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Addresses comments on Eigen pull request PR-174.
* Get rid of code-duplication for real vs. complex matrices. * Fix flipped arguments to select. * Make the condition estimation functions free functions. * Use Vector::Unit() to generate canonical unit vectors. * Misc. cleanup.
This commit is contained in:
Rasmus Munk Larsen
parent
30242b7565
commit
86e0ed81f8
@ -198,7 +198,7 @@ template<typename _MatrixType, int _UpLo> class LDLT
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RealScalar rcond() const
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{
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eigen_assert(m_isInitialized && "LDLT is not initialized.");
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return ConditionEstimator<LDLT<MatrixType, UpLo>, true >::rcond(m_l1_norm, *this);
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return ReciprocalConditionNumberEstimate(m_l1_norm, *this);
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}
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template <typename Derived>
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@ -216,6 +216,12 @@ template<typename _MatrixType, int _UpLo> class LDLT
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MatrixType reconstructedMatrix() const;
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/** \returns the decomposition itself to allow generic code to do
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* ldlt.transpose().solve(rhs).
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*/
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const LDLT<MatrixType, UpLo>& transpose() const { return *this; };
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const LDLT<MatrixType, UpLo>& adjoint() const { return *this; };
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inline Index rows() const { return m_matrix.rows(); }
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inline Index cols() const { return m_matrix.cols(); }
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@ -454,14 +460,14 @@ LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::compute(const EigenBase<InputTyp
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if (_UpLo == Lower) {
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for (int col = 0; col < size; ++col) {
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const RealScalar abs_col_sum = m_matrix.col(col).tail(size - col).cwiseAbs().sum() +
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m_matrix.row(col).tail(col).cwiseAbs().sum();
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m_matrix.row(col).head(col).cwiseAbs().sum();
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if (abs_col_sum > m_l1_norm) {
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m_l1_norm = abs_col_sum;
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}
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}
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} else {
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for (int col = 0; col < a.cols(); ++col) {
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const RealScalar abs_col_sum = m_matrix.col(col).tail(col).cwiseAbs().sum() +
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const RealScalar abs_col_sum = m_matrix.col(col).head(col).cwiseAbs().sum() +
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m_matrix.row(col).tail(size - col).cwiseAbs().sum();
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if (abs_col_sum > m_l1_norm) {
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m_l1_norm = abs_col_sum;
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@ -142,7 +142,7 @@ template<typename _MatrixType, int _UpLo> class LLT
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{
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eigen_assert(m_isInitialized && "LLT is not initialized.");
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eigen_assert(m_info == Success && "LLT failed because matrix appears to be negative");
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return ConditionEstimator<LLT<MatrixType, UpLo>, true >::rcond(m_l1_norm, *this);
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return ReciprocalConditionNumberEstimate(m_l1_norm, *this);
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}
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/** \returns the LLT decomposition matrix
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@ -169,6 +169,12 @@ template<typename _MatrixType, int _UpLo> class LLT
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return m_info;
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}
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/** \returns the decomposition itself to allow generic code to do
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* llt.transpose().solve(rhs).
