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Fix a lot in MatrixPower.h

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This commit is contained in:
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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229
unsupported/Eigen/src/MatrixFunctions/MatrixPower.h
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@@ -21,14 +21,12 @@ namespace Eigen {
*
* \brief Class for computing matrix powers.
*
* \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.
* \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,
int IsInteger = NumTraits<ExponentType>::IsInteger>
template <typename MatrixType, int IsInteger, typename PlainObject = MatrixType>
class MatrixPower
{
private:
@@ -93,7 +91,7 @@ class MatrixPower
void computeChainProduct(ResultType&);
/** \brief Compute the cost of binary powering. */
int computeCost(RealScalar);
static int computeCost(RealScalar);
/** \brief Solve the linear system repetitively. */
template <typename ResultType>
@@ -106,8 +104,8 @@ class MatrixPower
* \brief Split #m_p into integral part and fractional part.
*
* 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_pfrac, where #m_pfrac is in the interval \f$ (-1,1) \f$. To
* #m_pInt and the fractional part \f$ p_{\textrm frac} \f$ into
* #m_pFrac, where #m_pFrac is in the interval \f$ (-1,1) \f$. To
* choose between the possibilities below, it considers the computation
* of \f$ A^{p_1} \f$ and \f$ A^{p_2} \f$ and determines which of these
* computations is the better conditioned.
@@ -115,10 +113,10 @@ class MatrixPower
void getFractionalExponent();
/** \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. */
void compute2x2(const RealScalar& p);
void compute2x2(RealScalar p);
/**
* \brief Compute power of triangular matrices with size > 2.
@@ -159,16 +157,16 @@ class MatrixPower
void computeTmp(RealScalar);
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 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().
MatrixType m_tmp; ///< \brief Used for temporary storage.
RealScalar m_pint; ///< \brief Integer part of #m_p.
RealScalar m_pfrac; ///< \brief Fractional part of #m_p.
RealScalar m_pInt; ///< \brief Integral part of #m_p.
RealScalar m_pFrac; ///< \brief Fractional part of #m_p.
ComplexMatrix m_T; ///< \brief Triangular 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.
};
@@ -176,8 +174,8 @@ class MatrixPower
* \internal \ingroup MatrixFunctions_Module
* \brief Partial specialization for integral exponents.
*/
template <typename MatrixType, typename IntExponent, typename PlainObject>
class MatrixPower<MatrixType, IntExponent, PlainObject, 1>
template <typename MatrixType, typename PlainObject>
class MatrixPower<MatrixType, 1, PlainObject>
{
public:
/**
@@ -187,7 +185,7 @@ class MatrixPower<MatrixType, IntExponent, PlainObject, 1>
* \param[in] p the exponent of the matrix power.
* \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_p(p),
m_b(b),
@@ -213,7 +211,7 @@ class MatrixPower<MatrixType, IntExponent, PlainObject, 1>
typedef typename MatrixType::Index Index;
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 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().
@@ -230,48 +228,51 @@ class MatrixPower<MatrixType, IntExponent, PlainObject, 1>
void computeChainProduct(ResultType& result);
/** \brief Compute the cost of binary powering. */
int computeCost(const IntExponent& p);
static int computeCost(int);
/** \brief Solve the linear system repetitively. */
template <typename ResultType>
void partialPivLuSolve(ResultType&, IntExponent);
void partialPivLuSolve(ResultType&, int);
};
/******* Specialized for real exponents *******/
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger>
template <typename MatrixType, int IsInteger, typename PlainObject>
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::pow;
m_pint = floor(m_p);
m_pfrac = m_p - m_pint;
m_pInt = floor(m_p + RealScalar(0.5));
m_pFrac = m_p - m_pInt;
if (m_pfrac == RealScalar(0))
if (!m_pFrac) {
computeIntPower(result);
else if (m_dimA == 1)
} else if (m_dimA == 1)
result = pow(m_A(0,0), m_p) * m_b;
else {
computeSchurDecomposition();
getFractionalExponent();
computeIntPower(result);
if (m_dimA == 2)
compute2x2(m_pfrac);
else
if (m_dimA == 2) {
compute2x2(m_pFrac);
} else {
computeBig();
}
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>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeIntPower(ResultType& result)
void MatrixPower<MatrixType,IsInteger,PlainObject>::computeIntPower(ResultType& result)
{
MatrixType tmp;
if (m_dimb > m_dimA) {
MatrixType tmp = MatrixType::Identity(m_A.rows(), m_A.cols());
tmp = MatrixType::Identity(m_dimA, m_dimA);
computeChainProduct(tmp);
result = tmp * m_b;
