s);
						}
					}
					rv1[l] = 0.0;
					rv1[k] = f;
					Sigma(k,k) = x;
				}
			}
			//CHECK THIS!
			if( order_singular_values )
			{
				for(i = 0; i < (int)n; i++)
				{
					k = i;
					for(j = i+1; j < (int)n; ++j)
						if( Sigma(k,k) < Sigma(j,j) ) k = j;
					Var temp;
					if( Sigma(k,k) > Sigma(i,i) )
					{
						temp       = Sigma(k,k);
						Sigma(k,k) = Sigma(i,i);
						Sigma(i,i) = temp;
						// U is m by n
						for(int j = 0; j < (int)m; ++j)
						{
							temp   = U(j,k);
							U(j,k) = U(j,i);
							U(j,i) = temp;
						}
						// V is n by n
						for(int j = 0; j < (int)n; ++j)
						{
							temp   = V(j,k);
							V(j,k) = V(j,i);
							V(j,i) = temp;
						}
					}
				}
			}
			
			std::swap(n,m);
			return true;
		}
		Matrix() : space(0),n(0),m(0) {}
		Matrix(Var * pspace, enumerator pn, enumerator pm) : space(pspace,pspace+pn*pm), n(pn), m(pm) {}
		Matrix(enumerator pn, enumerator pm) : space(pn*pm), n(pn), m(pm) {}
		Matrix(const Matrix & other) : space(other.n*other.m), n(other.n), m(other.m)
		{
			for(enumerator i = 0; i < n*m; ++i)
				space[i] = other.space[i];
		}
		template<typename typeB>
		Matrix(const Matrix<typeB> & other) : space(other.Cols()*other.Rows()), n(other.Rows()), m(other.Cols())
		{
			for(enumerator i = 0; i < n; ++i)
				for(enumerator j = 0; j < m; ++j)
					(*this)(i,j) = get_value(other(i,j));
		}
		~Matrix() {}
		void Resize(enumerator nrows, enumerator mcols)
		{
			if( space.size() != mcols*nrows )
				space.resize(mcols*nrows);
			n = nrows;
			m = mcols;
		}
		Matrix & operator =(Matrix const & other)
		{
			if( n*m != other.n*other.m ) space.resize(other.n*other.m);
			for(enumerator i = 0; i < other.n*other.m; ++i)
				space[i] = other.space[i];
			n = other.n;
			m = other.m;
			return *this;
		}
		template<typename typeB>
		Matrix & operator =(Matrix<typeB> const & other)
		{
			if( n*m != other.n*other.m ) space.resize(other.n*other.m);
			for(enumerator i = 0; i < other.n*other.m; ++i)
				space[i] = get_value(other.space[i]);
			n = other.n;
			m = other.m;
			return *this;
		}
		// i is in [0,n] - row index
		// j is in [0,m] - column index
		Var & operator()(enumerator i, enumerator j)
		{
			assert(i >= 0 && i < n);
			assert(j >= 0 && j < m);
			assert(i*m+j < n*m); //overflow check?
			return space[i*m+j];
		}
		const Var & operator()(enumerator i, enumerator j) const
		{
			assert(i >= 0 && i < n);
			assert(j >= 0 && j < m);
			assert(i*m+j < n*m); //overflow check?
