Add condition estimation to Cholesky (LLT) factorization.

Eigen/src/Cholesky/LLT.h

@@ -135,6 +135,16 @@ template<typename _MatrixType, int _UpLo> class LLT
     template<typename InputType>
     LLT& compute(const EigenBase<InputType>& matrix);
 
+    /** \returns an estimate of the reciprocal condition number of the matrix of
+     *  which *this is the Cholesky decomposition.
+     */
+    RealScalar rcond() const
+    {
+      eigen_assert(m_isInitialized && "LLT is not initialized.");
+      eigen_assert(m_info == Success && "LLT failed because matrix appears to be negative");
+      return ConditionEstimator<LLT<MatrixType, UpLo>, true >::rcond(m_l1_norm, *this);
+    }
+
     /** \returns the LLT decomposition matrix
       *
       * TODO: document the storage layout
@@ -183,6 +193,7 @@ template<typename _MatrixType, int _UpLo> class LLT
       * The strict upper part is not used and even not initialized.
       */
     MatrixType m_matrix;
+    RealScalar m_l1_norm;
     bool m_isInitialized;
     ComputationInfo m_info;
 };
@@ -393,6 +404,26 @@ LLT<MatrixType,_UpLo>& LLT<MatrixType,_UpLo>::compute(const EigenBase<InputType>
   m_matrix.resize(size, size);
   m_matrix = a.derived();
 
+  // Compute matrix L1 norm = max abs column sum.
+  m_l1_norm = RealScalar(0);
+  if (_UpLo == Lower) {
+    for (int col = 0; col < size; ++col) {
+      const RealScalar abs_col_sum = m_matrix.col(col).tail(size - col).cwiseAbs().sum() +
+          m_matrix.row(col).tail(col).cwiseAbs().sum();
+      if (abs_col_sum > m_l1_norm) {
+        m_l1_norm = abs_col_sum;
+      }
+    }
+  } else {
+    for (int col = 0; col < a.cols(); ++col) {
+      const RealScalar abs_col_sum = m_matrix.col(col).tail(col).cwiseAbs().sum() +
+          m_matrix.row(col).tail(size - col).cwiseAbs().sum();
+      if (abs_col_sum > m_l1_norm) {
+        m_l1_norm = abs_col_sum;
+      }
+    }
+  }
+
   m_isInitialized = true;
   bool ok = Traits::inplace_decomposition(m_matrix);
   m_info = ok ? Success : NumericalIssue;

Eigen/src/Core/ConditionEstimator.h

@@ -13,11 +13,11 @@
 namespace Eigen {
 
 namespace internal {
-template <typename Decomposition, bool IsComplex>
+template <typename Decomposition, bool IsSelfAdjoint, bool IsComplex>
 struct EstimateInverseMatrixL1NormImpl {};
 }  // namespace internal
 
-template <typename Decomposition>
+template <typename Decomposition, bool IsSelfAdjoint = false>
 class ConditionEstimator {
  public:
   typedef typename Decomposition::MatrixType MatrixType;
@@ -101,7 +101,8 @@ class ConditionEstimator {
       return 0;
     }
     return internal::EstimateInverseMatrixL1NormImpl<
-        Decomposition, NumTraits<Scalar>::IsComplex>::compute(dec);
+        Decomposition, IsSelfAdjoint,
+        NumTraits<Scalar>::IsComplex != 0>::compute(dec);
   }
 
   /**
@@ -116,9 +117,27 @@ class ConditionEstimator {
 
 namespace internal {
 
+template <typename Decomposition, typename Vector, bool IsSelfAdjoint = false>
+struct solve_helper {
+  static inline Vector solve_adjoint(const Decomposition& dec,
+                                     const Vector& v) {
+    return dec.adjoint().solve(v);
+  }
+};
+
+// Partial specialization for self_adjoint matrices.
+template <typename Decomposition, typename Vector>
+struct solve_helper<Decomposition, Vector, true> {
+  static inline Vector solve_adjoint(const Decomposition& dec,
+                                     const Vector& v) {
+    return dec.solve(v);
+  }
+};
+
+
 // Partial specialization for real matrices.
-template <typename Decomposition>
+template <typename Decomposition, bool IsSelfAdjoint>
-struct EstimateInverseMatrixL1NormImpl<Decomposition, 0> {
+struct EstimateInverseMatrixL1NormImpl<Decomposition, IsSelfAdjoint, false> {
   typedef typename Decomposition::MatrixType MatrixType;
   typedef typename internal::traits<MatrixType>::Scalar Scalar;
   typedef typename internal::plain_col_type<MatrixType>::type Vector;
@@ -152,7 +171,7 @@ struct EstimateInverseMatrixL1NormImpl<Decomposition, 0> {
     int old_v_max_abs_index = v_max_abs_index;
     for (int k = 0; k < 4; ++k) {
       // argmax |inv(matrix)^T * sign_vector|
-      v = dec.adjoint().solve(sign_vector);
+      v = solve_helper<Decomposition, Vector, IsSelfAdjoint>::solve_adjoint(dec, sign_vector);
       v.cwiseAbs().maxCoeff(&v_max_abs_index);
       if (v_max_abs_index == old_v_max_abs_index) {
         // Break if the solution stagnated.
@@ -200,8 +219,8 @@ struct EstimateInverseMatrixL1NormImpl<Decomposition, 0> {
 };
 