} 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>
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::ldexp;
const bool pIsNegative = m_pint < RealScalar(0);
RealScalar p = pIsNegative? -m_pint: m_pint;
RealScalar p = abs(m_pInt);
int cost = computeCost(p);
if (pIsNegative) {
if (m_pInt < RealScalar(0)) {
if (p * m_dimb <= cost * m_dimA) {
partialPivLuSolve(result, p);
return;
@@ -314,12 +316,13 @@ void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeChainPro
result = m_tmp * result;
}
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger>
int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeCost(RealScalar p)
template <typename MatrixType, int IsInteger, typename PlainObject>
int MatrixPower<MatrixType,IsInteger,PlainObject>::computeCost(RealScalar p)
{
using std::frexp;
using std::ldexp;
int cost, tmp;
frexp(p, &cost);
while (frexp(p, &tmp), tmp > 0) {
p -= ldexp(RealScalar(0.5), tmp);
@@ -328,61 +331,49 @@ int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeCost(Real
return cost;
}
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger>
template <typename MatrixType, int IsInteger, typename PlainObject>
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);
for (; p >= RealScalar(1); p--)
result = Asolver.solve(result);
}
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeSchurDecomposition()
template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,IsInteger,PlainObject>::computeSchurDecomposition()
{
const ComplexSchur<MatrixType> schurOfA(m_A);
m_T = schurOfA.matrixT();
m_U = schurOfA.matrixU();
}
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getFractionalExponent()
template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,IsInteger,PlainObject>::getFractionalExponent()
{
using std::pow;
typedef Array<RealScalar, Rows, 1, ColMajor, MaxRows> RealArray;
const ComplexArray Tdiag = m_T.diagonal();
RealScalar maxAbsEival, minAbsEival, *begin, *end;
RealArray absTdiag;
const RealArray absTdiag = Tdiag.abs();
const RealScalar maxAbsEival = absTdiag.maxCoeff();
const RealScalar minAbsEival = absTdiag.minCoeff();
m_logTdiag = Tdiag.log();
absTdiag = Tdiag.abs();
maxAbsEival = minAbsEival = absTdiag[0];
begin = absTdiag.data();
end = begin + m_dimA;
// 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++;
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>
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::log;
using std::sqrt;
const ComplexScalar z = y / x;
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);
}
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::compute2x2(const RealScalar& p)
template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,IsInteger,PlainObject>::compute2x2(RealScalar p)
{
using std::abs;
using std::ceil;
@@ -407,7 +398,6 @@ void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::compute2x2(cons
ComplexScalar w;
m_fT(0,0) = pow(m_T(0,0), p);
for (j = 1; j < m_dimA; j++) {
i = j - 1;
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>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeBig()
template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,IsInteger,PlainObject>::computeBig()
{
using std::ldexp;
const int digits = std::numeric_limits<RealScalar>::digits;
@@ -441,7 +431,7 @@ void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeBig()
RealScalar normIminusT;
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();
if (normIminusT < maxNormForPade) {
degree = getPadeDegree(normIminusT);
@@ -457,14 +447,14 @@ void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeBig()
computePade(degree, IminusT);
for (; numberOfSquareRoots; numberOfSquareRoots--) {
compute2x2(ldexp(m_pfrac, -numberOfSquareRoots));
compute2x2(ldexp(m_pFrac, -numberOfSquareRoots));
m_fT *= m_fT;
}
compute2x2(m_pfrac);
compute2x2(m_pFrac);
}
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger>
inline int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getPadeDegree(float normIminusT)
template <typename MatrixType, int IsInteger, typename PlainObject>
inline int MatrixPower<MatrixType,IsInteger,PlainObject>::getPadeDegree(float normIminusT)
{
const float maxNormForPade[] = { 2.7996156e-1f /* degree = 3 */ , 4.3268868e-1f };
int degree = 3;
@@ -474,8 +464,8 @@ inline int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getPadeDe
return degree;
}
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger>
inline int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getPadeDegree(double normIminusT)
template <typename MatrixType, int IsInteger, typename PlainObject>
inline int MatrixPower<MatrixType,IsInteger,PlainObject>::getPadeDegree(double normIminusT)
{
const double maxNormForPade[] = { 1.882832775783710e-2 /* degree = 3 */ , 6.036100693089536e-2,
1.239372725584857e-1, 1.998030690604104e-1, 2.787629930861592e-1 };
