add partial-pivoting LU decomposition

the name 'PartialLU' is not meant to be definitive!
make inverse() and determinant() use it, so it's *almost* considered
well tested.

Eigen/LU

@@ -19,6 +19,7 @@ namespace Eigen {
   */
 
 #include "src/LU/LU.h"
+#include "src/LU/PartialLU.h"
 #include "src/LU/Determinant.h"
 #include "src/LU/Inverse.h"
 

Eigen/src/Core/MatrixBase.h

@@ -623,6 +623,7 @@ template<typename Derived> class MatrixBase
 /////////// LU module ///////////
 
     const LU<PlainMatrixType> lu() const;
+    const PartialLU<PlainMatrixType> partialLu() const;    
     const PlainMatrixType inverse() const;
     void computeInverse(PlainMatrixType *result) const;
     Scalar determinant() const;

Eigen/src/Core/util/ForwardDeclarations.h

@@ -109,6 +109,7 @@ template<typename MatrixType,int RowFactor,int ColFactor> class Replicate;
 template<typename MatrixType, int Direction = BothDirections> class Reverse;
 
 template<typename MatrixType> class LU;
+template<typename MatrixType> class PartialLU;
 template<typename MatrixType> class QR;
 template<typename MatrixType> class SVD;
 template<typename MatrixType> class LLT;

Eigen/src/LU/Determinant.h

@@ -51,7 +51,7 @@ template<typename Derived,
 {
   static inline typename ei_traits<Derived>::Scalar run(const Derived& m)
   {
-    return m.lu().determinant();
+    return m.partialLu().determinant();
   }
 };
 

Eigen/src/LU/Inverse.h

@@ -170,8 +170,7 @@ struct ei_compute_inverse
 {
   static inline void run(const MatrixType& matrix, MatrixType* result)
   {
-    LU<MatrixType> lu(matrix);
-    lu.computeInverse(result);
+    matrix.partialLu().computeInverse(result);
   }
 };
 

