Add internal method _solve_impl_transposed() to LU decomposition classes that solves A^T x = b or A^* x = b.

Eigen/src/LU/FullPivLU.h

@@ -10,7 +10,7 @@
 #ifndef EIGEN_LU_H
 #define EIGEN_LU_H
 
-namespace Eigen { 
+namespace Eigen {
 
 namespace internal {
 template<typename _MatrixType> struct traits<FullPivLU<_MatrixType> >
@@ -384,22 +384,26 @@ template<typename _MatrixType> class FullPivLU
 
     inline Index rows() const { return m_lu.rows(); }
     inline Index cols() const { return m_lu.cols(); }
-    
+
     #ifndef EIGEN_PARSED_BY_DOXYGEN
     template<typename RhsType, typename DstType>
     EIGEN_DEVICE_FUNC
     void _solve_impl(const RhsType &rhs, DstType &dst) const;
+
+    template<bool Conjugate, typename RhsType, typename DstType>
+    EIGEN_DEVICE_FUNC
+    void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const;
     #endif
 
   protected:
-    
+
     static void check_template_parameters()
     {
       EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
     }
-    
+
     void computeInPlace();
-    
+
     MatrixType m_lu;
     PermutationPType m_p;
     PermutationQType m_q;
@@ -447,15 +451,15 @@ template<typename InputType>
 FullPivLU<MatrixType>& FullPivLU<MatrixType>::compute(const EigenBase<InputType>& matrix)
 {
   check_template_parameters();
-  
+
   // the permutations are stored as int indices, so just to be sure:
   eigen_assert(matrix.rows()<=NumTraits<int>::highest() && matrix.cols()<=NumTraits<int>::highest());
-  
+
   m_isInitialized = true;
   m_lu = matrix.derived();
-  
+
   computeInPlace();
-  
+
   return *this;
 }
 
@@ -709,7 +713,7 @@ struct image_retval<FullPivLU<_MatrixType> >
 template<typename _MatrixType>
 template<typename RhsType, typename DstType>
 void FullPivLU<_MatrixType>::_solve_impl(const RhsType &rhs, DstType &dst) const
-{  
+{
   /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}.
   * So we proceed as follows:
   * Step 1: compute c = P * rhs.
@@ -753,6 +757,70 @@ void FullPivLU<_MatrixType>::_solve_impl(const RhsType &rhs, DstType &dst) const
   for(Index i = nonzero_pivots; i < m_lu.cols(); ++i)
     dst.row(permutationQ().indices().coeff(i)).setZero();
 }
+
+template<typename _MatrixType>
+template<bool Conjugate, typename RhsType, typename DstType>
+void FullPivLU<_MatrixType>::_solve_impl_transposed(const RhsType &rhs, DstType &dst) const
+{
+  /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1},
+   * and since permutations are real and unitary, we can write this
+   * as   A^T = Q U^T L^T P,
+   * So we proceed as follows:
+   * Step 1: compute c = Q^T rhs.
+   * Step 2: replace c by the solution x to U^T x = c. May or may not exist.
+   * Step 3: replace c by the solution x to L^T x = c.
+   * Step 4: result = P^T c.
+   * If Conjugate is true, replace "^T" by "^*" above.
+   */
+
+  const Index rows = this->rows(), cols = this->cols(),
+    nonzero_pivots = this->rank();
+   eigen_assert(rhs.rows() == cols);
+  const Index smalldim = (std::min)(rows, cols);
+
+  if(nonzero_pivots == 0)
+  {
+    dst.setZero();
+    return;
+  }
+
+  typename RhsType::PlainObject c(rhs.rows(), rhs.cols());
+
+  // Step 1
+  c = permutationQ().inverse() * rhs;
+
+  if (Conjugate) {
+    // Step 2
+    m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots)
+        .template triangularView<Upper>()
+        .adjoint()
+        .solveInPlace(c.topRows(nonzero_pivots));
+    // Step 3
+    m_lu.topLeftCorner(smalldim, smalldim)
+        .template triangularView<UnitLower>()
+        .adjoint()
+        .solveInPlace(c.topRows(smalldim));
+  } else {
+    // Step 2
+    m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots)
+        .template triangularView<Upper>()
+        .transpose()
+        .solveInPlace(c.topRows(nonzero_pivots));
+    // Step 3
+    m_lu.topLeftCorner(smalldim, smalldim)
+        .template triangularView<UnitLower>()
+        .transpose()
+        .solveInPlace(c.topRows(smalldim));
+  }
+
+  // Step 4
+  PermutationPType invp = permutationP().inverse().eval();
+  for(Index i = 0; i < smalldim; ++i)
+    dst.row(invp.indices().coeff(i)) = c.row(i);
+  for(Index i = smalldim; i < rows; ++i)
+    dst.row(invp.indices().coeff(i)).setZero();
+}
+
 #endif
 