			return space[i*m+j];
		}
		Matrix operator-() const
		{
			Matrix ret(n,m);
			for(enumerator k = 0; k < n*m; ++k) ret.space[k] = -space[k];
			return ret;
		}
		template<typename typeB>
		Matrix<typename Promote<Var,typeB>::type> operator-(const Matrix<typeB> & other) const
		{
			assert(Rows() == other.Rows());
			assert(Cols() == other.Cols());
			Matrix<typename Promote<Var,typeB>::type> ret(n,m); //check RVO
			for(enumerator i = 0; i < Rows(); ++i)
				for(enumerator j = 0; j < Cols(); ++j)
					ret(i,j) = (*this)(i,j)-other(i,j);
			return ret;
		}
		Matrix & operator-=(const Matrix & other)
		{
			assert(n == other.n);
			assert(m == other.m);
			for(enumerator k = 0; k < n*m; ++k) space[k] -= other.space[k];
			return *this;
		}
		template<typename typeB>
		Matrix<typename Promote<Var,typeB>::type> operator+(const Matrix<typeB> & other) const
		{
			assert(Rows() == other.Rows());
			assert(Cols() == other.Cols());
			Matrix<typename Promote<Var,typeB>::type> ret(n,m); //check RVO
			for(enumerator i = 0; i < Rows(); ++i)
				for(enumerator j = 0; j < Cols(); ++j)
					ret(i,j) = (*this)(i,j)+other(i,j);
			return ret;
		}
		Matrix & operator+=(const Matrix & other)
		{
			assert(n == other.n);
			assert(m == other.m);
			for(enumerator k = 0; k < n*m; ++k) space[k] += other.space[k];
			return *this;
		}
		template<typename typeB>
		Matrix<typename Promote<Var,typeB>::type> operator*(typeB coef) const
		{
			Matrix<typename Promote<Var,typeB>::type> ret(n,m); //check RVO
			for(enumerator i = 0; i < Rows(); ++i)
				for(enumerator j = 0; j < Cols(); ++j) ret(i,j) = (*this)(i,j)*coef;
			return ret;
		}
		Matrix & operator*=(Var coef)
		{
			for(enumerator k = 0; k < n*m; ++k) space[k] *= coef;
			return *this;
		}
		template<typename typeB>
		Matrix<typename Promote<Var,typeB>::type> operator/(typeB coef) const
		{
			Matrix<typename Promote<Var,typeB>::type> ret(n,m); //check RVO
			for(enumerator i = 0; i < Rows(); ++i)
				for(enumerator j = 0; j < Cols(); ++j) ret(i,j) = (*this)(i,j)/coef;
			return ret;
		}
		Matrix & operator/=(Var coef)
		{
			for(enumerator k = 0; k < n*m; ++k) space[k] /= coef;
			return *this;
		}
		template<typename typeB>
		Matrix<typename Promote<Var,typeB>::type> operator*(const Matrix<typeB> & other) const
		{
			assert(Cols() == other.Rows());
			Matrix<typename Promote<Var,typeB>::type> ret(Rows(),other.Cols()); //check RVO
			for(enumerator i = 0; i < Rows(); ++i) //loop rows
			{
				for(enumerator j = 0; j < other.Cols(); ++j) //loop columns
				{
					typename Promote<Var,typeB>::type tmp = 0.0;
					for(enumerator k = 0; k < Cols(); ++k)
						tmp += (*this)(i,k)*other(k,j);
					ret(i,j) = tmp;
				}
			}
			return ret;
		}
		/// performs A*B^{-1}
		/// checks existence of B^{-1} only in debug mode.
		template<typename typeB>
		Matrix<typename Promote<Var,typeB>::type> operator/(const Matrix<typeB> & other) const
		{
			std::pair<Matrix<typeB>,bool> other_inv = other.Invert();
			assert(other_inv.second);
			assert(Cols() == other_inv.Rows());
			Matrix<typename Promote<Var,typeB>::type> ret(n,other.m); //check RVO
			for(enumerator i = 0; i < Rows(); ++i) //loop rows
			{
				for(enumerator j = 0; j < other_inv.Cols(); ++j) //loop columns
				{
					typename Promote<Var,typeB>::type tmp = 0.0;
					for(enumerator k = 0; k < Cols(); ++k)
						tmp += (*this)(i,k)*other_inv.first(k,j);
					ret(i,j) = tmp;
				}
			}
			return ret;
		}
		Matrix Transpose() const
		{
			Matrix ret(m,n);
			for(enumerator i = 0; i < n; ++i)
			{
				for(enumerator j = 0; j < m; ++j)
				{
					ret(j,i) = (*this)(i,j);
				}
			}
			return ret;
		}
		std::pair<Matrix,bool> Invert(bool print_fail = false) const
		{
			std::pair<Matrix,bool> ret = std::make_pair(Matrix(m,n),true);
			Matrix At = Transpose(); //m by n matrix
			Matrix AtB = At; //m by n matrix
			Matrix AtA = At*(*this); //m by m matrix
			enumerator * order = new enumerator [m];
			for(enumerator i = 0; i < m; ++i) order[i] = i;
			for(enumerator i = 0; i < m; i++)
			{
				enumerator maxk = i, maxq = i, temp2;
				Var max, temp;
				max = fabs(AtA(maxk,maxq));
				//Find best pivot
				for(enumerator k = i; k < m; k++) // over rows
				{
