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Fix a lot in MatrixPower.h

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Chen-Pang He 2012-08-25 01:09:20 +08:00
parent edc7a09ee7
commit 1cd4279b03
2 changed files with 234 additions and 147 deletions
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225
unsupported/Eigen/src/MatrixFunctions/MatrixPower.h
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@ -23,12 +23,10 @@ namespace Eigen {
* *
* \tparam MatrixType type of the base, expected to be an instantiation * \tparam MatrixType type of the base, expected to be an instantiation
* of the Matrix class template. * of the Matrix class template.
* \tparam ExponentType type of the exponent, a real scalar.
* \tparam PlainObject type of the multiplier.
* \tparam IsInteger used internally to select correct specialization. * \tparam IsInteger used internally to select correct specialization.
* \tparam PlainObject type of the multiplier.
*/ */
template <typename MatrixType, typename ExponentType, typename PlainObject = MatrixType, template <typename MatrixType, int IsInteger, typename PlainObject = MatrixType>
int IsInteger = NumTraits<ExponentType>::IsInteger>
class MatrixPower class MatrixPower
{ {
private: private:
@ -93,7 +91,7 @@ class MatrixPower
void computeChainProduct(ResultType&); void computeChainProduct(ResultType&);
/** \brief Compute the cost of binary powering. */ /** \brief Compute the cost of binary powering. */
int computeCost(RealScalar); static int computeCost(RealScalar);
/** \brief Solve the linear system repetitively. */ /** \brief Solve the linear system repetitively. */
template <typename ResultType> template <typename ResultType>
@ -106,8 +104,8 @@ class MatrixPower
* \brief Split #m_p into integral part and fractional part. * \brief Split #m_p into integral part and fractional part.
* *
* This method stores the integral part \f$ p_{\textrm int} \f$ into * This method stores the integral part \f$ p_{\textrm int} \f$ into
* #m_pint and the fractional part \f$ p_{\textrm frac} \f$ into * #m_pInt and the fractional part \f$ p_{\textrm frac} \f$ into
* #m_pfrac, where #m_pfrac is in the interval \f$ (-1,1) \f$. To * #m_pFrac, where #m_pFrac is in the interval \f$ (-1,1) \f$. To
* choose between the possibilities below, it considers the computation * choose between the possibilities below, it considers the computation
* of \f$ A^{p_1} \f$ and \f$ A^{p_2} \f$ and determines which of these * of \f$ A^{p_1} \f$ and \f$ A^{p_2} \f$ and determines which of these
* computations is the better conditioned. * computations is the better conditioned.
@ -115,10 +113,10 @@ class MatrixPower
void getFractionalExponent(); void getFractionalExponent();
/** \brief Compute atanh (inverse hyperbolic tangent) for \f$ y / x \f$. */ /** \brief Compute atanh (inverse hyperbolic tangent) for \f$ y / x \f$. */
ComplexScalar atanh2(const ComplexScalar& y, const ComplexScalar& x); static ComplexScalar atanh2(const ComplexScalar& y, const ComplexScalar& x);
/** \brief Compute power of 2x2 triangular matrix. */ /** \brief Compute power of 2x2 triangular matrix. */
void compute2x2(const RealScalar& p); void compute2x2(RealScalar p);
/** /**
* \brief Compute power of triangular matrices with size > 2. * \brief Compute power of triangular matrices with size > 2.
@ -159,16 +157,16 @@ class MatrixPower
void computeTmp(RealScalar); void computeTmp(RealScalar);
const MatrixType& m_A; ///< \brief Reference to the matrix base. const MatrixType& m_A; ///< \brief Reference to the matrix base.
const RealScalar& m_p; ///< \brief Reference to the real exponent. const RealScalar m_p; ///< \brief The real exponent.
const PlainObject& m_b; ///< \brief Reference to the multiplier. const PlainObject& m_b; ///< \brief Reference to the multiplier.
const Index m_dimA; ///< \brief The dimension of #m_A, equivalent to %m_A.cols(). const Index m_dimA; ///< \brief The dimension of #m_A, equivalent to %m_A.cols().
const Index m_dimb; ///< \brief The dimension of #m_b, equivalent to %m_b.cols(). const Index m_dimb; ///< \brief The dimension of #m_b, equivalent to %m_b.cols().
MatrixType m_tmp; ///< \brief Used for temporary storage. MatrixType m_tmp; ///< \brief Used for temporary storage.
RealScalar m_pint; ///< \brief Integer part of #m_p. RealScalar m_pInt; ///< \brief Integral part of #m_p.
RealScalar m_pfrac; ///< \brief Fractional part of #m_p. RealScalar m_pFrac; ///< \brief Fractional part of #m_p.
ComplexMatrix m_T; ///< \brief Triangular part of Schur decomposition. ComplexMatrix m_T; ///< \brief Triangular part of Schur decomposition.