 // Partial specialization for complex matrices.
-template <typename Decomposition>
+template <typename Decomposition, bool IsSelfAdjoint>
-struct EstimateInverseMatrixL1NormImpl<Decomposition, 1> {
+struct EstimateInverseMatrixL1NormImpl<Decomposition, IsSelfAdjoint, true> {
   typedef typename Decomposition::MatrixType MatrixType;
   typedef typename internal::traits<MatrixType>::Scalar Scalar;
   typedef typename NumTraits<Scalar>::Real RealScalar;
@@ -238,7 +257,7 @@ struct EstimateInverseMatrixL1NormImpl<Decomposition, 1> {
       RealVector abs_v = v.cwiseAbs();
       const Vector psi =
           (abs_v.array() == 0).select(v.cwiseQuotient(abs_v), ones);
-      v = dec.adjoint().solve(psi);
+      v = solve_helper<Decomposition, Vector, IsSelfAdjoint>::solve_adjoint(dec, psi);
       const RealVector z = v.real();
       z.cwiseAbs().maxCoeff(&v_max_abs_index);
       if (v_max_abs_index == old_v_max_abs_index) {

test/cholesky.cpp

@@ -17,6 +17,12 @@
 #include <Eigen/Cholesky>
 #include <Eigen/QR>
 
+template<typename MatrixType, int UpLo>
+typename MatrixType::RealScalar matrix_l1_norm(const MatrixType& m) {
+  MatrixType symm = m.template selfadjointView<UpLo>();
+  return symm.cwiseAbs().colwise().sum().maxCoeff();
+}
+
 template<typename MatrixType,template <typename,int> class CholType> void test_chol_update(const MatrixType& symm)
 {
   typedef typename MatrixType::Scalar Scalar;
@@ -85,6 +91,14 @@ template<typename MatrixType> void cholesky(const MatrixType& m)
     matX = chollo.solve(matB);
     VERIFY_IS_APPROX(symm * matX, matB);
 
+    // Verify that the estimated condition number is within a factor of 10 of the
+    // truth.
+    const MatrixType symmLo_inverse = chollo.solve(MatrixType::Identity(rows,cols));
+    RealScalar rcond = (RealScalar(1) / matrix_l1_norm<MatrixType, Lower>(symmLo)) /
+                             matrix_l1_norm<MatrixType, Lower>(symmLo_inverse);
+    RealScalar rcond_est = chollo.rcond();
+    VERIFY(rcond_est > rcond / 10 && rcond_est < rcond * 10);
+
     // test the upper mode
     LLT<SquareMatrixType,Upper> cholup(symmUp);
     VERIFY_IS_APPROX(symm, cholup.reconstructedMatrix());
@@ -93,6 +107,15 @@ template<typename MatrixType> void cholesky(const MatrixType& m)
     matX = cholup.solve(matB);
     VERIFY_IS_APPROX(symm * matX, matB);
 
+    // Verify that the estimated condition number is within a factor of 10 of the
+    // truth.
+    const MatrixType symmUp_inverse = cholup.solve(MatrixType::Identity(rows,cols));
+    rcond = (RealScalar(1) / matrix_l1_norm<MatrixType, Upper>(symmUp)) /
+                             matrix_l1_norm<MatrixType, Upper>(symmUp_inverse);
+    rcond_est = cholup.rcond();
+    VERIFY(rcond_est > rcond / 10 && rcond_est < rcond * 10);
+
+
     MatrixType neg = -symmLo;
     chollo.compute(neg);
     VERIFY(chollo.info()==NumericalIssue);

test/lu.cpp

@@ -151,10 +151,11 @@ template<typename MatrixType> void lu_invertible()
   MatrixType m1_inverse = lu.inverse();
   VERIFY_IS_APPROX(m2, m1_inverse*m3);
 
-  // Test condition number estimation.
+  // Verify that the estimated condition number is within a factor of 10 of the
+  // truth.
   RealScalar rcond = (RealScalar(1) / matrix_l1_norm(m1)) / matrix_l1_norm(m1_inverse);
-  // Verify that the estimate is within a factor of 10 of the truth.
+  const RealScalar rcond_est = lu.rcond();
-  VERIFY(lu.rcond() > rcond / 10 && lu.rcond() < rcond * 10);
+  VERIFY(rcond_est > rcond / 10 && rcond_est < rcond * 10);
 
   // test solve with transposed
   lu.template _solve_impl_transposed<false>(m3, m2);
@@ -199,7 +200,8 @@ template<typename MatrixType> void lu_partial_piv()
   // Test condition number estimation.
   RealScalar rcond = (RealScalar(1) / matrix_l1_norm(m1)) / matrix_l1_norm(m1_inverse);
   // Verify that the estimate is within a factor of 10 of the truth.
-  VERIFY(plu.rcond() > rcond / 10 && plu.rcond() < rcond * 10);
+  const RealScalar rcond_est = plu.rcond();
+  VERIFY(rcond_est > rcond / 10 && rcond_est < rcond * 10);
 
   // test solve with transposed
   plu.template _solve_impl_transposed<false>(m3, m2);