					for(enumerator q = i; q < m; q++) // over columns
					{
						if( fabs(AtA(k,q)) > max )
						{
							max = fabs(AtA(k,q));
							maxk = k;
							maxq = q;
						}
					}
				}
				//Exchange rows
				if( maxk != i )
				{
					for(enumerator q = 0; q < m; q++) // over columns of A
					{
						temp = AtA(maxk,q);
						AtA(maxk,q) = AtA(i,q);
						AtA(i,q) = temp;
					}
					//exchange rhs
					for(enumerator q = 0; q < n; q++) // over columns of B
					{
						temp = AtB(maxk,q);
						AtB(maxk,q) = AtB(i,q);
						AtB(i,q) = temp;
					}
				}
				//Exchange columns
				if( maxq != i )
				{
					for(enumerator k = 0; k < m; k++) //over rows
					{
						temp = AtA(k,maxq);
						AtA(k,maxq) = AtA(k,i);
						AtA(k,i) = temp;
					}
					//remember order in sol
					{
						temp2 = order[maxq];
						order[maxq] = order[i];
						order[i] = temp2;
					}
				}
				
				if( fabs(AtA(i,i)) < 1.0e-54 )
				{
					bool ok = true;
					for(enumerator k = 0; k < n; k++) // over columns of B
					{
						if( fabs(AtB(i,k)/1.0e-54) > 1 )
						{
							ok = false;
							break;
						}
					}
					if( ok ) AtA(i,i) = AtA(i,i) < 0.0 ? - 1.0e-12 : 1.0e-12;
					else
					{
						if( print_fail ) std::cout << "Failed to invert matrix" << std::endl;
						ret.second = false;
						delete [] order;
						return ret;
					}
				}
				for(enumerator k = i+1; k < m; k++)
				{
					AtA(i,k) /= AtA(i,i);
					AtA(k,i) /= AtA(i,i);
				}
				for(enumerator k = i+1; k < m; k++)
					for(enumerator q = i+1; q < m; q++)
					{
						AtA(k,q) -= AtA(k,i) * AtA(i,i) * AtA(i,q);
					}
				for(enumerator k = 0; k < n; k++)
				{
					for(enumerator j = i+1; j < m; j++) //iterate over columns of L
					{
						AtB(j,k) -= AtB(i,k) * AtA(j,i);
					}
					AtB(i,k) /= AtA(i,i);
				}
			}
			
			for(enumerator k = 0; k < n; k++)
			{
				for(enumerator i = m; i-- > 0; ) //iterate over rows of U
					for(enumerator j = i+1; j < m; j++)
					{
						AtB(i,k) -= AtB(j,k) * AtA(i,j);
					}
				for(enumerator i = 0; i < m; i++)
					ret.first(order[i],k) = AtB(i,k);
			}
			delete [] order;
			return ret;
		}
		void Zero()
		{
			for(enumerator i = 0; i < n*m; ++i) space[i] = 0.0;
		}
		Var Trace() const
		{
			assert(n == m);
			Var ret = 0.0;
			for(enumerator i = 0; i < n; ++i) ret += (*this)(i,i);
			return ret;
		}
		Var * data() {return space.data();}
		const Var * data() const {return space.data();}
		enumerator Rows() const {return n;}
		enumerator Cols() const {return m;}
		void Print(INMOST_DATA_REAL_TYPE threshold = 1.0e-10) const
		{
			for(enumerator k = 0; k < n; ++k)
			{
				for(enumerator l = 0; l < m; ++l)
				{
					if( fabs(get_value((*this)(k,l))) > threshold )
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#if defined(USE_AUTODIFF)
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						std::cout << std::setw(10) << get_value((*this)(k,l));
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#else
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					std::cout << std::setw(10) << (*this)(k,l);
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#endif
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					else
						std::cout << std::setw(10) << 0;
					std::cout << " ";
				}
				std::cout << std::endl;
			}
		}
		bool isSymmetric() const
		{
			if( n != m ) return false;
			for(enumerator k = 0; k < n; ++k)
			{
				for(enumerator l = k+1; l < n; ++l)
					if( fabs((*this)(k,l)-(*this)(l,k)) > 1.0e-7 )
						return false;
			}
			return true;
		}
		template<typename typeB>
		typename Promote<Var,typeB>::type DotProduct(const Matrix<typeB> & other) const
		{
			assert(Cols() == other.Cols());
			assert(Rows() == other.Rows());
			typename Promote<Var,typeB>::type ret = 0.0;
			for(enumerator i = 0; i < n; ++i)
				for(enumerator j = 0; j < m; ++j)
					ret += ((*this)(i,j))*other(i,j);
			return ret;
		}
		template<typename typeB>
		typename Promote<Var,typeB>::type operator ^(const Matrix<typeB> & other) const
		{
			return DotProduct(other);
		}
		Var FrobeniusNorm()
		{
			Var ret = 0;
			for(enumerator i = 0; i < n*m; ++i) ret += space[i]*space[i];
			return sqrt(ret);
		}
		/// Convert values in array into matrix.
