Extend MatrixPowerTriangularAtomic for future implementation for triangular matrix power.

2 files changed, 205 insertions, 68 deletions

diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h b/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h
index 64afdaab8..7beb209eb 100644
--- a/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h
+++ b/unsupported/Eigen/src/MatrixFunctions/MatrixPower.h
@@ -74,11 +74,11 @@ class MatrixPower : public MatrixPowerBase<MatrixPower<MatrixType>,MatrixType>
     void compute(const Derived& b, ResultType& res, RealScalar p);
 
   private:
+    EIGEN_MATRIX_POWER_PROTECTED_MEMBERS(MatrixPower)
+
     typedef Matrix<std::complex<RealScalar>,   RowsAtCompileTime,   ColsAtCompileTime,
 				    Options,MaxRowsAtCompileTime,MaxColsAtCompileTime> ComplexMatrix;
-    MatrixType m_tmp1, m_tmp2;
     ComplexMatrix m_T, m_U, m_fT;
-    bool m_init;
 
     RealScalar modfAndInit(RealScalar, RealScalar*);
 
@@ -98,9 +98,8 @@ class MatrixPower : public MatrixPowerBase<MatrixPower<MatrixType>,MatrixType>
 template<typename MatrixType>
 template<typename MatrixExpression>
 MatrixPower<MatrixType>::MatrixPower(const MatrixExpression& A) :
-  Base(A),
-  m_init(false)
-{ /* empty body */ }
+  Base(A, 0)
+{ }
 
 template<typename MatrixType>
 void MatrixPower<MatrixType>::compute(MatrixType& res, RealScalar p)
@@ -113,7 +112,7 @@ void MatrixPower<MatrixType>::compute(MatrixType& res, RealScalar p)
       break;
     default:
       RealScalar intpart, x = modfAndInit(p, &intpart);
-      res = MatrixType::Identity(m_A.rows(), m_A.cols());
+      res = m_Id;
       computeIntPower(res, intpart);
       computeFracPower(res, x);
   }
@@ -139,22 +138,19 @@ void MatrixPower<MatrixType>::compute(const Derived& b, ResultType& res, RealSca
 template<typename MatrixType>
 typename MatrixPower<MatrixType>::Base::RealScalar MatrixPower<MatrixType>::modfAndInit(RealScalar x, RealScalar* intpart)
 {
-  static RealScalar maxAbsEival, minAbsEival;
   *intpart = std::floor(x);
   RealScalar res = x - *intpart;
 
-  if (!m_init && res) {
+  if (!m_conditionNumber && res) {
     const ComplexSchur<MatrixType> schurOfA(m_A);
     m_T = schurOfA.matrixT();
     m_U = schurOfA.matrixU();
-    m_init = true;
-
+    
     const RealArray absTdiag = m_T.diagonal().array().abs();
-    maxAbsEival = absTdiag.maxCoeff();
-    minAbsEival = absTdiag.minCoeff();
+    m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff();
   }
 
-  if (res > RealScalar(0.5) && res > (1-res) * std::pow(maxAbsEival/minAbsEival, res)) {
+  if (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) {
     --res;
     ++*intpart;
   }
@@ -195,7 +191,7 @@ template<typename Derived, typename ResultType>
 void MatrixPower<MatrixType>::computeIntPower(const Derived& b, ResultType& res, RealScalar p)
 {
   if (b.cols() >= m_A.cols()) {
-    m_tmp2 = MatrixType::Identity(m_A.rows(), m_A.cols());
+    m_tmp2 = m_Id;
     computeIntPower(m_tmp2, p);
     res.noalias() = m_tmp2 * b;
   }
diff --git a/unsupported/Eigen/src/MatrixFunctions/MatrixPowerBase.h b/unsupported/Eigen/src/MatrixFunctions/MatrixPowerBase.h
index b5e8ed7ed..e83d4abfd 100644
--- a/unsupported/Eigen/src/MatrixFunctions/MatrixPowerBase.h
+++ b/unsupported/Eigen/src/MatrixFunctions/MatrixPowerBase.h
@@ -18,7 +18,7 @@ struct recompose_complex_schur
 {
   template<typename ResultType, typename MatrixType>
   static inline void run(ResultType& res, const MatrixType& T, const MatrixType& U)
-  { res = U * (T.template triangularView<Upper>() * U.adjoint()); }
+  { res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); }
 };
 
