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* merge with mainline

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* adapt Eigenvalues module to the new rule that the RowMajorBit must have the proper value for vectors
* Fix RowMajorBit in ei_traits<ProductBase>
* Fix vectorizability logic in CoeffBasedProduct
This commit is contained in:
Benoit Jacob
2010-04-16 11:25:50 -04:00
parent ff6a46105d ea1a2df370
commit 0ab431d7b8
62 changed files with 3207 additions and 807 deletions
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2
unsupported/Eigen/CMakeLists.txt
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@@ -1,4 +1,4 @@
set(Eigen_HEADERS AdolcForward BVH IterativeSolvers MatrixFunctions MoreVectorization AutoDiff AlignedVector3)
set(Eigen_HEADERS AdolcForward BVH IterativeSolvers MatrixFunctions MoreVectorization AutoDiff AlignedVector3 Polynomials)
install(FILES
${Eigen_HEADERS}

2
unsupported/Eigen/FFT
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@@ -320,7 +320,7 @@ class FFT
// if the vector is strided, then we need to copy it to a packed temporary
Matrix<src_type,1,Dynamic> tmp;
if ( resize_input ) {
size_t ncopy = min(src.size(),src.size() + resize_input);
size_t ncopy = std::min(src.size(),src.size() + resize_input);
tmp.setZero(src.size() + resize_input);
if ( realfft && HasFlag(HalfSpectrum) ) {
// pad at the Nyquist bin

213
unsupported/Eigen/MatrixFunctions
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@@ -40,6 +40,22 @@ namespace Eigen {
* \brief This module aims to provide various methods for the computation of
* matrix functions.
*
* To use this module, add
* \code
* #include <unsupported/Eigen/MatrixFunctions>
* \endcode
* at the start of your source file.
*
* This module defines the following MatrixBase methods.
* - \ref matrixbase_cos "MatrixBase::cos()", for computing the matrix cosine
* - \ref matrixbase_cosh "MatrixBase::cosh()", for computing the matrix hyperbolic cosine
* - \ref matrixbase_exp "MatrixBase::exp()", for computing the matrix exponential
* - \ref matrixbase_matrixfunction "MatrixBase::matrixFunction()", for computing general matrix functions
* - \ref matrixbase_sin "MatrixBase::sin()", for computing the matrix sine
* - \ref matrixbase_sinh "MatrixBase::sinh()", for computing the matrix hyperbolic sine
*
* These methods are the main entry points to this module.
*
* %Matrix functions are defined as follows. Suppose that \f$ f \f$
* is an entire function (that is, a function on the complex plane
* that is everywhere complex differentiable). Then its Taylor
@@ -49,16 +65,205 @@ namespace Eigen {
* function by the same series:
* \f[ f(M) = f(0) + f'(0) M + \frac{f''(0)}{2} M^2 + \frac{f'''(0)}{3!} M^3 + \cdots \f]
*
* \code
* #include <unsupported/Eigen/MatrixFunctions>
* \endcode
*/
#include "src/MatrixFunctions/MatrixExponential.h"
#include "src/MatrixFunctions/MatrixFunction.h"
}
/**
\page matrixbaseextra MatrixBase methods defined in the MatrixFunctions module
\ingroup MatrixFunctions_Module
The remainder of the page documents the following MatrixBase methods
which are defined in the MatrixFunctions module.
\section matrixbase_cos MatrixBase::cos()
Compute the matrix cosine.
\code
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cos() const
\endcode
\param[in] M a square matrix.
\returns expression representing \f$ \cos(M) \f$.
This function calls \ref matrixbase_matrixfunction "matrixFunction()" with StdStemFunctions::cos().
\sa \ref matrixbase_sin "sin()" for an example.
\section matrixbase_cosh MatrixBase::cosh()
Compute the matrix hyberbolic cosine.
\code
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cosh() const
\endcode
\param[in] M a square matrix.
\returns expression representing \f$ \cosh(M) \f$
This function calls \ref matrixbase_matrixfunction "matrixFunction()" with StdStemFunctions::cosh().
\sa \ref matrixbase_sinh "sinh()" for an example.
\section matrixbase_exp MatrixBase::exp()
Compute the matrix exponential.
\code
const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const
\endcode
\param[in] M matrix whose exponential is to be computed.