ComplexMatrix m_U; ///< \brief Unitary part of Schur decomposition. ComplexMatrix m_U; ///< \brief Unitary part of Schur decomposition.
ComplexMatrix m_fT; ///< \brief #m_T to the power of #m_pfrac. ComplexMatrix m_fT; ///< \brief #m_T to the power of #m_pFrac.
ComplexArray m_logTdiag; ///< \brief Logarithm of the main diagonal of #m_T. ComplexArray m_logTdiag; ///< \brief Logarithm of the main diagonal of #m_T.
}; };
@ -176,8 +174,8 @@ class MatrixPower
* \internal \ingroup MatrixFunctions_Module * \internal \ingroup MatrixFunctions_Module
* \brief Partial specialization for integral exponents. * \brief Partial specialization for integral exponents.
*/ */
template <typename MatrixType, typename IntExponent, typename PlainObject> template <typename MatrixType, typename PlainObject>
class MatrixPower<MatrixType, IntExponent, PlainObject, 1> class MatrixPower<MatrixType, 1, PlainObject>
{ {
public: public:
/** /**
@ -187,7 +185,7 @@ class MatrixPower<MatrixType, IntExponent, PlainObject, 1>
* \param[in] p the exponent of the matrix power. * \param[in] p the exponent of the matrix power.
* \param[in] b the multiplier. * \param[in] b the multiplier.
*/ */
MatrixPower(const MatrixType& A, const IntExponent& p, const PlainObject& b) : MatrixPower(const MatrixType& A, int p, const PlainObject& b) :
m_A(A), m_A(A),
m_p(p), m_p(p),
m_b(b), m_b(b),
@ -213,7 +211,7 @@ class MatrixPower<MatrixType, IntExponent, PlainObject, 1>
typedef typename MatrixType::Index Index; typedef typename MatrixType::Index Index;
const MatrixType& m_A; ///< \brief Reference to the matrix base. const MatrixType& m_A; ///< \brief Reference to the matrix base.
const IntExponent& m_p; ///< \brief Reference to the real exponent. const int m_p; ///< \brief The integral exponent.
const PlainObject& m_b; ///< \brief Reference to the multiplier. const PlainObject& m_b; ///< \brief Reference to the multiplier.
const Index m_dimA; ///< \brief The dimension of #m_A, equivalent to %m_A.cols(). const Index m_dimA; ///< \brief The dimension of #m_A, equivalent to %m_A.cols().
const Index m_dimb; ///< \brief The dimension of #m_b, equivalent to %m_b.cols(). const Index m_dimb; ///< \brief The dimension of #m_b, equivalent to %m_b.cols().
@ -230,48 +228,51 @@ class MatrixPower<MatrixType, IntExponent, PlainObject, 1>
void computeChainProduct(ResultType& result); void computeChainProduct(ResultType& result);
/** \brief Compute the cost of binary powering. */ /** \brief Compute the cost of binary powering. */
int computeCost(const IntExponent& p); static int computeCost(int);
/** \brief Solve the linear system repetitively. */ /** \brief Solve the linear system repetitively. */
template <typename ResultType> template <typename ResultType>
void partialPivLuSolve(ResultType&, IntExponent); void partialPivLuSolve(ResultType&, int);
}; };
/******* Specialized for real exponents *******/ /******* Specialized for real exponents *******/
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
template <typename ResultType> template <typename ResultType>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::compute(ResultType& result) void MatrixPower<MatrixType,IsInteger,PlainObject>::compute(ResultType& result)
{ {
using std::abs;
using std::floor; using std::floor;
using std::pow; using std::pow;
m_pint = floor(m_p); m_pInt = floor(m_p + RealScalar(0.5));
m_pfrac = m_p - m_pint; m_pFrac = m_p - m_pInt;
if (m_pfrac == RealScalar(0)) if (!m_pFrac) {
computeIntPower(result); computeIntPower(result);
else if (m_dimA == 1) } else if (m_dimA == 1)
result = pow(m_A(0,0), m_p) * m_b; result = pow(m_A(0,0), m_p) * m_b;
else { else {
computeSchurDecomposition(); computeSchurDecomposition();
getFractionalExponent(); getFractionalExponent();
computeIntPower(result); computeIntPower(result);
if (m_dimA == 2) if (m_dimA == 2) {
compute2x2(m_pfrac); compute2x2(m_pFrac);
else } else {
computeBig(); computeBig();
}
computeTmp(Scalar()); computeTmp(Scalar());
result *= m_tmp; result = m_tmp * result;
} }
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