 template<>
@@ -26,7 +26,21 @@ struct recompose_complex_schur<0>
 {
   template<typename ResultType, typename MatrixType>
   static inline void run(ResultType& res, const MatrixType& T, const MatrixType& U)
-  { res = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
+  { res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }
+};
+
+template<typename Scalar, int IsComplex=NumTraits<Scalar>::IsComplex>
+struct matrix_power_unwinder
+{
+  static inline Scalar run(const Scalar& eival, const Scalar& eival0, int unwindingNumber)
+  { return internal::atanh2(eival-eival0, eival+eival0) + Scalar(0, M_PI*unwindingNumber); }
+};
+
+template<typename Scalar>
+struct matrix_power_unwinder<Scalar,0>
+{
+  static inline Scalar run(Scalar eival, Scalar eival0, int)
+  { return internal::atanh2(eival-eival0, eival+eival0); }
 };
 
 template<typename T>
@@ -68,21 +82,21 @@ inline int matrix_power_get_pade_degree(double normIminusT)
 inline int matrix_power_get_pade_degree(long double normIminusT)
 {
 #if   LDBL_MANT_DIG == 53
-  const int maxPadeDegree = 7;
+  enum { maxPadeDegree = 7 };
   const double maxNormForPade[] = { 1.884160592658218e-2L /* degree = 3 */ , 6.038881904059573e-2L, 1.239917516308172e-1L,
       1.999045567181744e-1L, 2.789358995219730e-1L };
 #elif LDBL_MANT_DIG <= 64
-  const int maxPadeDegree = 8;
+  enum { maxPadeDegree = 8 };
   const double maxNormForPade[] = { 6.3854693117491799460e-3L /* degree = 3 */ , 2.6394893435456973676e-2L,
       6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };
 #elif LDBL_MANT_DIG <= 106
-  const int maxPadeDegree = 10;
+  enum { maxPadeDegree = 10 };
   const double maxNormForPade[] = { 1.0007161601787493236741409687186e-4L /* degree = 3 */ ,
       1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,
       2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,
       1.1016843812851143391275867258512e-1L };
 #else
-  const int maxPadeDegree = 10;
+  enum { maxPadeDegree = 10 };
   const double maxNormForPade[] = { 5.524506147036624377378713555116378e-5L /* degree = 3 */ ,
       6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,
       9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,
@@ -97,6 +111,125 @@ inline int matrix_power_get_pade_degree(long double normIminusT)
 }
 } // namespace internal
 