\returns expression representing the matrix exponential of \p M.
The matrix exponential of \f$ M \f$ is defined by
\f[ \exp(M) = \sum_{k=0}^\infty \frac{M^k}{k!}. \f]
The matrix exponential can be used to solve linear ordinary
differential equations: the solution of \f$ y' = My \f$ with the
initial condition \f$ y(0) = y_0 \f$ is given by
\f$ y(t) = \exp(M) y_0 \f$.
The cost of the computation is approximately \f$ 20 n^3 \f$ for
matrices of size \f$ n \f$. The number 20 depends weakly on the
norm of the matrix.
The matrix exponential is computed using the scaling-and-squaring
method combined with Pad&eacute; approximation. The matrix is first
rescaled, then the exponential of the reduced matrix is computed
approximant, and then the rescaling is undone by repeated
squaring. The degree of the Pad&eacute; approximant is chosen such
that the approximation error is less than the round-off
error. However, errors may accumulate during the squaring phase.
Details of the algorithm can be found in: Nicholas J. Higham, "The
scaling and squaring method for the matrix exponential revisited,"
<em>SIAM J. %Matrix Anal. Applic.</em>, <b>26</b>:1179&ndash;1193,
2005.
Example: The following program checks that
\f[ \exp \left[ \begin{array}{ccc}
0 & \frac14\pi & 0 \\
-\frac14\pi & 0 & 0 \\
0 & 0 & 0
\end{array} \right] = \left[ \begin{array}{ccc}
\frac12\sqrt2 & -\frac12\sqrt2 & 0 \\
\frac12\sqrt2 & \frac12\sqrt2 & 0 \\
0 & 0 & 1
\end{array} \right]. \f]
This corresponds to a rotation of \f$ \frac14\pi \f$ radians around
the z-axis.
\include MatrixExponential.cpp
Output: \verbinclude MatrixExponential.out
\note \p M has to be a matrix of \c float, \c double,
\c complex<float> or \c complex<double> .
\section matrixbase_matrixfunction MatrixBase::matrixFunction()
Compute a matrix function.
\code
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::matrixFunction(typename ei_stem_function<typename ei_traits<Derived>::Scalar>::type f) const
\endcode
\param[in] M argument of matrix function, should be a square matrix.
\param[in] f an entire function; \c f(x,n) should compute the n-th
derivative of f at x.
\returns expression representing \p f applied to \p M.
Suppose that \p M is a matrix whose entries have type \c Scalar.
Then, the second argument, \p f, should be a function with prototype
\code
ComplexScalar f(ComplexScalar, int)
\endcode
where \c ComplexScalar = \c std::complex<Scalar> if \c Scalar is
real (e.g., \c float or \c double) and \c ComplexScalar =
\c Scalar if \c Scalar is complex. The return value of \c f(x,n)
should be \f$ f^{(n)}(x) \f$, the n-th derivative of f at x.
This routine uses the algorithm described in:
Philip Davies and Nicholas J. Higham,
"A Schur-Parlett algorithm for computing matrix functions",
<em>SIAM J. %Matrix Anal. Applic.</em>, <b>25</b>:464&ndash;485, 2003.
The actual work is done by the MatrixFunction class.
Example: The following program checks that
\f[ \exp \left[ \begin{array}{ccc}
0 & \frac14\pi & 0 \\
-\frac14\pi & 0 & 0 \\
0 & 0 & 0
\end{array} \right] = \left[ \begin{array}{ccc}
\frac12\sqrt2 & -\frac12\sqrt2 & 0 \\
\frac12\sqrt2 & \frac12\sqrt2 & 0 \\
0 & 0 & 1
\end{array} \right]. \f]
This corresponds to a rotation of \f$ \frac14\pi \f$ radians around
the z-axis. This is the same example as used in the documentation
of \ref matrixbase_exp "exp()".
\include MatrixFunction.cpp
Output: \verbinclude MatrixFunction.out
Note that the function \c expfn is defined for complex numbers
\c x, even though the matrix \c A is over the reals. Instead of
\c expfn, we could also have used StdStemFunctions::exp:
\code
A.matrixFunction(StdStemFunctions<std::complex<double> >::exp, &B);
\endcode
\section matrixbase_sin MatrixBase::sin()
Compute the matrix sine.