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*/
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const LLT<MatrixType, UpLo>& transpose() const { return *this; };
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const LLT<MatrixType, UpLo>& adjoint() const { return *this; };
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inline Index rows() const { return m_matrix.rows(); }
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inline Index cols() const { return m_matrix.cols(); }
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@ -409,14 +415,14 @@ LLT<MatrixType,_UpLo>& LLT<MatrixType,_UpLo>::compute(const EigenBase<InputType>
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if (_UpLo == Lower) {
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for (int col = 0; col < size; ++col) {
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const RealScalar abs_col_sum = m_matrix.col(col).tail(size - col).cwiseAbs().sum() +
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m_matrix.row(col).tail(col).cwiseAbs().sum();
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m_matrix.row(col).head(col).cwiseAbs().sum();
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if (abs_col_sum > m_l1_norm) {
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m_l1_norm = abs_col_sum;
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}
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}
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} else {
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for (int col = 0; col < a.cols(); ++col) {
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const RealScalar abs_col_sum = m_matrix.col(col).tail(col).cwiseAbs().sum() +
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const RealScalar abs_col_sum = m_matrix.col(col).head(col).cwiseAbs().sum() +
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m_matrix.row(col).tail(size - col).cwiseAbs().sum();
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if (abs_col_sum > m_l1_norm) {
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m_l1_norm = abs_col_sum;
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@ -13,19 +13,35 @@
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namespace Eigen {
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namespace internal {
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template <typename Decomposition, bool IsSelfAdjoint, bool IsComplex>
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struct EstimateInverseMatrixL1NormImpl {};
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template <typename MatrixType>
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inline typename MatrixType::RealScalar MatrixL1Norm(const MatrixType& matrix) {
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return matrix.cwiseAbs().colwise().sum().maxCoeff();
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}
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template <typename Vector>
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inline typename Vector::RealScalar VectorL1Norm(const Vector& v) {
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return v.template lpNorm<1>();
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}
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template <typename Vector, typename RealVector, bool IsComplex>
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struct SignOrUnity {
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static inline Vector run(const Vector& v) {
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const RealVector v_abs = v.cwiseAbs();
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return (v_abs.array() == 0).select(Vector::Ones(v.size()), v.cwiseQuotient(v_abs));
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}
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};
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// Partial specialization to avoid elementwise division for real vectors.
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template <typename Vector>
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struct SignOrUnity<Vector, Vector, false> {
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static inline Vector run(const Vector& v) {
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return (v.array() < 0).select(-Vector::Ones(v.size()), Vector::Ones(v.size()));
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}
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};
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} // namespace internal
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template <typename Decomposition, bool IsSelfAdjoint = false>
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class ConditionEstimator {
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public:
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typedef typename Decomposition::MatrixType MatrixType;
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typedef typename internal::traits<MatrixType>::Scalar Scalar;
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typedef typename NumTraits<Scalar>::Real RealScalar;
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typedef typename internal::plain_col_type<MatrixType>::type Vector;
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/** \class ConditionEstimator
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/** \class ConditionEstimator
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* \ingroup Core_Module
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*
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* \brief Condition number estimator.
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@ -41,265 +57,148 @@ class ConditionEstimator {
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*
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* \sa FullPivLU, PartialPivLU.
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*/
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static RealScalar rcond(const MatrixType& matrix, const Decomposition& dec) {
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eigen_assert(matrix.rows() == dec.rows());
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eigen_assert(matrix.cols() == dec.cols());
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eigen_assert(matrix.rows() == matrix.cols());
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if (dec.rows() == 0) {
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return RealScalar(1);
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}
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return rcond(MatrixL1Norm(matrix), dec);
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template <typename Decomposition>
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typename Decomposition::RealScalar ReciprocalConditionNumberEstimate(
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const typename Decomposition::MatrixType& matrix,
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const Decomposition& dec) {
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eigen_assert(matrix.rows() == dec.rows());
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eigen_assert(matrix.cols() == dec.cols());
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eigen_assert(matrix.rows() == matrix.cols());
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if (dec.rows() == 0) {
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return Decomposition::RealScalar(1);
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}
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return ReciprocalConditionNumberEstimate(MatrixL1Norm(matrix), dec);
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}
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/** \class ConditionEstimator
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* \ingroup Core_Module
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*
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* \brief Condition number estimator.
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*
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* Computing a decomposition of a dense matrix takes O(n^3) operations, while
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* this method estimates the condition number quickly and reliably in O(n^2)
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* operations.
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*
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* \returns an estimate of the reciprocal condition number
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* (1 / (||matrix||_1 * ||inv(matrix)||_1)) of matrix, given ||matrix||_1 and
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* its decomposition. Supports the following decompositions: FullPivLU,
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* PartialPivLU.