Eigen/src/LU/PartialLU.h

@@ -0,0 +1,262 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra. Eigen itself is part of the KDE project.
+//
+// Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_PARTIALLU_H
+#define EIGEN_PARTIALLU_H
+
+/** \ingroup LU_Module
+  *
+  * \class PartialLU
+  *
+  * \brief LU decomposition of a matrix with partial pivoting, and related features
+  *
+  * \param MatrixType the type of the matrix of which we are computing the LU decomposition
+  *
+  * This class represents a LU decomposition of a \b square \b invertible matrix, with partial pivoting: the matrix A
+  * is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P
+  * is a permutation matrix.
+  *
+  * Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible matrices.
+  * So in this class, we plainly require that and take advantage of that to do some simplifications and optimizations.
+  * This class will assert that the matrix is square, but it won't (actually it can't) check that the matrix is invertible:
+  * it is your task to check that you only use this decomposition on invertible matrices.
+  *
+  * The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided by class LU.
+  *
+  * This is \b not a rank-revealing LU decomposition. Many features are intentionally absent from this class,
+  * such as rank computation. If you need these features, use class LU.
+  *
+  * This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses. On the other hand,
+  * it is \b not suitable to determine whether a given matrix is invertible.
+  *
+  * The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP().
+  *
+  * \sa MatrixBase::partialLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class LU
+  */
+template<typename MatrixType> class PartialLU
+{
+  public:
+
+    typedef typename MatrixType::Scalar Scalar;
+    typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
+    typedef Matrix<int, 1, MatrixType::ColsAtCompileTime> IntRowVectorType;
+    typedef Matrix<int, MatrixType::RowsAtCompileTime, 1> IntColVectorType;
+    typedef Matrix<Scalar, 1, MatrixType::ColsAtCompileTime> RowVectorType;
+    typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> ColVectorType;
+
+    enum { MaxSmallDimAtCompileTime = EIGEN_ENUM_MIN(
+             MatrixType::MaxColsAtCompileTime,
+             MatrixType::MaxRowsAtCompileTime)
+    };
+
+    /** Constructor.
+      *
+      * \param matrix the matrix of which to compute the LU decomposition.
+      *
+      * \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
+      * If you need to deal with non-full rank, use class LU instead.
+      */
+    PartialLU(const MatrixType& matrix);
+
+    /** \returns the LU decomposition matrix: the upper-triangular part is U, the
+      * unit-lower-triangular part is L (at least for square matrices; in the non-square
+      * case, special care is needed, see the documentation of class LU).
+      *
+      * \sa matrixL(), matrixU()
+      */
+    inline const MatrixType& matrixLU() const
+    {
+      return m_lu;
+    }
+
+    /** \returns a vector of integers, whose size is the number of rows of the matrix being decomposed,
+      * representing the P permutation i.e. the permutation of the rows. For its precise meaning,
+      * see the examples given in the documentation of class LU.
+      */
+    inline const IntColVectorType& permutationP() const
+    {
+      return m_p;
+    }
+
+    /** This method finds the solution x to the equation Ax=b, where A is the matrix of which
+      * *this is the LU decomposition. Since if this partial pivoting decomposition the matrix is assumed
+      * to have full rank, such a solution is assumed to exist and to be unique.
+      *
+      * \warning Again, if your matrix may not have full rank, use class LU instead. See LU::solve().
+      *
+      * \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
+      *          the only requirement in order for the equation to make sense is that
+      *          b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
+      * \param result a pointer to the vector or matrix in which to store the solution, if any exists.
+      *          Resized if necessary, so that result->rows()==A.cols() and result->cols()==b.cols().
+      *          If no solution exists, *result is left with undefined coefficients.
+      *
+      * Example: \include PartialLU_solve.cpp
+      * Output: \verbinclude PartialLU_solve.out
+      *
+      * \sa MatrixBase::solveTriangular(), inverse(), computeInverse()
+      */
+    template<typename OtherDerived, typename ResultType>
+    void solve(const MatrixBase<OtherDerived>& b, ResultType *result) const;
+
+    /** \returns the determinant of the matrix of which
+      * *this is the LU decomposition. It has only linear complexity
+      * (that is, O(n) where n is the dimension of the square matrix)
+      * as the LU decomposition has already been computed.
+      *
+      * \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
+      *       optimized paths.
+      *
+      * \warning a determinant can be very big or small, so for matrices
+      * of large enough dimension, there is a risk of overflow/underflow.
+      *
+      * \sa MatrixBase::determinant()
+      */
+    typename ei_traits<MatrixType>::Scalar determinant() const;
+
+    /** Computes the inverse of the matrix of which *this is the LU decomposition.
+      *
+      * \param result a pointer to the matrix into which to store the inverse. Resized if needed.
+      *
+      * \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
+      *          invertibility, use class LU instead.
+      *
+      * \sa MatrixBase::computeInverse(), inverse()
+      */
+    inline void computeInverse(MatrixType *result) const
+    {
+      solve(MatrixType::Identity(m_lu.rows(), m_lu.cols()), result);
+    }
+
+    /** \returns the inverse of the matrix of which *this is the LU decomposition.
+      *
+      * \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
+      *          invertibility, use class LU instead.
+      *
+      * \sa computeInverse(), MatrixBase::inverse()
+      */
+    inline MatrixType inverse() const
+    {
+      MatrixType result;
+      computeInverse(&result);
+      return result;
+    }
+
+  protected:
+    const MatrixType& m_originalMatrix;
+    MatrixType m_lu;
+    IntColVectorType m_p;
+    int m_det_p;
+};
+
+template<typename MatrixType>
+PartialLU<MatrixType>::PartialLU(const MatrixType& matrix)
+  : m_originalMatrix(matrix),
+    m_lu(matrix),
+    m_p(matrix.rows())
+{
+  ei_assert(matrix.rows() == matrix.cols() && "PartialLU is only for square (and moreover invertible) matrices");
+  const int size = matrix.rows();
+
+  IntColVectorType rows_transpositions(size);
+  int number_of_transpositions = 0;
+
+  for(int k = 0; k < size; ++k)
+  {
+    int row_of_biggest_in_col;
+    m_lu.block(k,k,size-k,1).cwise().abs().maxCoeff(&row_of_biggest_in_col);
+    row_of_biggest_in_col += k;
+
+    rows_transpositions.coeffRef(k) = row_of_biggest_in_col;
+
+    if(k != row_of_biggest_in_col) {
+      m_lu.row(k).swap(m_lu.row(row_of_biggest_in_col));
+      ++number_of_transpositions;
+    }
+
+    if(k<size-1) {
+      m_lu.col(k).end(size-k-1) /= m_lu.coeff(k,k);
+      for(int col = k + 1; col < size; ++col)
+        m_lu.col(col).end(size-k-1) -= m_lu.col(k).end(size-k-1) * m_lu.coeff(k,col); 
+    }
+  }
+
+  for(int k = 0; k < size; ++k) m_p.coeffRef(k) = k;
+  for(int k = size-1; k >= 0; --k)
+    std::swap(m_p.coeffRef(k), m_p.coeffRef(rows_transpositions.coeff(k)));
+
+  m_det_p = (number_of_transpositions%2) ? -1 : 1;
+}
+
+template<typename MatrixType>
+typename ei_traits<MatrixType>::Scalar PartialLU<MatrixType>::determinant() const
+{
+  return Scalar(m_det_p) * m_lu.diagonal().prod();
+}
+
+template<typename MatrixType>
+template<typename OtherDerived, typename ResultType>
+void PartialLU<MatrixType>::solve(
+  const MatrixBase<OtherDerived>& b,
+  ResultType *result
+) const
+{
+  /* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
+   * So we proceed as follows:
+   * Step 1: compute c = Pb.
+   * Step 2: replace c by the solution x to Lx = c. Exists because L is invertible.
+   * Step 3: replace c by the solution x to Ux = c. Check if a solution really exists.
+   */
+
+  const int size = m_lu.rows();
+  ei_assert(b.rows() == size);
+
+  result->resize(size, b.cols());
+
+  // Step 1
+  for(int i = 0; i < size; ++i) result->row(m_p.coeff(i)) = b.row(i);
+
+  // Step 2
+  m_lu.template marked<UnitLowerTriangular>()
+      .solveTriangularInPlace(*result);
+
+  // Step 3
+  m_lu.template marked<UpperTriangular>()
+      .solveTriangularInPlace(*result);
+}
+
+/** \lu_module
+  *
+  * \return the LU decomposition of \c *this.
+  *
+  * \sa class LU
+  */
+template<typename Derived>
+inline const PartialLU<typename MatrixBase<Derived>::PlainMatrixType>
+MatrixBase<Derived>::partialLu() const
+{
+  return PartialLU<PlainMatrixType>(eval());
+}
+
+#endif // EIGEN_PARTIALLU_H