\code
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sin() const
\endcode
\param[in] M a square matrix.
\returns expression representing \f$ \sin(M) \f$.
This function calls \ref matrixbase_matrixfunction "matrixFunction()" with StdStemFunctions::sin().
Example: \include MatrixSine.cpp
Output: \verbinclude MatrixSine.out
\section matrixbase_sinh const MatrixBase::sinh()
Compute the matrix hyperbolic sine.
\code
MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sinh() const
\endcode
\param[in] M a square matrix.
\returns expression representing \f$ \sinh(M) \f$
This function calls \ref matrixbase_matrixfunction "matrixFunction()" with StdStemFunctions::sinh().
Example: \include MatrixSinh.cpp
Output: \verbinclude MatrixSinh.out
*/
}
#endif // EIGEN_MATRIX_FUNCTIONS

137
unsupported/Eigen/Polynomials Normal file
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@@ -0,0 +1,137 @@
#ifndef EIGEN_POLYNOMIALS_MODULE_H
#define EIGEN_POLYNOMIALS_MODULE_H
#include <Eigen/Core>
#include <Eigen/src/Core/util/DisableMSVCWarnings.h>
#include <Eigen/QR>
// Note that EIGEN_HIDE_HEAVY_CODE has to be defined per module
#if (defined EIGEN_EXTERN_INSTANTIATIONS) && (EIGEN_EXTERN_INSTANTIATIONS>=2)
#ifndef EIGEN_HIDE_HEAVY_CODE
#define EIGEN_HIDE_HEAVY_CODE
#endif
#elif defined EIGEN_HIDE_HEAVY_CODE
#undef EIGEN_HIDE_HEAVY_CODE
#endif
namespace Eigen {
/** \ingroup Unsupported_modules
* \defgroup Polynomials_Module Polynomials module
*
* \nonstableyet
*
* \brief This module provides a QR based polynomial solver.
*
* To use this module, add
* \code
* #include <unsupported/Eigen/Polynomials>
* \endcode
* at the start of your source file.
*/
#include "src/Polynomials/PolynomialUtils.h"
#include "src/Polynomials/Companion.h"
#include "src/Polynomials/PolynomialSolver.h"
/**
\page polynomials Polynomials defines functions for dealing with polynomials
and a QR based polynomial solver.
\ingroup Polynomials_Module
The remainder of the page documents first the functions for evaluating, computing
polynomials, computing estimates about polynomials and next the QR based polynomial
solver.
\section polynomialUtils convenient functions to deal with polynomials
\subsection roots_to_monicPolynomial
The function
\code
void roots_to_monicPolynomial( const RootVector& rv, Polynomial& poly )
\endcode
computes the coefficients \f$ a_i \f$ of
\f$ p(x) = a_0 + a_{1}x + ... + a_{n-1}x^{n-1} + x^n \f$
where \f$ p \f$ is known through its roots i.e. \f$ p(x) = (x-r_1)(x-r_2)...(x-r_n) \f$.
\subsection poly_eval
The function
\code
T poly_eval( const Polynomials& poly, const T& x )
\endcode
evaluates a polynomial at a given point using stabilized H&ouml;rner method.
The following code: first computes the coefficients in the monomial basis of the monic polynomial that has the provided roots;
then, it evaluates the computed polynomial, using a stabilized H&ouml;rner method.
\include PolynomialUtils1.cpp
Output: \verbinclude PolynomialUtils1.out
\subsection Cauchy bounds
The function
\code
Real cauchy_max_bound( const Polynomial& poly )
\endcode
provides a maximum bound (the Cauchy one: \f$C(p)\f$) for the absolute value of a root of the given polynomial i.e.
\f$ \forall r_i \f$ root of \f$ p(x) = \sum_{k=0}^d a_k x^k \f$,
\f$ |r_i| \le C(p) = \sum_{k=0}^{d} \left | \frac{a_k}{a_d} \right | \f$
The leading coefficient \f$ p \f$: should be non zero \f$a_d \neq 0\f$.
The function
\code
Real cauchy_min_bound( const Polynomial& poly )
\endcode
provides a minimum bound (the Cauchy one: \f$c(p)\f$) for the absolute value of a non zero root of the given polynomial i.e.