+#define MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(Mode) \
+  template<typename MatrixType> \
+  class MatrixPowerTriangular2x2<MatrixType,Mode> \
+  { \
+    private: \
+      enum { \
+	RowsAtCompileTime = MatrixType::RowsAtCompileTime, \
+	MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime \
+      }; \
+      typedef typename MatrixType::Scalar Scalar; \
+      typedef typename MatrixType::RealScalar RealScalar; \
+      typedef Array<Scalar,RowsAtCompileTime,1,ColMajor,MaxRowsAtCompileTime> ArrayType; \
+      const MatrixType& m_T; \
+    public: \
+      explicit MatrixPowerTriangular2x2(const MatrixType& T) : m_T(T) { } \
+      void compute(MatrixType& res, RealScalar p) const; \
+  };
+
+template<typename MatrixType, unsigned int Mode>
+class MatrixPowerTriangular2x2;
+
+MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(Upper)
+MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(Lower)
+MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(UnitUpper)
+MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(UnitLower)
+MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(StrictlyUpper)
+MATRIX_POWER_TRIANGULAR_2x2_SPECIALIZATION(StrictlyLower)
+
+template<typename MatrixType>
+void MatrixPowerTriangular2x2<MatrixType,Upper>::compute(MatrixType& res, RealScalar p) const
+{
+  using std::abs;
+  using std::pow;
+  
+  ArrayType logTdiag = m_T.diagonal().array().log();
+  res.coeffRef(0,0) = pow(m_T.coeff(0,0), p);
+
+  for (int i=1; i < m_T.cols(); ++i) {
+    res.coeffRef(i,i) = pow(m_T.coeff(i,i), p);
+    if (m_T.coeff(i-1,i-1) == m_T.coeff(i,i)) {
+      res.coeffRef(i-1,i) = p * pow(m_T.coeff(i-1,i), p-1);
+    }
+    else if (2*abs(m_T.coeff(i-1,i-1)) < abs(m_T.coeff(i,i)) || 2*abs(m_T.coeff(i,i)) < abs(m_T.coeff(i-1,i-1))) {
+      res.coeffRef(i-1,i) = m_T.coeff(i-1,i) * (res.coeff(i,i)-res.coeff(i-1,i-1)) / (m_T.coeff(i,i)-m_T.coeff(i-1,i-1));
+    }
+    else {
+      int unwindingNumber = std::ceil((internal::imag(logTdiag[i]-logTdiag[i-1]) - M_PI) / (2*M_PI));
+      Scalar w = internal::matrix_power_unwinder<Scalar>::run(m_T.coeff(i,i), m_T.coeff(i-1,i-1), unwindingNumber);
+      res.coeffRef(i-1,i) = m_T.coeff(i-1,i) * RealScalar(2) * std::exp(RealScalar(0.5)*p*(logTdiag[i]+logTdiag[i-1])) *
+	  std::sinh(p * w) / (m_T.coeff(i,i) - m_T.coeff(i-1,i-1));
+    }
+  }
+}
+
+template<typename MatrixType>
+void MatrixPowerTriangular2x2<MatrixType,Lower>::compute(MatrixType& res, RealScalar p) const
+{
+  using std::abs;
+  using std::pow;
+  
+  ArrayType logTdiag = m_T.diagonal().array().log();
+  res.coeffRef(0,0) = pow(m_T.coeff(0,0), p);
+
+  for (int i=1; i < m_T.cols(); ++i) {
+    res.coeffRef(i,i) = pow(m_T.coeff(i,i), p);
+    if (m_T.coeff(i-1,i-1) == m_T.coeff(i,i)) {
+      res.coeffRef(i,i-1) = p * pow(m_T.coeff(i,i-1), p-1);
+    }
+    else if (2*abs(m_T.coeff(i-1,i-1)) < abs(m_T.coeff(i,i)) || 2*abs(m_T.coeff(i,i)) < abs(m_T.coeff(i-1,i-1))) {
+      res.coeffRef(i,i-1) = m_T.coeff(i,i-1) * (res.coeff(i,i)-res.coeff(i-1,i-1)) / (m_T.coeff(i,i)-m_T.coeff(i-1,i-1));
+    }
+    else {
+      int unwindingNumber = std::ceil((internal::imag(logTdiag[i]-logTdiag[i-1]) - M_PI) / (2*M_PI));
+      Scalar w = internal::matrix_power_unwinder<Scalar>::run(m_T.coeff(i,i), m_T.coeff(i-1,i-1), unwindingNumber);
+      res.coeffRef(i,i-1) = m_T.coeff(i,i-1) * RealScalar(2) * std::exp(RealScalar(0.5)*p*(logTdiag[i]+logTdiag[i-1])) *
+	  std::sinh(p * w) / (m_T.coeff(i,i) - m_T.coeff(i-1,i-1));
+    }
+  }
+}
+
+template<typename MatrixType>
+void MatrixPowerTriangular2x2<MatrixType,UnitUpper>::compute(MatrixType& res, RealScalar p) const
+{
+  for (int i=1; i < m_T.cols(); ++i)
+    res.coeffRef(i-1,i) = p * std::pow(m_T.coeff(i-1,i), p-1);
+}
+
+template<typename MatrixType>
+void MatrixPowerTriangular2x2<MatrixType,UnitLower>::compute(MatrixType& res, RealScalar p) const
+{
+  for (int i=1; i < m_T.cols(); ++i) {
+    res.coeffRef(i,i-1) = p * std::pow(m_T.coeff(i,i-1), p-1);
+  }
+}
+
+template<typename MatrixType>
+void MatrixPowerTriangular2x2<MatrixType,StrictlyUpper>::compute(MatrixType& res, RealScalar p) const
+{
+  RealScalar diag   = !p ? 1 : 0;
+  res.coeffRef(0,0) = diag;
+
+  for (int i=1; i < m_T.cols(); ++i) {
+    res.coeffRef(i,i)   = diag;
+    res.coeffRef(i-1,i) = p * std::pow(m_T.coeff(i-1,i), p-1);
+  }
+}
+
+template<typename MatrixType>
+void MatrixPowerTriangular2x2<MatrixType,StrictlyLower>::compute(MatrixType& res, RealScalar p) const
+{
+  RealScalar diag   = !p ? 1 : 0;
+  res.coeffRef(0,0) = diag;
+
+  for (int i=1; i < m_T.cols(); ++i) {
+    res.coeffRef(i,i)   = diag;
+    res.coeffRef(i,i-1) = p * std::pow(m_T.coeff(i,i-1), p-1);
+  }
+}
+
 template<typename MatrixType, unsigned int Mode=Upper>
 class MatrixPowerTriangularAtomic
 {
@@ -113,7 +246,6 @@ class MatrixPowerTriangularAtomic
     const MatrixType m_Id;
 