 namespace internal {
@@ -765,7 +833,7 @@ struct Assignment<DstXprType, Inverse<FullPivLU<MatrixType> >, internal::assign_
   typedef FullPivLU<MatrixType> LuType;
   typedef Inverse<LuType> SrcXprType;
   static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<Scalar> &)
-  {    
+  {
     dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
   }
 };

Eigen/src/LU/PartialPivLU.h

@@ -11,7 +11,7 @@
 #ifndef EIGEN_PARTIALLU_H
 #define EIGEN_PARTIALLU_H
 
-namespace Eigen { 
+namespace Eigen {
 
 namespace internal {
 template<typename _MatrixType> struct traits<PartialPivLU<_MatrixType> >
@@ -185,7 +185,7 @@ template<typename _MatrixType> class PartialPivLU
 
     inline Index rows() const { return m_lu.rows(); }
     inline Index cols() const { return m_lu.cols(); }
-    
+
     #ifndef EIGEN_PARSED_BY_DOXYGEN
     template<typename RhsType, typename DstType>
     EIGEN_DEVICE_FUNC
@@ -206,17 +206,44 @@ template<typename _MatrixType> class PartialPivLU
       m_lu.template triangularView<UnitLower>().solveInPlace(dst);
 
       // Step 3
-      m_lu.template triangularView<Upper>().solveInPlace(dst); 
+      m_lu.template triangularView<Upper>().solveInPlace(dst);
+    }
+
+    template<bool Conjugate, typename RhsType, typename DstType>
+    EIGEN_DEVICE_FUNC
+    void _solve_impl_transposed(const RhsType &rhs, DstType &dst) 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.
+      * Step 3: replace c by the solution x to Ux = c.
+      */
+
+      eigen_assert(rhs.rows() == m_lu.cols());
+
+      if (Conjugate) {
+        // Step 1
+        dst = m_lu.template triangularView<Upper>().adjoint().solve(rhs);
+        // Step 2
+        m_lu.template triangularView<UnitLower>().adjoint().solveInPlace(dst);
+      } else {
+        // Step 1
+        dst = m_lu.template triangularView<Upper>().transpose().solve(rhs);
+        // Step 2
+        m_lu.template triangularView<UnitLower>().transpose().solveInPlace(dst);
+      }
+      // Step 3
+      dst = permutationP().transpose() * dst;
     }
     #endif
 
   protected:
-    
+
     static void check_template_parameters()
     {
       EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
     }
-    
+
     MatrixType m_lu;
     PermutationType m_p;
     TranspositionType m_rowsTranspositions;
@@ -295,7 +322,7 @@ struct partial_lu_impl
     {
       Index rrows = rows-k-1;
       Index rcols = cols-k-1;
-        
+
       Index row_of_biggest_in_col;
       Score biggest_in_corner
         = lu.col(k).tail(rows-k).unaryExpr(Scoring()).maxCoeff(&row_of_biggest_in_col);
@@ -436,10 +463,10 @@ template<typename InputType>
 PartialPivLU<MatrixType>& PartialPivLU<MatrixType>::compute(const EigenBase<InputType>& matrix)
 {
   check_template_parameters();
-  
+
   // the row permutation is stored as int indices, so just to be sure:
   eigen_assert(matrix.rows()<NumTraits<int>::highest());
-  
+
   m_lu = matrix.derived();
 