template <typename ResultType> template <typename ResultType>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeIntPower(ResultType& result) void MatrixPower<MatrixType,IsInteger,PlainObject>::computeIntPower(ResultType& result)
{ {
MatrixType tmp;
if (m_dimb > m_dimA) { if (m_dimb > m_dimA) {
MatrixType tmp = MatrixType::Identity(m_A.rows(), m_A.cols()); tmp = MatrixType::Identity(m_dimA, m_dimA);
computeChainProduct(tmp); computeChainProduct(tmp);
result = tmp * m_b; result = tmp * m_b;
} else { } else {
@ -280,18 +281,19 @@ void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeIntPower
} }
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
template <typename ResultType> template <typename ResultType>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeChainProduct(ResultType& result) void MatrixPower<MatrixType,IsInteger,PlainObject>::computeChainProduct(ResultType& result)
{ {
using std::abs;
using std::fmod;
using std::frexp; using std::frexp;
using std::ldexp; using std::ldexp;
const bool pIsNegative = m_pint < RealScalar(0); RealScalar p = abs(m_pInt);
RealScalar p = pIsNegative? -m_pint: m_pint;
int cost = computeCost(p); int cost = computeCost(p);
if (pIsNegative) { if (m_pInt < RealScalar(0)) {
if (p * m_dimb <= cost * m_dimA) { if (p * m_dimb <= cost * m_dimA) {
partialPivLuSolve(result, p); partialPivLuSolve(result, p);
return; return;
@ -314,12 +316,13 @@ void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeChainPro
result = m_tmp * result; result = m_tmp * result;
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeCost(RealScalar p) int MatrixPower<MatrixType,IsInteger,PlainObject>::computeCost(RealScalar p)
{ {
using std::frexp; using std::frexp;
using std::ldexp; using std::ldexp;
int cost, tmp; int cost, tmp;
frexp(p, &cost); frexp(p, &cost);
while (frexp(p, &tmp), tmp > 0) { while (frexp(p, &tmp), tmp > 0) {
p -= ldexp(RealScalar(0.5), tmp); p -= ldexp(RealScalar(0.5), tmp);
@ -328,61 +331,49 @@ int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeCost(Real
return cost; return cost;
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
template <typename ResultType> template <typename ResultType>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::partialPivLuSolve(ResultType& result, RealScalar p) void MatrixPower<MatrixType,IsInteger,PlainObject>::partialPivLuSolve(ResultType& result, RealScalar p)
{ {
const PartialPivLU<MatrixType> Asolver(m_A); const PartialPivLU<MatrixType> Asolver(m_A);
for (; p >= RealScalar(1); p--) for (; p >= RealScalar(1); p--)
result = Asolver.solve(result); result = Asolver.solve(result);
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeSchurDecomposition() void MatrixPower<MatrixType,IsInteger,PlainObject>::computeSchurDecomposition()
{ {
const ComplexSchur<MatrixType> schurOfA(m_A); const ComplexSchur<MatrixType> schurOfA(m_A);
m_T = schurOfA.matrixT(); m_T = schurOfA.matrixT();
m_U = schurOfA.matrixU(); m_U = schurOfA.matrixU();
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getFractionalExponent() void MatrixPower<MatrixType,IsInteger,PlainObject>::getFractionalExponent()
{ {
using std::pow; using std::pow;
typedef Array<RealScalar, Rows, 1, ColMajor, MaxRows> RealArray; typedef Array<RealScalar, Rows, 1, ColMajor, MaxRows> RealArray;
const ComplexArray Tdiag = m_T.diagonal(); const ComplexArray Tdiag = m_T.diagonal();
RealScalar maxAbsEival, minAbsEival, *begin, *end; const RealArray absTdiag = Tdiag.abs();
RealArray absTdiag; const RealScalar maxAbsEival = absTdiag.maxCoeff();
const RealScalar minAbsEival = absTdiag.minCoeff();
m_logTdiag = Tdiag.log(); m_logTdiag = Tdiag.log();
absTdiag = Tdiag.abs(); if (m_pFrac > RealScalar(0.5) && // This is just a shortcut.
maxAbsEival = minAbsEival = absTdiag[0]; m_pFrac > (RealScalar(1) - m_pFrac) * pow(maxAbsEival/minAbsEival, m_pFrac)) {
begin = absTdiag.data(); m_pFrac--;
end = begin + m_dimA; m_pInt++;
// This avoids traversing the array twice.
for (RealScalar *ptr = begin + 1; ptr < end; ptr++) {
if (*ptr > maxAbsEival)
maxAbsEival = *ptr;
else if (*ptr < minAbsEival)
minAbsEival = *ptr;
}
if (m_pfrac > RealScalar(0.5) && // This is just a shortcut.