\f$ \forall r_i \neq 0 \f$ root of \f$ p(x) = \sum_{k=0}^d a_k x^k \f$,
\f$ |r_i| \ge c(p) = \left( \sum_{k=0}^{d} \left | \frac{a_k}{a_0} \right | \right)^{-1} \f$
\section QR polynomial solver class
Computes the complex roots of a polynomial by computing the eigenvalues of the associated companion matrix with the QR algorithm.
The roots of \f$ p(x) = a_0 + a_1 x + a_2 x^2 + a_{3} x^3 + x^4 \f$ are the eigenvalues of
\f$
\left [
\begin{array}{cccc}
0 & 0 & 0 & a_0 \\
1 & 0 & 0 & a_1 \\
0 & 1 & 0 & a_2 \\
0 & 0 & 1 & a_3
\end{array} \right ]
\f$
However, the QR algorithm is not guaranteed to converge when there are several eigenvalues with same modulus.
Therefore the current polynomial solver is guaranteed to provide a correct result only when the complex roots \f$r_1,r_2,...,r_d\f$ have distinct moduli i.e.
\f$ \forall i,j \in [1;d],~ \| r_i \| \neq \| r_j \| \f$.
With 32bit (float) floating types this problem shows up frequently.
However, almost always, correct accuracy is reached even in these cases for 64bit
(double) floating types and small polynomial degree (<20).
\include PolynomialSolver1.cpp
In the above example:
-# a simple use of the polynomial solver is shown;
-# the accuracy problem with the QR algorithm is presented: a polynomial with almost conjugate roots is provided to the solver.
Those roots have almost same module therefore the QR algorithm failed to converge: the accuracy
of the last root is bad;
-# a simple way to circumvent the problem is shown: use doubles instead of floats.
Output: \verbinclude PolynomialSolver1.out
*/
} // namespace Eigen
#include <Eigen/src/Core/util/EnableMSVCWarnings.h>
#endif // EIGEN_POLYNOMIALS_MODULE_H
/* vim: set filetype=cpp et sw=2 ts=2 ai: */

1
unsupported/Eigen/src/CMakeLists.txt
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@@ -5,3 +5,4 @@ ADD_SUBDIRECTORY(MoreVectorization)
# ADD_SUBDIRECTORY(FFT)
# ADD_SUBDIRECTORY(Skyline)
ADD_SUBDIRECTORY(MatrixFunctions)
ADD_SUBDIRECTORY(Polynomials)

50
unsupported/Eigen/src/MatrixFunctions/MatrixExponential.h
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@@ -330,56 +330,6 @@ struct ei_traits<MatrixExponentialReturnValue<Derived> >
typedef typename Derived::PlainObject ReturnType;
};
/** \ingroup MatrixFunctions_Module
*
* \brief Compute the matrix exponential.
*
* \param[in] M matrix whose exponential is to be computed.
* \returns expression representing the matrix exponential of \p M.
*
* The matrix exponential of \f$ M \f$ is defined by
* \f[ \exp(M) = \sum_{k=0}^\infty \frac{M^k}{k!}. \f]
* The matrix exponential can be used to solve linear ordinary
* differential equations: the solution of \f$ y' = My \f$ with the
* initial condition \f$ y(0) = y_0 \f$ is given by
* \f$ y(t) = \exp(M) y_0 \f$.
*
* The cost of the computation is approximately \f$ 20 n^3 \f$ for
* matrices of size \f$ n \f$. The number 20 depends weakly on the
* norm of the matrix.
*
* The matrix exponential is computed using the scaling-and-squaring
* method combined with Pad&eacute; approximation. The matrix is first
* rescaled, then the exponential of the reduced matrix is computed
* approximant, and then the rescaling is undone by repeated
* squaring. The degree of the Pad&eacute; approximant is chosen such
* that the approximation error is less than the round-off
* error. However, errors may accumulate during the squaring phase.
*
* Details of the algorithm can be found in: Nicholas J. Higham, "The
* scaling and squaring method for the matrix exponential revisited,"
* <em>SIAM J. %Matrix Anal. Applic.</em>, <b>26</b>:1179&ndash;1193,
* 2005.
