jama_lu.h

////////////////////////////////////////////////////////////////////////////////
//                         HYPERJAMA LU DECOMPOSITION                         //
//                         By Alex Christensen                                //
//                         based on NIST's jama_lu.h                          //
//                         http://math.nist.gov/tnt/                          //
////////////////////////////////////////////////////////////////////////////////
//                                                                            //
// Copyright (C) 2012 Alex Christensen                                        //
//                                                                            //
// Permission is hereby granted, free of charge, to any person obtaining a    //
// copy of this software and associated documentation files (the "Software"), //
// to deal in the Software without restriction, including without limitation  //
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// and/or sell copies of the Software, and to permit persons to whom the      //
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//                                                                            //
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//                                                                            //
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// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,   //
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// DEALINGS IN THE SOFTWARE.                                                  //
//                                                                            //
////////////////////////////////////////////////////////////////////////////////
//                                                                            //
// Changes:                                                                   //
//                                                                            //
// include tnt_math_utils.h <cmath> and <omp.h>                               //
// constructor optimized                                                      //
// changed constants to template constructors in getL, getU, and isNonsingular//
//                                                                            //
////////////////////////////////////////////////////////////////////////////////

#ifndef JAMA_LU_H
#define JAMA_LU_H

#include "tnt.h"
#include <algorithm>
//for min(), max() below

#include "tnt_math_utils.h"
#include <omp.h>
#include <cmath>

using namespace TNT;
using namespace std;

namespace JAMA
{

   /** LU Decomposition.
   <P>
   For an m-by-n matrix A with m >= n, the LU decomposition is an m-by-n
   unit lower triangular matrix L, an n-by-n upper triangular matrix U,
   and a permutation vector piv of length m so that A(piv,:) = L*U.
   If m < n, then L is m-by-m and U is m-by-n.
   <P>
   The LU decompostion with pivoting always exists, even if the matrix is
   singular, so the constructor will never fail.  The primary use of the
   LU decomposition is in the solution of square systems of simultaneous
   linear equations.  This will fail if isNonsingular() returns false.
   */
template <class Real>
class LU
{



   /* Array for internal storage of decomposition.  */
   Array2D<Real>  LU_;
   int m, n, pivsign; 
   Array1D<int> piv;


   Array2D<Real> permute_copy(const Array2D<Real> &A, 
   			const Array1D<int> &piv, int j0, int j1)
	{
		int piv_length = piv.dim();

		Array2D<Real> X(piv_length, j1-j0+1);


         for (int i = 0; i < piv_length; i++) 
            for (int j = j0; j <= j1; j++) 
               X[i][j-j0] = A[piv[i]][j];

		return X;
	}

   Array1D<Real> permute_copy(const Array1D<Real> &A, 
   		const Array1D<int> &piv)
	{
		int piv_length = piv.dim();
		if (piv_length != A.dim())
			return Array1D<Real>();

		Array1D<Real> x(piv_length);


         for (int i = 0; i < piv_length; i++) 
               x[i] = A[piv[i]];

		return x;
	}


	public :

   /** LU Decomposition
   @param  A   Rectangular matrix
   @return     LU Decomposition object to access L, U and piv.
   */

    LU (const Array2D<Real> &A) : LU_(A.copy()), m(A.dim1()), n(A.dim2()), 
		piv(A.dim1())
	
	{

		// Use a parallel "left-looking", row-operation-based, block Crout/Doolittle algorithm.
		for (int i = 0; i < m; i++)
			piv[i] = i;
		pivsign = 1;
		Real *LUrowi = 0;
		Array1D<Real> LUcolj(m);

		// Outer loop.

		const int cpuCacheLineSize=64;
		const int maxBlockSize=cpuCacheLineSize/sizeof(Real);

		int blockSize;
		for(int j = 0; j < n; j+=blockSize){
			blockSize=min(n-j,maxBlockSize);

			//do Gaussian elimination with partial pivoting in the block
			//(in the first few columns of the matrix)
			for(int bi=0;bi<blockSize;bi++){

				// Find pivot and exchange if necessary.
				int p = j+bi;
				for (int i = j+bi+1; i < m; i++)
					if (abs(LU_[i][j+bi]) > abs(LU_[p][j+bi]))
						p = i;

				if (p != j+bi) {
					for (int k = 0; k < n; k++) {
						Real t = LU_[p][k]; 
						LU_[p][k] = LU_[j+bi][k]; 
						LU_[j+bi][k] = t;
					}

					//swap pivot vector
					int k = piv[p]; 
					piv[p] = piv[j+bi]; 
					piv[j+bi] = k;
					pivsign = -pivsign;
				}

				//eliminate below the diagonal in the block
				Real elimVal=LU_[j+bi][j+bi];
				if(elimVal==Real(0)){
					//apply previous transformations from the block to the top of the submatrix
					for(int row=j+1;row<j+bi;row++)
						for(int bi_=0;bi_<row-j;bi_++)
							if(LU_[row][j+bi_]!=Real(0))
								_axpy(&(LU_[j+bi_][j+blockSize]),&(LU_[row][j+blockSize]),n-j-blockSize,-LU_[row][j+bi_]);
					return;
				}
				for(int row=j+bi+1;row<m;row++){
					Real multiplier=LU_[row][j+bi]/elimVal;
					for(int col=j+bi;col<j+blockSize;col++)
						LU_[row][col]-=multiplier*LU_[j+bi][col];
					LU_[row][j+bi]=multiplier;
				}
			}