m_pfrac > (RealScalar(1) - m_pfrac) * pow(maxAbsEival/minAbsEival, m_pfrac)) {
m_pfrac--;
m_pint++;
} }
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
std::complex<typename MatrixType::RealScalar> std::complex<typename MatrixType::RealScalar>
MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::atanh2(const ComplexScalar& y, const ComplexScalar& x) MatrixPower<MatrixType,IsInteger,PlainObject>::atanh2(const ComplexScalar& y, const ComplexScalar& x)
{ {
using std::abs; using std::abs;
using std::log; using std::log;
using std::sqrt; using std::sqrt;
const ComplexScalar z = y / x; const ComplexScalar z = y / x;
if (abs(z) > sqrt(NumTraits<RealScalar>::epsilon())) if (abs(z) > sqrt(NumTraits<RealScalar>::epsilon()))
@ -391,8 +382,8 @@ MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::atanh2(const Complex
return z + z*z*z / RealScalar(3); return z + z*z*z / RealScalar(3);
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::compute2x2(const RealScalar& p) void MatrixPower<MatrixType,IsInteger,PlainObject>::compute2x2(RealScalar p)
{ {
using std::abs; using std::abs;
using std::ceil; using std::ceil;
@ -407,7 +398,6 @@ void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::compute2x2(cons
ComplexScalar w; ComplexScalar w;
m_fT(0,0) = pow(m_T(0,0), p); m_fT(0,0) = pow(m_T(0,0), p);
for (j = 1; j < m_dimA; j++) { for (j = 1; j < m_dimA; j++) {
i = j - 1; i = j - 1;
m_fT(j,j) = pow(m_T(j,j), p); m_fT(j,j) = pow(m_T(j,j), p);
@ -426,8 +416,8 @@ void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::compute2x2(cons
} }
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeBig() void MatrixPower<MatrixType,IsInteger,PlainObject>::computeBig()
{ {
using std::ldexp; using std::ldexp;
const int digits = std::numeric_limits<RealScalar>::digits; const int digits = std::numeric_limits<RealScalar>::digits;
@ -441,7 +431,7 @@ void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeBig()
RealScalar normIminusT; RealScalar normIminusT;
while (true) { while (true) {
IminusT = ComplexMatrix::Identity(m_A.rows(), m_A.cols()) - T; IminusT = ComplexMatrix::Identity(m_dimA, m_dimA) - T;
normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff(); normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();
if (normIminusT < maxNormForPade) { if (normIminusT < maxNormForPade) {
degree = getPadeDegree(normIminusT); degree = getPadeDegree(normIminusT);
@ -457,14 +447,14 @@ void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeBig()
computePade(degree, IminusT); computePade(degree, IminusT);
for (; numberOfSquareRoots; numberOfSquareRoots--) { for (; numberOfSquareRoots; numberOfSquareRoots--) {
compute2x2(ldexp(m_pfrac, -numberOfSquareRoots)); compute2x2(ldexp(m_pFrac, -numberOfSquareRoots));
m_fT *= m_fT; m_fT *= m_fT;
} }
compute2x2(m_pfrac); compute2x2(m_pFrac);
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
inline int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getPadeDegree(float normIminusT) inline int MatrixPower<MatrixType,IsInteger,PlainObject>::getPadeDegree(float normIminusT)
{ {
const float maxNormForPade[] = { 2.7996156e-1f /* degree = 3 */ , 4.3268868e-1f }; const float maxNormForPade[] = { 2.7996156e-1f /* degree = 3 */ , 4.3268868e-1f };
int degree = 3; int degree = 3;
@ -474,8 +464,8 @@ inline int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getPadeDe
return degree; return degree;
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
inline int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getPadeDegree(double normIminusT) inline int MatrixPower<MatrixType,IsInteger,PlainObject>::getPadeDegree(double normIminusT)
{ {
const double maxNormForPade[] = { 1.882832775783710e-2 /* degree = 3 */ , 6.036100693089536e-2, const double maxNormForPade[] = { 1.882832775783710e-2 /* degree = 3 */ , 6.036100693089536e-2,