*
* Example: The following program checks that
* \f[ \exp \left[ \begin{array}{ccc}
* 0 & \frac14\pi & 0 \\
* -\frac14\pi & 0 & 0 \\
* 0 & 0 & 0
* \end{array} \right] = \left[ \begin{array}{ccc}
* \frac12\sqrt2 & -\frac12\sqrt2 & 0 \\
* \frac12\sqrt2 & \frac12\sqrt2 & 0 \\
* 0 & 0 & 1
* \end{array} \right]. \f]
* This corresponds to a rotation of \f$ \frac14\pi \f$ radians around
* the z-axis.
*
* \include MatrixExponential.cpp
* Output: \verbinclude MatrixExponential.out
*
* \note \p M has to be a matrix of \c float, \c double,
* \c complex<float> or \c complex<double> .
*/
template <typename Derived>
const MatrixExponentialReturnValue<Derived> MatrixBase<Derived>::exp() const
{

96
unsupported/Eigen/src/MatrixFunctions/MatrixFunction.h
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@@ -536,56 +536,6 @@ struct ei_traits<MatrixFunctionReturnValue<Derived> >
/********** MatrixBase methods **********/
/** \ingroup MatrixFunctions_Module
*
* \brief Compute a matrix function.
*
* \param[in] M argument of matrix function, should be a square matrix.
* \param[in] f an entire function; \c f(x,n) should compute the n-th
* derivative of f at x.
* \returns expression representing \p f applied to \p M.
*
* Suppose that \p M is a matrix whose entries have type \c Scalar.
* Then, the second argument, \p f, should be a function with prototype
* \code
* ComplexScalar f(ComplexScalar, int)
* \endcode
* where \c ComplexScalar = \c std::complex<Scalar> if \c Scalar is
* real (e.g., \c float or \c double) and \c ComplexScalar =
* \c Scalar if \c Scalar is complex. The return value of \c f(x,n)
* should be \f$ f^{(n)}(x) \f$, the n-th derivative of f at x.
*
* This routine uses the algorithm described in:
* Philip Davies and Nicholas J. Higham,
* "A Schur-Parlett algorithm for computing matrix functions",
* <em>SIAM J. %Matrix Anal. Applic.</em>, <b>25</b>:464&ndash;485, 2003.
*
* The actual work is done by the MatrixFunction class.
*
* Example: The following program checks that
* \f[ \exp \left[ \begin{array}{ccc}
* 0 & \frac14\pi & 0 \\
* -\frac14\pi & 0 & 0 \\
* 0 & 0 & 0
* \end{array} \right] = \left[ \begin{array}{ccc}
* \frac12\sqrt2 & -\frac12\sqrt2 & 0 \\
* \frac12\sqrt2 & \frac12\sqrt2 & 0 \\
* 0 & 0 & 1
* \end{array} \right]. \f]
* This corresponds to a rotation of \f$ \frac14\pi \f$ radians around
* the z-axis. This is the same example as used in the documentation
* of MatrixBase::exp().
*
* \include MatrixFunction.cpp
* Output: \verbinclude MatrixFunction.out
*
* Note that the function \c expfn is defined for complex numbers
* \c x, even though the matrix \c A is over the reals. Instead of
* \c expfn, we could also have used StdStemFunctions::exp:
* \code
* A.matrixFunction(StdStemFunctions<std::complex<double> >::exp, &B);
* \endcode
*/
template <typename Derived>
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::matrixFunction(typename ei_stem_function<typename ei_traits<Derived>::Scalar>::type f) const
{
@@ -593,18 +543,6 @@ const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::matrixFunction(typ
return MatrixFunctionReturnValue<Derived>(derived(), f);
}
/** \ingroup MatrixFunctions_Module
*
* \brief Compute the matrix sine.
*
* \param[in] M a square matrix.
* \returns expression representing \f$ \sin(M) \f$.
*
* This function calls matrixFunction() with StdStemFunctions::sin().
*
* \include MatrixSine.cpp
* Output: \verbinclude MatrixSine.out
*/
template <typename Derived>
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sin() const
{
@@ -613,17 +551,6 @@ const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sin() const
return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::sin);
}
/** \ingroup MatrixFunctions_Module
*
* \brief Compute the matrix cosine.
*
* \param[in] M a square matrix.