@@ -486,8 +476,8 @@ inline int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getPadeDe
return degree;
}
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger>
inline int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getPadeDegree(long double normIminusT)
template <typename MatrixType, int IsInteger, typename PlainObject>
inline int MatrixPower<MatrixType,IsInteger,PlainObject>::getPadeDegree(long double normIminusT)
{
#if LDBL_MANT_DIG == 53
const int maxPadeDegree = 7;
@@ -519,45 +509,46 @@ inline int MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::getPadeDe
break;
return degree;
}
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computePade(const int& degree, const ComplexMatrix& IminusT)
template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,IsInteger,PlainObject>::computePade(const int& degree, const ComplexMatrix& IminusT)
{
int i = degree << 1;
m_fT = coeff(i) * IminusT;
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();
}
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>
inline typename MatrixType::RealScalar MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::coeff(const int& i)
template <typename MatrixType, int IsInteger, typename PlainObject>
inline typename MatrixType::RealScalar MatrixPower<MatrixType,IsInteger,PlainObject>::coeff(const int& i)
{
if (i == 1)
return -m_pfrac;
return -m_pFrac;
else if (i & 1)
return (-m_pfrac - RealScalar(i >> 1)) / RealScalar(i << 1);
return (-m_pFrac - RealScalar(i >> 1)) / RealScalar(i << 1);
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>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeTmp(RealScalar)
template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,IsInteger,PlainObject>::computeTmp(RealScalar)
{ m_tmp = (m_U * m_fT * m_U.adjoint()).real(); }
template <typename MatrixType, typename ExponentType, typename PlainObject, int IsInteger>
void MatrixPower<MatrixType,ExponentType,PlainObject,IsInteger>::computeTmp(ComplexScalar)
template <typename MatrixType, int IsInteger, typename PlainObject>
void MatrixPower<MatrixType,IsInteger,PlainObject>::computeTmp(ComplexScalar)
{ m_tmp = m_U * m_fT * m_U.adjoint(); }
/******* Specialized for integral exponents *******/
template <typename MatrixType, typename IntExponent, typename PlainObject>
template <typename MatrixType, typename PlainObject>
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) {
MatrixType tmp = MatrixType::Identity(m_dimA, m_dimA);
tmp = MatrixType::Identity(m_dimA, m_dimA);
computeChainProduct(tmp);
result = tmp * m_b;
} else {
@@ -566,41 +557,43 @@ void MatrixPower<MatrixType,IntExponent,PlainObject,1>::compute(ResultType& resu
}
}
template <typename MatrixType, typename IntExponent, typename PlainObject>
int MatrixPower<MatrixType,IntExponent,PlainObject,1>::computeCost(const IntExponent& p)
template <typename MatrixType, typename PlainObject>
int MatrixPower<MatrixType,1,PlainObject>::computeCost(int p)
{
int cost = 0;
IntExponent tmp = p;
for (tmp = p >> 1; tmp; tmp >>= 1)
int cost = 0, tmp;
for (tmp = p; tmp; tmp >>= 1)
cost++;
for (tmp = IntExponent(1); tmp <= p; tmp <<= 1)
for (tmp = 1; tmp <= p; tmp <<= 1)
if (tmp & p) cost++;
return cost;
}
template <typename MatrixType, typename IntExponent, typename PlainObject>
template <typename MatrixType, typename PlainObject>
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);
for(; p; p--)
result = Asolver.solve(result);
}
template <typename MatrixType, typename IntExponent, typename PlainObject>
template <typename MatrixType, typename PlainObject>
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);
IntExponent p = pIsNegative? -m_p: m_p;
int cost = computeCost(p);
using std::abs;
int p = abs(m_p), cost = computeCost(p);
if (pIsNegative) {
if (m_p < 0) {
if (p * m_dimb <= cost * m_dimA) {
partialPivLuSolve(result, p);
return;
} else { m_tmp = m_A.inverse(); }
} else { m_tmp = m_A; }
} else {
m_tmp = m_A.inverse();
}
} else {
m_tmp = m_A;
}
while (p * m_dimb > cost * m_dimA) {
if (p & 1) {
@@ -658,9 +651,10 @@ template<typename MatrixType, typename ExponentType, typename Derived> class Mat
inline void evalTo(ResultType& result) const
{
typedef typename Derived::PlainObject PlainObject;
const int IsInteger = NumTraits<ExponentType>::IsInteger;
const typename MatrixType::PlainObject Aevaluated = m_A.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);
}
@@ -726,9 +720,10 @@ template<typename Derived, typename ExponentType> class MatrixPowerReturnValue
inline void evalTo(ResultType& result) const
{
typedef typename Derived::PlainObject PlainObject;
const int IsInteger = NumTraits<ExponentType>::IsInteger;
const PlainObject Aevaluated = m_A.eval();
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);
}