1.239372725584857e-1, 1.998030690604104e-1, 2.787629930861592e-1 }; 1.239372725584857e-1, 1.998030690604104e-1, 2.787629930861592e-1 };
@ -486,8 +476,8 @@ inline int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getPadeDe
return degree; return degree;
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
inline int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getPadeDegree(long double normIminusT) inline int MatrixPower<MatrixType,IsInteger,PlainObject>::getPadeDegree(long double normIminusT)
{ {
#if LDBL_MANT_DIG == 53 #if LDBL_MANT_DIG == 53
const int maxPadeDegree = 7; const int maxPadeDegree = 7;
@ -519,45 +509,46 @@ inline int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getPadeDe
break; break;
return degree; return degree;
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computePade(const int& degree, const ComplexMatrix& IminusT) void MatrixPower<MatrixType,IsInteger,PlainObject>::computePade(const int& degree, const ComplexMatrix& IminusT)
{ {
int i = degree << 1; int i = degree << 1;
m_fT = coeff(i) * IminusT; m_fT = coeff(i) * IminusT;
for (i--; i; i--) { for (i--; i; i--) {
m_fT = (ComplexMatrix::Identity(m_A.rows(), m_A.cols()) + m_fT).template triangularView<Upper>() m_fT = (ComplexMatrix::Identity(m_dimA, m_dimA) + m_fT).template triangularView<Upper>()
.solve(coeff(i) * IminusT).eval(); .solve(coeff(i) * IminusT).eval();
} }
m_fT += ComplexMatrix::Identity(m_A.rows(), m_A.cols()); m_fT += ComplexMatrix::Identity(m_dimA, m_dimA);
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
inline typename MatrixType::RealScalar MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::coeff(const int& i) inline typename MatrixType::RealScalar MatrixPower<MatrixType,IsInteger,PlainObject>::coeff(const int& i)
{ {
if (i == 1) if (i == 1)
return -m_pfrac; return -m_pFrac;
else if (i & 1) else if (i & 1)
return (-m_pfrac - RealScalar(i >> 1)) / RealScalar(i << 1); return (-m_pFrac - RealScalar(i >> 1)) / RealScalar(i << 1);
else else
return (m_pfrac - RealScalar(i >> 1)) / RealScalar(i-1 << 1); return (m_pFrac - RealScalar(i >> 1)) / RealScalar((i - 1) << 1);
} }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeTmp(RealScalar) void MatrixPower<MatrixType,IsInteger,PlainObject>::computeTmp(RealScalar)
{ m_tmp = (m_U * m_fT * m_U.adjoint()).real(); } { m_tmp = (m_U * m_fT * m_U.adjoint()).real(); }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger> template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeTmp(ComplexScalar) void MatrixPower<MatrixType,IsInteger,PlainObject>::computeTmp(ComplexScalar)
{ m_tmp = m_U * m_fT * m_U.adjoint(); } { m_tmp = m_U * m_fT * m_U.adjoint(); }
/******* Specialized for integral exponents *******/ /******* Specialized for integral exponents *******/
template <typename MatrixType, typename IntExponent, typename PlainObject> template <typename MatrixType, typename PlainObject>
template <typename ResultType> template <typename ResultType>
void MatrixPower<MatrixType,IntExponent,PlainObject,1>::compute(ResultType& result) void MatrixPower<MatrixType,1,PlainObject>::compute(ResultType& result)
{ {
MatrixType tmp;
if (m_dimb > m_dimA) { if (m_dimb > m_dimA) {
MatrixType tmp = MatrixType::Identity(m_dimA, m_dimA); tmp = MatrixType::Identity(m_dimA, m_dimA);
computeChainProduct(tmp); computeChainProduct(tmp);
result = tmp * m_b; result = tmp * m_b;
} else { } else {
@ -566,41 +557,43 @@ void MatrixPower<MatrixType,IntExponent,PlainObject,1>::compute(ResultType& resu
} }
} }
template <typename MatrixType, typename IntExponent, typename PlainObject> template <typename MatrixType, typename PlainObject>
int MatrixPower<MatrixType,IntExponent,PlainObject,1>::computeCost(const IntExponent& p) int MatrixPower<MatrixType,1,PlainObject>::computeCost(int p)
{ {
int cost = 0; int cost = 0, tmp;