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*
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* \sa FullPivLU, PartialPivLU.
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*/
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static RealScalar rcond(RealScalar matrix_norm, const Decomposition& dec) {
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eigen_assert(dec.rows() == dec.cols());
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if (dec.rows() == 0) {
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return 1;
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}
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if (matrix_norm == 0) {
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return 0;
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}
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const RealScalar inverse_matrix_norm = EstimateInverseMatrixL1Norm(dec);
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return inverse_matrix_norm == 0 ? 0
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: (1 / inverse_matrix_norm) / matrix_norm;
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/** \class ConditionEstimator
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* \ingroup Core_Module
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*
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* \brief Condition number estimator.
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*
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* Computing a decomposition of a dense matrix takes O(n^3) operations, while
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* this method estimates the condition number quickly and reliably in O(n^2)
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* operations.
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*
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* \returns an estimate of the reciprocal condition number
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* (1 / (||matrix||_1 * ||inv(matrix)||_1)) of matrix, given ||matrix||_1 and
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* its decomposition. Supports the following decompositions: FullPivLU,
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* PartialPivLU.
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*
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* \sa FullPivLU, PartialPivLU.
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*/
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template <typename Decomposition>
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typename Decomposition::RealScalar ReciprocalConditionNumberEstimate(
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typename Decomposition::RealScalar matrix_norm, const Decomposition& dec) {
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eigen_assert(dec.rows() == dec.cols());
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if (dec.rows() == 0) {
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return 1;
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}
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/**
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* \returns an estimate of ||inv(matrix)||_1 given a decomposition of
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* matrix that implements .solve() and .adjoint().solve() methods.
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*
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* The method implements Algorithms 4.1 and 5.1 from
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* http://www.maths.manchester.ac.uk/~higham/narep/narep135.pdf
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* which also forms the basis for the condition number estimators in
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* LAPACK. Since at most 10 calls to the solve method of dec are
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* performed, the total cost is O(dims^2), as opposed to O(dims^3)
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* needed to compute the inverse matrix explicitly.
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*
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* The most common usage is in estimating the condition number
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* ||matrix||_1 * ||inv(matrix)||_1. The first term ||matrix||_1 can be
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* computed directly in O(n^2) operations.
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*/
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static RealScalar EstimateInverseMatrixL1Norm(const Decomposition& dec) {
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eigen_assert(dec.rows() == dec.cols());
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if (dec.rows() == 0) {
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return 0;
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}
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return internal::EstimateInverseMatrixL1NormImpl<
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Decomposition, IsSelfAdjoint,
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NumTraits<Scalar>::IsComplex != 0>::compute(dec);
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if (matrix_norm == 0) {
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return 0;
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}
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const typename Decomposition::RealScalar inverse_matrix_norm = InverseMatrixL1NormEstimate(dec);
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return inverse_matrix_norm == 0 ? 0 : (1 / inverse_matrix_norm) / matrix_norm;
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}
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/**
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* \returns the induced matrix l1-norm
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* ||matrix||_1 = sup ||matrix * v||_1 / ||v||_1, which is equal to
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* the greatest absolute column sum.
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*/
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static inline Scalar MatrixL1Norm(const MatrixType& matrix) {
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return matrix.cwiseAbs().colwise().sum().maxCoeff();
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}
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};
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namespace internal {
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template <typename Decomposition, typename Vector, bool IsSelfAdjoint = false>
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struct solve_helper {
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static inline Vector solve_adjoint(const Decomposition& dec,
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const Vector& v) {
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return dec.adjoint().solve(v);
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}
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};
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// Partial specialization for self_adjoint matrices.
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template <typename Decomposition, typename Vector>
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struct solve_helper<Decomposition, Vector, true> {
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static inline Vector solve_adjoint(const Decomposition& dec,
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const Vector& v) {
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return dec.solve(v);
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}
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};
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// Partial specialization for real matrices.