     void computePade(int degree, const MatrixType& IminusT, MatrixType& res, RealScalar p) const;
-    void compute2x2(MatrixType& res, RealScalar p) const;
     void computeBig(MatrixType& res, RealScalar p) const;
 
   public:
@@ -125,7 +257,7 @@ template<typename MatrixType, unsigned int Mode>
 MatrixPowerTriangularAtomic<MatrixType,Mode>::MatrixPowerTriangularAtomic(const MatrixType& T) :
   m_T(T),
   m_Id(MatrixType::Identity(T.rows(), T.cols()))
-{ /* empty body */ }
+{ }
 
 template<typename MatrixType, unsigned int Mode>
 void MatrixPowerTriangularAtomic<MatrixType,Mode>::compute(MatrixType& res, RealScalar p) const
@@ -137,7 +269,7 @@ void MatrixPowerTriangularAtomic<MatrixType,Mode>::compute(MatrixType& res, Real
       res(0,0) = std::pow(m_T(0,0), p);
       break;
     case 2:
-      compute2x2(res, p);
+      MatrixPowerTriangular2x2<MatrixType,Mode>(m_T).compute(res, p);
       break;
     default:
       computeBig(res, p);
@@ -158,41 +290,15 @@ void MatrixPowerTriangularAtomic<MatrixType,Mode>::computePade(int degree, const
 }
 