IntExponent tmp = p; for (tmp = p; tmp; tmp >>= 1)
for (tmp = p >> 1; tmp; tmp >>= 1)
cost++; cost++;
for (tmp = IntExponent(1); tmp <= p; tmp <<= 1) for (tmp = 1; tmp <= p; tmp <<= 1)
if (tmp & p) cost++; if (tmp & p) cost++;
return cost; return cost;
} }
template <typename MatrixType, typename IntExponent, typename PlainObject> template <typename MatrixType, typename PlainObject>
template <typename ResultType> template <typename ResultType>
void MatrixPower<MatrixType,IntExponent,PlainObject,1>::partialPivLuSolve(ResultType& result, IntExponent p) void MatrixPower<MatrixType,1,PlainObject>::partialPivLuSolve(ResultType& result, int p)
{ {
const PartialPivLU<MatrixType> Asolver(m_A); const PartialPivLU<MatrixType> Asolver(m_A);
for(; p; p--) for(; p; p--)
result = Asolver.solve(result); result = Asolver.solve(result);
} }
template <typename MatrixType, typename IntExponent, typename PlainObject> template <typename MatrixType, typename PlainObject>
template <typename ResultType> template <typename ResultType>
void MatrixPower<MatrixType,IntExponent,PlainObject,1>::computeChainProduct(ResultType& result) void MatrixPower<MatrixType,1,PlainObject>::computeChainProduct(ResultType& result)
{ {
const bool pIsNegative = m_p < IntExponent(0); using std::abs;
IntExponent p = pIsNegative? -m_p: m_p; int p = abs(m_p), cost = computeCost(p);
int cost = computeCost(p);
if (pIsNegative) { if (m_p < 0) {
if (p * m_dimb <= cost * m_dimA) { if (p * m_dimb <= cost * m_dimA) {
partialPivLuSolve(result, p); partialPivLuSolve(result, p);
return; return;
} else { m_tmp = m_A.inverse(); } } else {
} else { m_tmp = m_A; } m_tmp = m_A.inverse();
}
} else {
m_tmp = m_A;
}
while (p * m_dimb > cost * m_dimA) { while (p * m_dimb > cost * m_dimA) {
if (p & 1) { if (p & 1) {
@ -658,9 +651,10 @@ template<typename MatrixType, typename ExponentType, typename Derived> class Mat
inline void evalTo(ResultType& result) const inline void evalTo(ResultType& result) const
{ {
typedef typename Derived::PlainObject PlainObject; typedef typename Derived::PlainObject PlainObject;
const int IsInteger = NumTraits<ExponentType>::IsInteger;
const typename MatrixType::PlainObject Aevaluated = m_A.eval(); const typename MatrixType::PlainObject Aevaluated = m_A.eval();
const PlainObject bevaluated = m_b.eval(); const PlainObject bevaluated = m_b.eval();
MatrixPower<MatrixType, ExponentType, PlainObject> mp(Aevaluated, m_p, bevaluated); MatrixPower<MatrixType, IsInteger, PlainObject> mp(Aevaluated, m_p, bevaluated);
mp.compute(result); mp.compute(result);
} }
@ -726,9 +720,10 @@ template<typename Derived, typename ExponentType> class MatrixPowerReturnValue
inline void evalTo(ResultType& result) const inline void evalTo(ResultType& result) const
{ {
typedef typename Derived::PlainObject PlainObject; typedef typename Derived::PlainObject PlainObject;
const int IsInteger = NumTraits<ExponentType>::IsInteger;
const PlainObject Aevaluated = m_A.eval(); const PlainObject Aevaluated = m_A.eval();
const PlainObject Identity = PlainObject::Identity(m_A.rows(), m_A.cols()); const PlainObject Identity = PlainObject::Identity(m_A.rows(), m_A.cols());
MatrixPower<PlainObject, ExponentType> mp(Aevaluated, m_p, Identity); MatrixPower<PlainObject, IsInteger> mp(Aevaluated, m_p, Identity);
mp.compute(result); mp.compute(result);
} }

152
unsupported/test/matrix_power.cpp
View File

@ -23,7 +23,7 @@ void test2dRotation(double tol)
B << c, s, -s, c; B << c, s, -s, c;
C = A.pow(std::ldexp(angle, 1) / M_PI); C = A.pow(std::ldexp(angle, 1) / M_PI);
std::cout << "test2dRotation: i = " << i << " error powerm = " << relerr(C, B) << "\n"; std::cout << "test2dRotation: i = " << i << " error powerm = " << relerr(C, B) << '\n';
VERIFY(C.isApprox(B, T(tol))); VERIFY(C.isApprox(B, T(tol)));
} }
} }
@ -43,44 +43,117 @@ void test2dHyperbolicRotation(double tol)