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template <typename Decomposition, bool IsSelfAdjoint>
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struct EstimateInverseMatrixL1NormImpl<Decomposition, IsSelfAdjoint, false> {
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/**
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* \returns an estimate of ||inv(matrix)||_1 given a decomposition of
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* matrix that implements .solve() and .adjoint().solve() methods.
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*
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* The method implements Algorithms 4.1 and 5.1 from
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* http://www.maths.manchester.ac.uk/~higham/narep/narep135.pdf
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* which also forms the basis for the condition number estimators in
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* LAPACK. Since at most 10 calls to the solve method of dec are
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* performed, the total cost is O(dims^2), as opposed to O(dims^3)
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* needed to compute the inverse matrix explicitly.
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*
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* The most common usage is in estimating the condition number
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* ||matrix||_1 * ||inv(matrix)||_1. The first term ||matrix||_1 can be
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* computed directly in O(n^2) operations.
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*/
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template <typename Decomposition>
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typename Decomposition::RealScalar InverseMatrixL1NormEstimate(
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const Decomposition& dec) {
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typedef typename Decomposition::MatrixType MatrixType;
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typedef typename internal::traits<MatrixType>::Scalar Scalar;
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typedef typename Decomposition::Scalar Scalar;
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typedef typename Decomposition::RealScalar RealScalar;
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typedef typename internal::plain_col_type<MatrixType>::type Vector;
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typedef typename internal::plain_col_type<MatrixType, RealScalar>::type RealVector;
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const bool is_complex = (NumTraits<Scalar>::IsComplex != 0);
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// Shorthand for vector L1 norm in Eigen.
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static inline Scalar VectorL1Norm(const Vector& v) {
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return v.template lpNorm<1>();
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eigen_assert(dec.rows() == dec.cols());
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const int n = dec.rows();
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if (n == 0) {
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return 0;
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}
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Vector v = Vector::Ones(n) / n;
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v = dec.solve(v);
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static inline Scalar compute(const Decomposition& dec) {
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const int n = dec.rows();
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const Vector plus = Vector::Ones(n);
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Vector v = plus / n;
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v = dec.solve(v);
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Scalar lower_bound = VectorL1Norm(v);
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if (n == 1) {
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return lower_bound;
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}
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// lower_bound is a lower bound on
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// ||inv(matrix)||_1 = sup_v ||inv(matrix) v||_1 / ||v||_1
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// and is the objective maximized by the ("super-") gradient ascent
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// algorithm.
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// Basic idea: We know that the optimum is achieved at one of the simplices
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// v = e_i, so in each iteration we follow a super-gradient to move towards
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// the optimal one.
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Scalar old_lower_bound = lower_bound;
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const Vector minus = -Vector::Ones(n);
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Vector sign_vector = (v.cwiseAbs().array() == 0).select(plus, minus);
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Vector old_sign_vector = sign_vector;
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int v_max_abs_index = -1;
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int old_v_max_abs_index = v_max_abs_index;
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for (int k = 0; k < 4; ++k) {
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// argmax |inv(matrix)^T * sign_vector|
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v = solve_helper<Decomposition, Vector, IsSelfAdjoint>::solve_adjoint(dec, sign_vector);
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v.cwiseAbs().maxCoeff(&v_max_abs_index);
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if (v_max_abs_index == old_v_max_abs_index) {
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// Break if the solution stagnated.
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break;
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}
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// Move to the new simplex e_j, where j = v_max_abs_index.
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v.setZero();
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v[v_max_abs_index] = 1;
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v = dec.solve(v); // v = inv(matrix) * e_j.
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lower_bound = VectorL1Norm(v);
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if (lower_bound <= old_lower_bound) {
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// Break if the gradient step did not increase the lower_bound.
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break;
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}
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sign_vector = (v.array() < 0).select(plus, minus);
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// lower_bound is a lower bound on
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// ||inv(matrix)||_1 = sup_v ||inv(matrix) v||_1 / ||v||_1
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// and is the objective maximized by the ("super-") gradient ascent
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// algorithm below.