		/// Depending on
		static Matrix<Var> FromTensor(Var * K, enumerator size, enumerator matsize = 3)
		{
			Matrix<Var> Kc(matsize,matsize);
			if( matsize == 1 )
			{
				assert(size == 1);
				Kc(0,0) = K[0];
			}
			if( matsize == 2 )
			{
				assert(size == 1 || size == 2 || size == 3 || size == 4);
				switch(size)
				{
					case 1: //scalar
						Kc(0,0) = Kc(1,1) = K[0];
						break;
					case 2: //diagonal
						Kc(0,0) = K[0]; // KXX
						Kc(1,1) = K[1]; // KYY
						break;
					case 3: //symmetric
						Kc(0,0) = K[0]; // KXX
						Kc(0,1) = Kc(1,0) = K[1]; //KXY
						Kc(1,1) = K[2]; //KYY
						break;
					case 4: //full
						Kc(0,0) = K[0]; //KXX
						Kc(0,1) = K[1]; //KXY
						Kc(1,0) = K[2]; //KYX
						Kc(1,1) = K[3]; //KYY
						break;
				}
			}
			else if( matsize == 3 )
			{
				assert(size == 1 || size == 3 || size == 6 || size == 9);
				switch(size)
				{
					case 1: //scalar permeability tensor
						Kc(0,0) = Kc(1,1) = Kc(2,2) = K[0];
						break;
					case 3: //diagonal permeability tensor
						Kc(0,0) = K[0]; //KXX
						Kc(1,1) = K[1]; //KYY
						Kc(2,2) = K[2]; //KZZ
						break;
					case 6: //symmetric permeability tensor
						Kc(0,0) = K[0]; //KXX
						Kc(0,1) = Kc(1,0) = K[1]; //KXY
						Kc(0,2) = Kc(2,0) = K[2]; //KXZ
						Kc(1,1) = K[3]; //KYY
						Kc(1,2) = Kc(2,1) = K[4]; //KYZ
						Kc(2,2) = K[5]; //KZZ
						break;
					case 9: //full permeability tensor
						Kc(0,0) = K[0]; //KXX
						Kc(0,1) = K[1]; //KXY
						Kc(0,2) = K[2]; //KXZ
						Kc(1,0) = K[3]; //KYX
						Kc(1,1) = K[4]; //KYY
						Kc(1,2) = K[5]; //KYZ
						Kc(2,0) = K[6]; //KZX
						Kc(2,1) = K[7]; //KZY
						Kc(2,2) = K[8]; //KZZ
						break;
				}
			}
			else if( matsize == 6 )
			{
				assert(size == 1 || size == 6 || size == 21 || size == 36);
				switch(size)
				{
					case 1: //scalar elasticity tensor
						Kc(0,0) = Kc(1,1) = Kc(2,2) = Kc(3,3) = Kc(4,4) = Kc(5,5) = K[0];
						break;
					case 6: //diagonal elasticity tensor
						Kc(0,0) = K[0]; //KXX
						Kc(1,1) = K[1]; //KYY
						Kc(2,2) = K[2]; //KZZ
						break;
					case 21: //symmetric elasticity tensor (note - diagonal first, then off-diagonal rows)
					{
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						Kc(0,0) = K[0]; //c11
						Kc(0,1) = Kc(1,0) = K[1]; //c12
						Kc(0,2) = Kc(2,0) = K[2]; //c13
						Kc(0,3) = Kc(3,0) = K[3]; //c14
						Kc(0,4) = Kc(4,0) = K[4]; //c15
						Kc(0,5) = Kc(5,0) = K[5]; //c16
						Kc(1,1) = K[6]; //c22
						Kc(1,2) = Kc(2,1) = K[7]; //c23
						Kc(1,3) = Kc(3,1) = K[8]; //c24
						Kc(1,4) = Kc(4,1) = K[9]; //c25
						Kc(1,5) = Kc(5,1) = K[10]; //c26
						Kc(2,2) = K[11]; //c33
						Kc(2,3) = Kc(3,2) = K[12]; //c34
						Kc(2,4) = Kc(4,2) = K[13]; //c35
						Kc(2,5) = Kc(5,2) = K[14]; //c36
						Kc(3,3) = K[15]; //c44
						Kc(3,4) = Kc(4,3) = K[16]; //c45
						Kc(3,5) = Kc(5,3) = K[17]; //c46
						Kc(4,4) = K[18]; //c55
						Kc(4,5) = Kc(5,3) = K[19]; //c56
						Kc(5,5) = K[20]; //c66
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						break;