B << ch, ish, -ish, ch; B << ch, ish, -ish, ch;
C = A.pow(angle); C = A.pow(angle);
std::cout << "test2dHyperbolicRotation: i = " << i << " error powerm = " << relerr(C, B) << "\n"; std::cout << "test2dHyperbolicRotation: i = " << i << " error powerm = " << relerr(C, B) << '\n';
VERIFY(C.isApprox(B, T(tol))); VERIFY(C.isApprox(B, T(tol)));
} }
} }
template <typename MatrixType>
void testIntPowers(const MatrixType& m, double tol)
{
typedef typename MatrixType::RealScalar RealScalar;
const MatrixType m1 = MatrixType::Random(m.rows(), m.cols());
const MatrixType identity = MatrixType::Identity(m.rows(), m.cols());
const PartialPivLU<MatrixType> solver(m1);
MatrixType m2, m3, m4;
m3 = m1.pow(0);
m4 = m1.pow(0.);
std::cout << "testIntPower: i = 0 error powerm = " << relerr(identity, m3) << " " << relerr(identity, m4) << '\n';
VERIFY(identity == m3 && identity == m4);
m3 = m1.pow(1);
m4 = m1.pow(1.);
std::cout << "testIntPower: i = 1 error powerm = " << relerr(m1, m3) << " " << relerr(m1, m4) << '\n';
VERIFY(m1 == m3 && m1 == m4);
m2 = m1 * m1;
m3 = m1.pow(2);
m4 = m1.pow(2.);
std::cout << "testIntPower: i = 2 error powerm = " << relerr(m2, m3) << " " << relerr(m2, m4) << '\n';
VERIFY(m2.isApprox(m3, RealScalar(tol)) && m2.isApprox(m4, RealScalar(tol)));
for (int i = 3; i <= 20; i++) {
m2 *= m1;
m3 = m1.pow(i);
m4 = m1.pow(RealScalar(i));
std::cout << "testIntPower: i = " << i << " error powerm = " << relerr(m2, m3) << " " << relerr (m2, m4) << '\n';
VERIFY(m2.isApprox(m3, RealScalar(tol)) && m2.isApprox(m4, RealScalar(tol)));
}
m2 = solver.inverse();
m3 = m1.pow(-1);
m4 = m1.pow(-1.);
std::cout << "testIntPower: i = -1 error powerm = " << relerr(m2, m3) << " " << relerr (m2, m4) << '\n';
VERIFY(m2.isApprox(m3, RealScalar(tol)) && m2.isApprox(m4, RealScalar(tol)));
for (int i = -2; i >= -20; i--) {
m2 = solver.solve(m2);
m3 = m1.pow(i);
m4 = m1.pow(RealScalar(i));
std::cout << "testIntPower: i = " << i << " error powerm = " << relerr(m2, m3) << " " << relerr (m2, m4) << '\n';
VERIFY(m2.isApprox(m3, RealScalar(tol)) && m2.isApprox(m4, RealScalar(tol)));
}
}
template <typename MatrixType> template <typename MatrixType>
void testExponentLaws(const MatrixType& m, double tol) void testExponentLaws(const MatrixType& m, double tol)
{ {
typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar;
typedef typename NumTraits<Scalar>::Real RealScalar; MatrixType m1, m2, m3, m4, m5;
RealScalar x, y;
typename MatrixType::Index rows = m.rows(); for (int i = 0; i < g_repeat; i++) {
typename MatrixType::Index cols = m.cols();
MatrixType m1, m1x, m1y, m2, m3;
RealScalar x = internal::random<RealScalar>(), y = internal::random<RealScalar>();
double err[3];
for(int i = 0; i < g_repeat; i++) {
generateTestMatrix<MatrixType>::run(m1, m.rows()); generateTestMatrix<MatrixType>::run(m1, m.rows());
m1x = m1.pow(x); x = internal::random<RealScalar>();
m1y = m1.pow(y); y = internal::random<RealScalar>();
m2 = m1.pow(x);
m3 = m1.pow(y);
m2 = m1.pow(x + y); m4 = m1.pow(x + y);
m3 = m1x * m1y; m5 = m2 * m3;
err[0] = relerr(m2, m3); std::cout << "testExponentLaws: error powerm = " << relerr(m4, m5);
VERIFY(m2.isApprox(m3, static_cast<RealScalar>(tol))); VERIFY(m4.isApprox(m5, RealScalar(tol)));
m2 = m1.pow(x * y); if (!NumTraits<typename MatrixType::Scalar>::IsComplex) {
m3 = m1x.pow(y); m4 = m1.pow(x * y);
err[1] = relerr(m2, m3); m5 = m2.pow(y);
VERIFY(m2.isApprox(m3, static_cast<RealScalar>(tol))); std::cout << " " << relerr(m4, m5);
VERIFY(m4.isApprox(m5, RealScalar(tol)));
}
m2 = (std::abs(x) * m1).pow(y); m4 = (std::abs(x) * m1).pow(y);
m3 = std::pow(std::abs(x), y) * m1y; m5 = std::pow(std::abs(x), y) * m3;
err[2] = relerr(m2, m3); std::cout << " " << relerr(m4, m5) << '\n';
VERIFY(m2.isApprox(m3, static_cast<RealScalar>(tol))); VERIFY(m4.isApprox(m5, RealScalar(tol)));