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RealScalar lower_bound = internal::VectorL1Norm(v);
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if (n == 1) {
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return lower_bound;
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}
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// Gradient ascent algorithm follows: We know that the optimum is achieved at
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// one of the simplices v = e_i, so in each iteration we follow a
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// super-gradient to move towards the optimal one.
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RealScalar old_lower_bound = lower_bound;
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Vector sign_vector(n);
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Vector old_sign_vector;
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int v_max_abs_index = -1;
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int old_v_max_abs_index = v_max_abs_index;
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for (int k = 0; k < 4; ++k) {
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sign_vector = internal::SignOrUnity<Vector, RealVector, is_complex>::run(v);
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if (k > 0 && !is_complex) {
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if (sign_vector == old_sign_vector) {
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// Break if the solution stagnated.
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break;
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}
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}
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// v_max_abs_index = argmax |real( inv(matrix)^T * sign_vector )|
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v = dec.adjoint().solve(sign_vector);
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v.real().cwiseAbs().maxCoeff(&v_max_abs_index);
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if (v_max_abs_index == old_v_max_abs_index) {
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// Break if the solution stagnated.
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break;
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}
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// Move to the new simplex e_j, where j = v_max_abs_index.
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v = dec.solve(Vector::Unit(n, v_max_abs_index)); // v = inv(matrix) * e_j.
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lower_bound = internal::VectorL1Norm(v);
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if (lower_bound <= old_lower_bound) {
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// Break if the gradient step did not increase the lower_bound.
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break;
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}
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if (!is_complex) {
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old_sign_vector = sign_vector;
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old_v_max_abs_index = v_max_abs_index;
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old_lower_bound = lower_bound;
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}
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// The following calculates an independent estimate of ||matrix||_1 by
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// multiplying matrix by a vector with entries of slowly increasing
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// magnitude and alternating sign:
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// v_i = (-1)^{i} (1 + (i / (dim-1))), i = 0,...,dim-1.
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// This improvement to Hager's algorithm above is due to Higham. It was
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// added to make the algorithm more robust in certain corner cases where
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// large elements in the matrix might otherwise escape detection due to
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// exact cancellation (especially when op and op_adjoint correspond to a
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// sequence of backsubstitutions and permutations), which could cause
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// Hager's algorithm to vastly underestimate ||matrix||_1.
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Scalar alternating_sign = 1;
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for (int i = 0; i < n; ++i) {
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v[i] = alternating_sign * static_cast<Scalar>(1) +
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(static_cast<Scalar>(i) / (static_cast<Scalar>(n - 1)));
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alternating_sign = -alternating_sign;
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}
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v = dec.solve(v);
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const Scalar alternate_lower_bound =
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(2 * VectorL1Norm(v)) / (3 * static_cast<Scalar>(n));
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return numext::maxi(lower_bound, alternate_lower_bound);
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old_v_max_abs_index = v_max_abs_index;
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old_lower_bound = lower_bound;
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}
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};
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// Partial specialization for complex matrices.
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template <typename Decomposition, bool IsSelfAdjoint>
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struct EstimateInverseMatrixL1NormImpl<Decomposition, IsSelfAdjoint, true> {
|
||||
typedef typename Decomposition::MatrixType MatrixType;
|
||||
typedef typename internal::traits<MatrixType>::Scalar Scalar;
|
||||
typedef typename NumTraits<Scalar>::Real RealScalar;
|
||||
typedef typename internal::plain_col_type<MatrixType>::type Vector;
|
||||
typedef typename internal::plain_col_type<MatrixType, RealScalar>::type
|
||||
RealVector;
|
||||
|
||||
// Shorthand for vector L1 norm in Eigen.