					}
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					case 36: //full elasticity tensor
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						for(int i = 0; i < 6; ++i)
							for(int j = 0; j < 6; ++j)
								Kc(i,j) = K[6*i+j];
						break;
				}
			}
			return Kc;
		}
		static Matrix<Var> FromElasticTensor(Var * K, enumerator size)
		{
			Matrix<Var> Kc(6,6);
			switch(size)
			{
				case 1: //scalar permeability tensor
					Kc(0,0) = Kc(1,1) = Kc(2,2) = Kc(3,3) = Kc(4,4) = Kc(5,5) = K[0];
					break;
				case 6: //diagonal tensor
					Kc.Zero();
					Kc(0,0) = K[0]; //KXX
					Kc(1,1) = K[1]; //KYY
					Kc(2,2) = K[2]; //KZZ
					break;
				case 21: //symmetric tensor
					Kc(0,0) = K[0]; //KXX
					Kc(0,1) = Kc(1,0) = K[1]; //KXY
					Kc(0,2) = Kc(2,0) = K[2]; //KXZ
					Kc(1,1) = K[3]; //KYY
					Kc(1,2) = Kc(2,1) = K[4]; //KYZ
					Kc(2,2) = K[5]; //KZZ
					break;
				case 36: //full tensor
					Kc(0,0) = K[0]; //KXX
					Kc(0,1) = K[1]; //KXY
					Kc(0,2) = K[2]; //KXZ
					Kc(1,0) = K[3]; //KYX
					Kc(1,1) = K[4]; //KYY
					Kc(1,2) = K[5]; //KYZ
					Kc(2,0) = K[6]; //KZX
					Kc(2,1) = K[7]; //KZY
					Kc(2,2) = K[8]; //KZZ
					break;
			}
			return Kc;
		}
		///Retrive vector in matrix form from array
		static Matrix<Var> FromVector(Var * n, enumerator size)
		{
			return Matrix(n,size,1);
		}
		///Create diagonal matrix from array
		static Matrix<Var> FromDiagonal(Var * r, enumerator size)
		{
			Matrix ret(size,size);
			ret.Zero();
			for(enumerator k = 0; k < size; ++k) ret(k,k) = r[k];
			return ret;
		}
		///Create diagonal matrix from array of inversed values
		static Matrix<Var> FromDiagonalInverse(Var * r, enumerator size)
		{
			Matrix ret(size,size);
			ret.Zero();
			for(enumerator k = 0; k < size; ++k) ret(k,k) = 1.0/r[k];
			return ret;
		}
		static Matrix CrossProduct(Var vec[3])
		{
			// |  0  -z   y |
			// |  z   0  -x |
			// | -y   x   0 |
			Matrix ret(3,3);
			ret(0,0) = 0.0;
			ret(0,1) = -vec[2]; //-z
			ret(0,2) = vec[1]; //y
			ret(1,0) = vec[2]; //z
			ret(1,1) = 0;
			ret(1,2) = -vec[0]; //-x
			ret(2,0) = -vec[1]; //-y
			ret(2,1) = vec[0]; //x
			ret(2,2) = 0;
			return ret;
		}
		///Unit matrix
		static Matrix Unit(enumerator pn)
		{
			Matrix ret(pn,pn);
			ret.Zero();
			for(enumerator i = 0; i < pn; ++i) ret(i,i) = 1.0;
			return ret;
		}
		/// Concatenate B matrix as columns of current matrix.
		/// Assumes that number of rows of current matrix is
		/// equal to number of rows of B matrix.
		Matrix ConcatCols(const Matrix & B)
		{
			assert(Rows() == B.Rows());
			Matrix ret(Rows(),Cols()+B.Cols());
			Matrix & A = *this;
			for(enumerator i = 0; i < Rows(); ++i)
			{
				for(enumerator j = 0; j < Cols(); ++j)
					ret(i,j) = A(i,j);
				for(enumerator j = 0; j < B.Cols(); ++j)
					ret(i,j+Cols()) = B(i,j);
			}
			return ret;
		}
		/// Concatenate B matrix as rows of current matrix.
		/// Assumes that number of colums of current matrix is
		/// equal to number of columns of B matrix.