}
}
std::cout << "testExponentLaws: error powerm = " << err[0] << " " << err[1] << " " << err[2] << "\n"; template <typename MatrixType, typename VectorType>
void testMatrixVectorProduct(const MatrixType& m, const VectorType& v, double tol)
{
typedef typename MatrixType::RealScalar RealScalar;
MatrixType m1;
VectorType v1, v2, v3;
RealScalar pReal;
signed char pInt;
for (int i = 0; i < g_repeat; i++) {
generateTestMatrix<MatrixType>::run(m1, m.rows());
v1 = VectorType::Random(v.rows(), v.cols());
pReal = internal::random<RealScalar>();
pInt = rand();
pInt >>= 2;
v2 = m1.pow(pReal).eval() * v1;
v3 = m1.pow(pReal) * v1;
std::cout << "testMatrixVectorProduct: error powerm = " << relerr(v2, v3);
VERIFY(v2.isApprox(v3, RealScalar(tol)));
v2 = m1.pow(pInt).eval() * v1;
v3 = m1.pow(pInt) * v1;
std::cout << " " << relerr(v2, v3) << '\n';
VERIFY(v2.isApprox(v3, RealScalar(tol)) || v2 == v3);
} }
} }
@ -88,17 +161,36 @@ void test_matrix_power()
{ {
CALL_SUBTEST_2(test2dRotation<double>(1e-13)); CALL_SUBTEST_2(test2dRotation<double>(1e-13));
CALL_SUBTEST_1(test2dRotation<float>(2e-5)); // was 1e-5, relaxed for clang 2.8 / linux / x86-64 CALL_SUBTEST_1(test2dRotation<float>(2e-5)); // was 1e-5, relaxed for clang 2.8 / linux / x86-64
CALL_SUBTEST_8(test2dRotation<long double>(1e-13)); CALL_SUBTEST_9(test2dRotation<long double>(1e-13));
CALL_SUBTEST_2(test2dHyperbolicRotation<double>(1e-14)); CALL_SUBTEST_2(test2dHyperbolicRotation<double>(1e-14));
CALL_SUBTEST_1(test2dHyperbolicRotation<float>(1e-5)); CALL_SUBTEST_1(test2dHyperbolicRotation<float>(1e-5));
CALL_SUBTEST_8(test2dHyperbolicRotation<long double>(1e-14)); CALL_SUBTEST_9(test2dHyperbolicRotation<long double>(1e-14));
CALL_SUBTEST_2(testIntPowers(Matrix2d(), 1e-13));
CALL_SUBTEST_7(testIntPowers(Matrix<double,3,3,RowMajor>(), 1e-13));
CALL_SUBTEST_3(testIntPowers(Matrix4cd(), 1e-13));
CALL_SUBTEST_4(testIntPowers(MatrixXd(8,8), 1e-13));
CALL_SUBTEST_1(testIntPowers(Matrix2f(), 1e-4));
CALL_SUBTEST_5(testIntPowers(Matrix3cf(), 1e-4));
CALL_SUBTEST_8(testIntPowers(Matrix4f(), 1e-4));
CALL_SUBTEST_6(testIntPowers(MatrixXf(8,8), 1e-4));
CALL_SUBTEST_2(testExponentLaws(Matrix2d(), 1e-13)); CALL_SUBTEST_2(testExponentLaws(Matrix2d(), 1e-13));
CALL_SUBTEST_7(testExponentLaws(Matrix<double,3,3,RowMajor>(), 1e-13)); CALL_SUBTEST_7(testExponentLaws(Matrix<double,3,3,RowMajor>(), 1e-13));
CALL_SUBTEST_3(testExponentLaws(Matrix4cd(), 1e-13)); CALL_SUBTEST_3(testExponentLaws(Matrix4cd(), 1e-13));
CALL_SUBTEST_4(testExponentLaws(MatrixXd(8,8), 1e-13)); CALL_SUBTEST_4(testExponentLaws(MatrixXd(8,8), 1e-13));
CALL_SUBTEST_1(testExponentLaws(Matrix2f(), 1e-4)); CALL_SUBTEST_1(testExponentLaws(Matrix2f(), 1e-4));
CALL_SUBTEST_5(testExponentLaws(Matrix3cf(), 1e-4)); CALL_SUBTEST_5(testExponentLaws(Matrix3cf(), 1e-4));
CALL_SUBTEST_1(testExponentLaws(Matrix4f(), 1e-4)); CALL_SUBTEST_8(testExponentLaws(Matrix4f(), 1e-4));
CALL_SUBTEST_6(testExponentLaws(MatrixXf(8,8), 1e-4)); CALL_SUBTEST_6(testExponentLaws(MatrixXf(8,8), 1e-4));
CALL_SUBTEST_9(testExponentLaws(Matrix<long double,Dynamic,Dynamic>(7,7), 1e-13));
CALL_SUBTEST_2(testMatrixVectorProduct(Matrix2d(), Vector2d(), 1e-13));
CALL_SUBTEST_7(testMatrixVectorProduct(Matrix<double,3,3,RowMajor>(), Vector3d(), 1e-13));
CALL_SUBTEST_3(testMatrixVectorProduct(Matrix4cd(), Vector4cd(), 1e-13));
CALL_SUBTEST_4(testMatrixVectorProduct(MatrixXd(8,8), MatrixXd(8,2), 1e-13));
CALL_SUBTEST_1(testMatrixVectorProduct(Matrix2f(), Vector2f(), 1e-4));
CALL_SUBTEST_5(testMatrixVectorProduct(Matrix3cf(), Vector3cf(), 1e-4));
CALL_SUBTEST_8(testMatrixVectorProduct(Matrix4f(), Vector4f(), 1e-4));
CALL_SUBTEST_6(testMatrixVectorProduct(MatrixXf(8,8), VectorXf(8), 1e-4));
CALL_SUBTEST_10(testMatrixVectorProduct(Matrix<long double,Dynamic,Dynamic>(7,7), Matrix<long double,7,9>(), 1e-13));
} }