|
||||
static inline RealScalar VectorL1Norm(const Vector& v) {
|
||||
return v.template lpNorm<1>();
|
||||
// The following calculates an independent estimate of ||matrix||_1 by
|
||||
// multiplying matrix by a vector with entries of slowly increasing
|
||||
// magnitude and alternating sign:
|
||||
// v_i = (-1)^{i} (1 + (i / (dim-1))), i = 0,...,dim-1.
|
||||
// This improvement to Hager's algorithm above is due to Higham. It was
|
||||
// added to make the algorithm more robust in certain corner cases where
|
||||
// large elements in the matrix might otherwise escape detection due to
|
||||
// exact cancellation (especially when op and op_adjoint correspond to a
|
||||
// sequence of backsubstitutions and permutations), which could cause
|
||||
// Hager's algorithm to vastly underestimate ||matrix||_1.
|
||||
Scalar alternating_sign = 1;
|
||||
for (int i = 0; i < n; ++i) {
|
||||
v[i] = alternating_sign * static_cast<RealScalar>(1) +
|
||||
(static_cast<RealScalar>(i) / (static_cast<RealScalar>(n - 1)));
|
||||
alternating_sign = -alternating_sign;
|
||||
}
|
||||
v = dec.solve(v);
|
||||
const RealScalar alternate_lower_bound =
|
||||
(2 * internal::VectorL1Norm(v)) / (3 * static_cast<RealScalar>(n));
|
||||
return numext::maxi(lower_bound, alternate_lower_bound);
|
||||
}
|
||||
|
||||
static inline RealScalar compute(const Decomposition& dec) {
|
||||
const int n = dec.rows();
|
||||
const Vector ones = Vector::Ones(n);
|
||||
Vector v = ones / n;
|
||||
v = dec.solve(v);
|
||||
RealScalar lower_bound = VectorL1Norm(v);
|
||||
if (n == 1) {
|
||||
return lower_bound;
|
||||
}
|
||||
// lower_bound is a lower bound on
|
||||
// ||inv(matrix)||_1 = sup_v ||inv(matrix) v||_1 / ||v||_1
|
||||
// and is the objective maximized by the ("super-") gradient ascent
|
||||
// algorithm.
|
||||
// Basic idea: We know that the optimum is achieved at one of the simplices
|
||||
// v = e_i, so in each iteration we follow a super-gradient to move towards
|
||||
// the optimal one.
|
||||
RealScalar old_lower_bound = lower_bound;
|
||||
int v_max_abs_index = -1;
|
||||
int old_v_max_abs_index = v_max_abs_index;
|
||||
for (int k = 0; k < 4; ++k) {
|
||||
// argmax |inv(matrix)^* * sign_vector|
|
||||
RealVector abs_v = v.cwiseAbs();
|
||||
const Vector psi =
|
||||
(abs_v.array() == 0).select(v.cwiseQuotient(abs_v), ones);
|
||||
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) {
|
||||
// Break if the solution stagnated.
|
||||
break;
|
||||
}
|
||||
// Move to the new simplex e_j, where j = v_max_abs_index.
|
||||
v.setZero();
|
||||
v[v_max_abs_index] = 1;
|
||||
v = dec.solve(v); // v = inv(matrix) * e_j.
|
||||
lower_bound = VectorL1Norm(v);
|
||||
if (lower_bound <= old_lower_bound) {
|
||||
// Break if the gradient step did not increase the lower_bound.
|
||||
break;
|
||||
}
|
||||
old_v_max_abs_index = v_max_abs_index;
|
||||
old_lower_bound = lower_bound;
|
||||
}
|
||||
// The following calculates an independent estimate of ||matrix||_1 by
|
||||
// multiplying matrix by a vector with entries of slowly increasing
|
||||
// magnitude and alternating sign:
|
||||
// v_i = (-1)^{i} (1 + (i / (dim-1))), i = 0,...,dim-1.