			//at this point, the matrix looks something like this (if block size were 4)
			//
			//U U U U * * * *
			//L U U U * * * *
			//L L U U * * * *
			//L L L U * * * *
			//L L L L * * * *
			//L L L L * * * *
			//L L L L * * * *
			//L L L L * * * *

			//apply previous transformations from the block to the top of the submatrix
			for(int row=j+1;row<j+blockSize;row++)
				for(int bi=0;bi<row-j;bi++)
					if(LU_[row][j+bi]!=Real(0))
						_axpy(&(LU_[j+bi][j+blockSize]),&(LU_[row][j+blockSize]),n-j-blockSize,-LU_[row][j+bi]);

			//at this point, the matrix looks something like this (if block size were 4)
			//
			//U U U U U U U U
			//L U U U U U U U
			//L L U U U U U U
			//L L L U U U U U
			//L L L L * * * *
			//L L L L * * * *
			//L L L L * * * *
			//L L L L * * * *

			//apply previous transformations from the block in parallel to the rest of the submatrix
#pragma omp parallel for
			for(int row=j+blockSize;row<m;row++)
				for(int bi=0;bi<blockSize;bi++)
					if(LU_[row][j+bi]!=Real(0))
						_axpy(&(LU_[j+bi][j+blockSize]),&(LU_[row][j+blockSize]),n-j-blockSize,-LU_[row][j+bi]);
		}
   }


   /** Is the matrix nonsingular?
   @return     1 (true)  if upper triangular factor U (and hence A) 
   				is nonsingular, 0 otherwise.
   */

   int isNonsingular () {
      for (int j = 0; j < n; j++) {
         if (LU_[j][j] == Real(0))
            return 0;
      }
      return 1;
   }

   /** Return lower triangular factor
   @return     L
   */

   Array2D<Real> getL () {
      Array2D<Real> L_(m,n);
      for (int i = 0; i < m; i++) {
         for (int j = 0; j < n; j++) {
            if (i > j) {
               L_[i][j] = LU_[i][j];
            } else if (i == j) {
               L_[i][j] = Real(1);
            } else {
               L_[i][j] = Real(0);
            }
         }
      }
      return L_;
   }

   /** Return upper triangular factor
   @return     U portion of LU factorization.
   */

   Array2D<Real> getU () {
   	  Array2D<Real> U_(n,n);
      for (int i = 0; i < n; i++) {
         for (int j = 0; j < n; j++) {
            if (i <= j) {
               U_[i][j] = LU_[i][j];
            } else {
               U_[i][j] = Real(0);
            }
         }
      }
      return U_;
   }

   /** Return pivot permutation vector
   @return     piv
   */

   Array1D<int> getPivot () {
      return piv;
   }


   /** Compute determinant using LU factors.
   @return     determinant of A, or 0 if A is not square.
   */

   Real det () {
      if (m != n) {
         return Real(0);
      }
      Real d = Real(pivsign);
      for (int j = 0; j < n; j++) {
         d *= LU_[j][j];
      }
      return d;
   }

   /** Solve A*X = B
   @param  B   A Matrix with as many rows as A and any number of columns.
   @return     X so that L*U*X = B(piv,:), if B is nonconformant, returns
   					0x0 (null) array.
   */

   Array2D<Real> solve (const Array2D<Real> &B) 
   {

	  /* Dimensions: A is mxn, X is nxk, B is mxk */
      
      if (B.dim1() != m) {
	  	return Array2D<Real>(0,0);
      }
      if (!isNonsingular()) {
        return Array2D<Real>(0,0);
      }

      // Copy right hand side with pivoting
      int nx = B.dim2();


	  Array2D<Real> X = permute_copy(B, piv, 0, nx-1);

      // Solve L*Y = B(piv,:)
      for (int k = 0; k < n; k++) {
         for (int i = k+1; i < n; i++) {
            for (int j = 0; j < nx; j++) {
               X[i][j] -= X[k][j]*LU_[i][k];
            }
         }
      }
      // Solve U*X = Y;
      for (int k = n-1; k >= 0; k--) {
         for (int j = 0; j < nx; j++) {
            X[k][j] /= LU_[k][k];
         }
         for (int i = 0; i < k; i++) {
            for (int j = 0; j < nx; j++) {
               X[i][j] -= X[k][j]*LU_[i][k];
            }
         }
      }
      return X;
   }


   /** Solve A*x = b, where x and b are vectors of length equal	
   		to the number of rows in A.

   @param  b   a vector (Array1D> of length equal to the first dimension
   						of A.
   @return x a vector (Array1D> so that L*U*x = b(piv), if B is nonconformant,
   					returns 0x0 (null) array.
   */

   Array1D<Real> solve (const Array1D<Real> &b) 
   {

	  /* Dimensions: A is mxn, X is nxk, B is mxk */
      
      if (b.dim1() != m) {
	  	return Array1D<Real>();
      }
      if (!isNonsingular()) {
        return Array1D<Real>();
      }


	  Array1D<Real> x = permute_copy(b, piv);

      // Solve L*Y = B(piv)
      for (int k = 0; k < n; k++) {
         for (int i = k+1; i < n; i++) {
               x[i] -= x[k]*LU_[i][k];
            }
         }
      
	  // Solve U*X = Y;
      for (int k = n-1; k >= 0; k--) {
            x[k] /= LU_[k][k];
      		for (int i = 0; i < k; i++) 
            	x[i] -= x[k]*LU_[i][k];
      }
     

      return x;
   }

}; /* class LU */

} /* namespace JAMA */

#endif
/* JAMA_LU_H */