   eigen_assert(matrix.rows() == matrix.cols() && "PartialPivLU is only for square (and moreover invertible) matrices");
@@ -492,7 +519,7 @@ struct Assignment<DstXprType, Inverse<PartialPivLU<MatrixType> >, internal::assi
   typedef PartialPivLU<MatrixType> LuType;
   typedef Inverse<LuType> SrcXprType;
   static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<Scalar> &)
-  {    
+  {
     dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
   }
 };

test/lu.cpp

@@ -92,6 +92,20 @@ template<typename MatrixType> void lu_non_invertible()
   // test that the code, which does resize(), may be applied to an xpr
   m2.block(0,0,m2.rows(),m2.cols()) = lu.solve(m3);
   VERIFY_IS_APPROX(m3, m1*m2);
+
+  // test solve with transposed
+  m3 = MatrixType::Random(rows,cols2);
+  m2 = m1.transpose()*m3;
+  m3 = MatrixType::Random(rows,cols2);
+  lu.template _solve_impl_transposed<false>(m2, m3);
+  VERIFY_IS_APPROX(m2, m1.transpose()*m3);
+
+  // test solve with conjugate transposed
+  m3 = MatrixType::Random(rows,cols2);
+  m2 = m1.adjoint()*m3;
+  m3 = MatrixType::Random(rows,cols2);
+  lu.template _solve_impl_transposed<true>(m2, m3);
+  VERIFY_IS_APPROX(m2, m1.adjoint()*m3);
 }
 
 template<typename MatrixType> void lu_invertible()
@@ -124,6 +138,12 @@ template<typename MatrixType> void lu_invertible()
   m2 = lu.solve(m3);
   VERIFY_IS_APPROX(m3, m1*m2);
   VERIFY_IS_APPROX(m2, lu.inverse()*m3);
+  // test solve with transposed
+  lu.template _solve_impl_transposed<false>(m3, m2);
+  VERIFY_IS_APPROX(m3, m1.transpose()*m2);
+  // test solve with conjugate transposed
+  lu.template _solve_impl_transposed<true>(m3, m2);
+  VERIFY_IS_APPROX(m3, m1.adjoint()*m2);
 
   // Regression test for Bug 302
   MatrixType m4 = MatrixType::Random(size,size);
@@ -136,14 +156,24 @@ template<typename MatrixType> void lu_partial_piv()
      PartialPivLU.h
   */
   typedef typename MatrixType::Index Index;
-  Index rows = internal::random<Index>(1,4);
+  Index size = internal::random<Index>(1,4);
-  Index cols = rows;
 
-  MatrixType m1(cols, rows);
+  MatrixType m1(size, size), m2(size, size), m3(size, size);
   m1.setRandom();
   PartialPivLU<MatrixType> plu(m1);
 
   VERIFY_IS_APPROX(m1, plu.reconstructedMatrix());
+
+  m3 = MatrixType::Random(size,size);
+  m2 = plu.solve(m3);
+  VERIFY_IS_APPROX(m3, m1*m2);
+  VERIFY_IS_APPROX(m2, plu.inverse()*m3);
+  // test solve with transposed
+  plu.template _solve_impl_transposed<false>(m3, m2);
+  VERIFY_IS_APPROX(m3, m1.transpose()*m2);
+  // test solve with conjugate transposed
+  plu.template _solve_impl_transposed<true>(m3, m2);
+  VERIFY_IS_APPROX(m3, m1.adjoint()*m2);
 }
 
 template<typename MatrixType> void lu_verify_assert()