|
||||
// This improvement to Hager's algorithm above is due to Higham. It was
|
||||
// added to make the algorithm more robust in certain corner cases where
|
||||
// large elements in the matrix might otherwise escape detection due to
|
||||
// exact cancellation (especially when op and op_adjoint correspond to a
|
||||
// sequence of backsubstitutions and permutations), which could cause
|
||||
// Hager's algorithm to vastly underestimate ||matrix||_1.
|
||||
RealScalar alternating_sign = 1;
|
||||
for (int i = 0; i < n; ++i) {
|
||||
v[i] = alternating_sign * static_cast<RealScalar>(1) +
|
||||
(static_cast<RealScalar>(i) / (static_cast<RealScalar>(n - 1)));
|
||||
alternating_sign = -alternating_sign;
|
||||
}
|
||||
v = dec.solve(v);
|
||||
const RealScalar alternate_lower_bound =
|
||||
(2 * VectorL1Norm(v)) / (3 * static_cast<RealScalar>(n));
|
||||
return numext::maxi(lower_bound, alternate_lower_bound);
|
||||
}
|
||||
};
|
||||
|
||||
} // namespace internal
|
||||
} // namespace Eigen
|
||||
|
||||
#endif
|
||||
|
||||
@ -237,7 +237,7 @@ template<typename _MatrixType> class FullPivLU
|
||||
inline RealScalar rcond() const
|
||||
{
|
||||
eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
|
||||
return ConditionEstimator<FullPivLU<_MatrixType> >::rcond(m_l1_norm, *this);
|
||||
return ReciprocalConditionNumberEstimate(m_l1_norm, *this);
|
||||
}
|
||||
|
||||
/** \returns the determinant of the matrix of which
|
||||
|
||||
@ -157,7 +157,7 @@ template<typename _MatrixType> class PartialPivLU
|
||||
inline RealScalar rcond() const
|
||||
{
|
||||
eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
|
||||
return ConditionEstimator<PartialPivLU<_MatrixType> >::rcond(m_l1_norm, *this);
|
||||
return ReciprocalConditionNumberEstimate(m_l1_norm, *this);
|
||||
}
|
||||
|
||||
/** \returns the inverse of the matrix of which *this is the LU decomposition.
|
||||
|
||||
@ -91,12 +91,12 @@ 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 that the estimated condition number is within a factor of 10 of the
|
||||
// truth.
|
||||
VERIFY(rcond_est > rcond / 10 && rcond_est < rcond * 10);
|
||||
|
||||
// test the upper mode
|
||||
@ -160,12 +160,12 @@ template<typename MatrixType> void cholesky(const MatrixType& m)
|
||||
matX = ldltlo.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 = ldltlo.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 = ldltlo.rcond();
|
||||
// Verify that the estimated condition number is within a factor of 10 of the
|
||||
// truth.
|
||||
VERIFY(rcond_est > rcond / 10 && rcond_est < rcond * 10);
|
||||
|
||||
|
||||
|
||||
@ -151,10 +151,10 @@ template<typename MatrixType> void lu_invertible()
|
||||
MatrixType m1_inverse = lu.inverse();
|
||||
VERIFY_IS_APPROX(m2, m1_inverse*m3);
|
||||
|
||||
// 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);
|
||||
const RealScalar rcond_est = lu.rcond();
|
||||
// Verify that the estimated condition number is within a factor of 10 of the
|
||||
// truth.
|
||||
VERIFY(rcond_est > rcond / 10 && rcond_est < rcond * 10);
|
||||
|
||||
// test solve with transposed
|
||||
@ -197,10 +197,9 @@ template<typename MatrixType> void lu_partial_piv()
|
||||
MatrixType m1_inverse = plu.inverse();
|
||||
VERIFY_IS_APPROX(m2, m1_inverse*m3);
|
||||
|
||||
// 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.
|
||||
const RealScalar rcond_est = plu.rcond();
|
||||
// Verify that the estimate is within a factor of 10 of the truth.
|
||||
VERIFY(rcond_est > rcond / 10 && rcond_est < rcond * 10);
|
||||
|
||||
// test solve with transposed
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user