 template<typename MatrixType, unsigned int Mode>
-void MatrixPowerTriangularAtomic<MatrixType,Mode>::compute2x2(MatrixType& res, RealScalar p) const
-{
-  using std::abs;
-  using std::pow;
-  
-  ArrayType logTdiag = m_T.diagonal().array().log();
-  res(0,0) = pow(m_T(0,0), p);
-
-  for (int i=1; i < m_T.cols(); ++i) {
-    res(i,i) = pow(m_T(i,i), p);
-    if (m_T(i-1,i-1) == m_T(i,i)) {
-      res(i-1,i) = p * pow(m_T(i-1,i), p-1);
-    }
-    else if (2*abs(m_T(i-1,i-1)) < abs(m_T(i,i)) || 2*abs(m_T(i,i)) < abs(m_T(i-1,i-1))) {
-      res(i-1,i) = m_T(i-1,i) * (res(i,i)-res(i-1,i-1)) / (m_T(i,i)-m_T(i-1,i-1));
-    }
-    else {
-      // computation in previous branch is inaccurate if abs(m_T(i,i)) \approx abs(m_T(i-1,i-1))
-      int unwindingNumber = std::ceil(((logTdiag[i]-logTdiag[i-1]).imag() - M_PI) / (2*M_PI));
-      Scalar w = internal::atanh2(m_T(i,i)-m_T(i-1,i-1), m_T(i,i)+m_T(i-1,i-1)) + Scalar(0, M_PI*unwindingNumber);
-      res(i-1,i) = m_T(i-1,i) * RealScalar(2) * std::exp(RealScalar(0.5) * p * (logTdiag[i]+logTdiag[i-1])) *
-	  std::sinh(p * w) / (m_T(i,i) - m_T(i-1,i-1));
-    }
-  }
-}
-
-template<typename MatrixType, unsigned int Mode>
 void MatrixPowerTriangularAtomic<MatrixType,Mode>::computeBig(MatrixType& res, RealScalar p) const
 {
-  const int digits = std::numeric_limits<RealScalar>::digits;
+  enum { digits = std::numeric_limits<RealScalar>::digits };
   const RealScalar maxNormForPade = digits <=  24? 4.3386528e-1f:                           // sigle precision
 				    digits <=  53? 2.789358995219730e-1:                    // double precision
 				    digits <=  64? 2.4471944416607995472e-1L:               // extended precision
-				    digits <= 106? 1.1016843812851143391275867258512e-01:   // double-double
-						   9.134603732914548552537150753385375e-02; // quadruple precision
+				    digits <= 106? 1.1016843812851143391275867258512e-1L:   // double-double
+						   9.134603732914548552537150753385375e-2L; // quadruple precision
+  const MatrixPowerTriangular2x2<MatrixType,Mode> atomic2x2(m_T);
   MatrixType IminusT, sqrtT, T=m_T;
   RealScalar normIminusT;
   int degree, degree2, numberOfSquareRoots=0, numberOfExtraSquareRoots=0;
@@ -214,14 +320,14 @@ void MatrixPowerTriangularAtomic<MatrixType,Mode>::computeBig(MatrixType& res, R
   computePade(degree, IminusT, res, p);
 
   for (; numberOfSquareRoots; --numberOfSquareRoots) {
-    compute2x2(res, std::ldexp(p,-numberOfSquareRoots));
+    atomic2x2.compute(res, std::ldexp(p,-numberOfSquareRoots));
     res *= res;
   }
-  compute2x2(res, p);
+  atomic2x2.compute(res, p);
 }
 
 #define EIGEN_MATRIX_POWER_PUBLIC_INTERFACE(Derived) \
-  typedef MatrixPowerBase<Derived<MatrixType>, MatrixType> Base; \
+  typedef MatrixPowerBase<Derived, MatrixType> Base; \
   using Base::RowsAtCompileTime; \
   using Base::ColsAtCompileTime; \
   using Base::Options; \
@@ -229,8 +335,14 @@ void MatrixPowerTriangularAtomic<MatrixType,Mode>::computeBig(MatrixType& res, R
   using Base::MaxColsAtCompileTime; \
   typedef typename Base::Scalar     Scalar; \
   typedef typename Base::RealScalar RealScalar; \
-  typedef typename Base::RealArray  RealArray; \
-  using Base::m_A;
+  typedef typename Base::RealArray  RealArray;
+
+#define EIGEN_MATRIX_POWER_PROTECTED_MEMBERS(Derived) \
+  using Base::m_A; \
+  using Base::m_Id; \
+  using Base::m_tmp1; \
+  using Base::m_tmp2; \
+  using Base::m_conditionNumber;
 