		Matrix ConcatRows(const Matrix & B)
		{
			assert(Cols() == B.Cols());
			Matrix ret(Rows()+B.Rows(),Cols());
			Matrix & A = *this;
			for(enumerator i = 0; i < Rows(); ++i)
			{
				for(enumerator j = 0; j < Cols(); ++j)
					ret(i,j) = A(i,j);
			}
			for(enumerator i = 0; i < B.Rows(); ++i)
			{
				for(enumerator j = 0; j < Cols(); ++j)
					ret(i+Rows(),j) = B(i,j);
			}
			return ret;
		}
		
		/// Joint diagonalization algorithm by Cardoso
		/// http://perso.telecom-paristech.fr/~cardoso/Algo/Joint_Diag/joint_diag_r.m
		/// Current matrix should have size n by n*m
		/// And represent concatination of m n by n matrices
		Matrix JointDiagonalization(INMOST_DATA_REAL_TYPE threshold = 1.0e-7)
		{
			enumerator N = Rows();
			enumerator M = Cols() / Rows();
			Matrix V = Matrix::Unit(m);
			Matrix R(2,M);
			Matrix G(2,2);
			Matrix & A = *this;
			Var ton, toff, theta, c, s, Ap, Aq, Vp, Vq;
			bool repeat;
			do
			{
				repeat = false;
				for(enumerator p = 0; p < N-1; ++p)
				{
					for(enumerator q = p+1; q < N; ++q)
					{
						for(enumerator k = 0; k < M; ++k)
						{
							R(0,k) = A(p,p + k*N) - A(q,q + k*N);
							R(1,k) = A(p,q + k*N) + A(q,p + k*N);
						}
						G = R*R.Transpose();
						Var ton  = G(0,0) - G(1,1);
						Var toff = G(0,1) + G(1,0);
						Var theta = 0.5 * atan2( toff, ton + sqrt(ton*ton + toff*toff) );
						Var c = cos(theta);
						Var s = sin(theta);
						if( fabs(s) > threshold )
						{
							//std::cout << "p,q: " << p << "," << q << " c,s: " << c << "," << s << std::endl;
							repeat = true;
							for(enumerator k = 0; k < M; ++k)
							{
								for(enumerator i = 0; i < N; ++i)
								{
									Ap = A(i,p + k*N);
									Aq = A(i,q + k*N);
									A(i,p + k*N) = Ap*c + Aq*s;
									A(i,q + k*N) = Aq*c - Ap*s;
								}
							}
							for(enumerator k = 0; k < M; ++k)
							{
								for(enumerator j = 0; j < N; ++j)
								{
									Ap = A(p,j + k*N);
									Aq = A(q,j + k*N);
									A(p,j + k*N) = Ap*c + Aq*s;
									A(q,j + k*N) = Aq*c - Ap*s;
								}
							}
							for(enumerator i = 0; i < N; ++i)
							{
								Vp = V(i,p);
								Vq = V(i,q);
								V(i,p) = Vp*c + Vq*s;
								V(i,q) = Vq*c - Vp*s;
							}
						}
					}
				}
				//Print();
			} while( repeat );
			return V;
		}
		Matrix SubMatrix(enumerator ibeg, enumerator jbeg, enumerator iend, enumerator jend)
		{
			Matrix ret(iend-ibeg,jend-jbeg);
			for(enumerator i = ibeg; i < iend; ++i)
			{
				for(enumerator j = jbeg; j < jend; ++j)
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					ret(i-ibeg,j-jbeg) = (*this)(i,j);
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			}
			return ret;
		}
	};
	
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	typedef Matrix<INMOST_DATA_REAL_TYPE> rMatrix; //shortcut for real matrix
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#if defined(USE_AUTODIFF)
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	typedef Matrix<variable> vMatrix; //shortcut for matrix with variations
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	typedef Matrix<hessian_variable> hMatrix; //shortcut for matrix with second variations
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#endif
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}
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template<typename typeB>
INMOST::Matrix<typename INMOST::Promote<INMOST_DATA_REAL_TYPE,typeB>::type> operator *(INMOST_DATA_REAL_TYPE coef, const INMOST::Matrix<typeB> & other)
{return other*coef;}
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#if defined(USE_AUTODIFF)
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template<typename typeB>
INMOST::Matrix<typename INMOST::Promote<INMOST::variable,typeB>::type> operator *(const INMOST::variable & coef, const INMOST::Matrix<typeB> & other)
{return other*coef;}
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#endif


#endif //INMOST_DENSE_INCLUDED