 #define	EIGEN_MATRIX_POWER_PRODUCT_PUBLIC_INTERFACE(Derived) \
   typedef MatrixPowerProductBase<Derived, Lhs, Rhs> Base; \
@@ -277,34 +389,63 @@ class MatrixPowerBase
     typedef typename MatrixType::RealScalar RealScalar;
     typedef typename MatrixType::Index Index;
 
-    explicit MatrixPowerBase(const MatrixType& A)
-    : m_A(A), m_del(false) { }
+    explicit MatrixPowerBase(const MatrixType& A, RealScalar cond);
 
     template<typename OtherDerived>
-    explicit MatrixPowerBase(const MatrixBase<OtherDerived>& A)
-    : m_A(*new MatrixType(A)), m_del(true) { }
+    explicit MatrixPowerBase(const MatrixBase<OtherDerived>& A, RealScalar cond);
 
-    ~MatrixPowerBase()
-    { if (m_del)  delete &m_A; }
+    ~MatrixPowerBase();
 
-    void compute(MatrixType& res, RealScalar p)
-    { static_cast<Derived*>(this)->compute(res,p); }
+    void compute(MatrixType& res, RealScalar p);
 
     template<typename OtherDerived, typename ResultType>
-    void compute(const OtherDerived& b, ResultType& res, RealScalar p)
-    { static_cast<Derived*>(this)->compute(b,res,p); }
+    void compute(const OtherDerived& b, ResultType& res, RealScalar p);
     
     Index rows() const { return m_A.rows(); }
     Index cols() const { return m_A.cols(); }
 
   protected:
     typedef Array<RealScalar,RowsAtCompileTime,1,ColMajor,MaxRowsAtCompileTime> RealArray;
+
     const MatrixType& m_A;
+    const MatrixType m_Id;
+    MatrixType m_tmp1, m_tmp2;
+    RealScalar m_conditionNumber;
 
   private:
     const bool m_del;  // whether to delete the pointer at destruction
 };
 
+template<typename Derived, typename MatrixType>
+MatrixPowerBase<Derived,MatrixType>::MatrixPowerBase(const MatrixType& A, RealScalar cond) :
+  m_A(A),
+  m_Id(MatrixType::Identity(A.rows(),A.cols())),
+  m_conditionNumber(cond),
+  m_del(false)
+{ }
+
+template<typename Derived, typename MatrixType>
+template<typename OtherDerived>
+MatrixPowerBase<Derived,MatrixType>::MatrixPowerBase(const MatrixBase<OtherDerived>& A, RealScalar cond) :
+  m_A(*new MatrixType(A)),
+  m_Id(MatrixType::Identity(A.rows(),A.cols())),
+  m_conditionNumber(cond),
+  m_del(true)
+{ }
+
+template<typename Derived, typename MatrixType>
+MatrixPowerBase<Derived,MatrixType>::~MatrixPowerBase()
+{ if (m_del)  delete &m_A; }
+
+template<typename Derived, typename MatrixType>
+void MatrixPowerBase<Derived,MatrixType>::compute(MatrixType& res, RealScalar p)
+{ static_cast<Derived*>(this)->compute(res,p); }
+
+template<typename Derived, typename MatrixType>
+template<typename OtherDerived, typename ResultType>
+void MatrixPowerBase<Derived,MatrixType>::compute(const OtherDerived& b, ResultType& res, RealScalar p)
+{ static_cast<Derived*>(this)->compute(b,res,p); }
+
 template<typename Derived, typename Lhs, typename Rhs>
 class MatrixPowerProductBase : public MatrixBase<Derived>
 {