* \returns expression representing \f$ \cos(M) \f$.
*
* This function calls matrixFunction() with StdStemFunctions::cos().
*
* \sa ei_matrix_sin() for an example.
*/
template <typename Derived>
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cos() const
{
@@ -632,18 +559,6 @@ const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cos() const
return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::cos);
}
/** \ingroup MatrixFunctions_Module
*
* \brief Compute the matrix hyperbolic sine.
*
* \param[in] M a square matrix.
* \returns expression representing \f$ \sinh(M) \f$
*
* This function calls matrixFunction() with StdStemFunctions::sinh().
*
* \include MatrixSinh.cpp
* Output: \verbinclude MatrixSinh.out
*/
template <typename Derived>
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sinh() const
{
@@ -652,17 +567,6 @@ const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sinh() const
return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::sinh);
}
/** \ingroup MatrixFunctions_Module
*
* \brief Compute the matrix hyberbolic cosine.
*
* \param[in] M a square matrix.
* \returns expression representing \f$ \cosh(M) \f$
*
* This function calls matrixFunction() with StdStemFunctions::cosh().
*
* \sa ei_matrix_sinh() for an example.
*/
template <typename Derived>
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cosh() const
{

6
unsupported/Eigen/src/Polynomials/CMakeLists.txt Normal file
View File

@@ -0,0 +1,6 @@
FILE(GLOB Eigen_Polynomials_SRCS "*.h")
INSTALL(FILES
${Eigen_Polynomials_SRCS}
DESTINATION ${INCLUDE_INSTALL_DIR}/unsupported/Eigen/src/Polynomials COMPONENT Devel
)

281
unsupported/Eigen/src/Polynomials/Companion.h Normal file
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@@ -0,0 +1,281 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.
#ifndef EIGEN_COMPANION_H
#define EIGEN_COMPANION_H
// This file requires the user to include
// * Eigen/Core
// * Eigen/src/PolynomialSolver.h
#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename T>
T ei_radix(){ return 2; }
template <typename T>
T ei_radix2(){ return ei_radix<T>()*ei_radix<T>(); }
template<int Size>
struct ei_decrement_if_fixed_size
{
enum {
ret = (Size == Dynamic) ? Dynamic : Size-1 };
};
#endif
template< typename _Scalar, int _Deg >
class ei_companion
{
public:
EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_Deg==Dynamic ? Dynamic : _Deg)
enum {
Deg = _Deg,
Deg_1=ei_decrement_if_fixed_size<Deg>::ret
};
typedef _Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef Matrix<Scalar, Deg, 1> RightColumn;
//typedef DiagonalMatrix< Scalar, Deg_1, Deg_1 > BottomLeftDiagonal;
typedef Matrix<Scalar, Deg_1, 1> BottomLeftDiagonal;
typedef Matrix<Scalar, Deg, Deg> DenseCompanionMatrixType;
typedef Matrix< Scalar, _Deg, Deg_1 > LeftBlock;
typedef Matrix< Scalar, Deg_1, Deg_1 > BottomLeftBlock;
typedef Matrix< Scalar, 1, Deg_1 > LeftBlockFirstRow;
public:
EIGEN_STRONG_INLINE const _Scalar operator()( int row, int col ) const
{
if( m_bl_diag.rows() > col )
{
if( 0 < row ){ return m_bl_diag[col]; }
else{ return 0; }
}
else{ return m_monic[row]; }
}
public:
template<typename VectorType>
void setPolynomial( const VectorType& poly )
{
const int deg = poly.size()-1;
m_monic = -1/poly[deg] * poly.head(deg);
//m_bl_diag.setIdentity( deg-1 );
m_bl_diag.setOnes(deg-1);
}
template<typename VectorType>
ei_companion( const VectorType& poly ){
setPolynomial( poly ); }
public:
DenseCompanionMatrixType denseMatrix() const
{
const int deg = m_monic.size();
const int deg_1 = deg-1;
DenseCompanionMatrixType companion(deg,deg);
companion <<
( LeftBlock(deg,deg_1)
<< LeftBlockFirstRow::Zero(1,deg_1),
BottomLeftBlock::Identity(deg-1,deg-1)*m_bl_diag.asDiagonal() ).finished()
, m_monic;
return companion;
}
protected:
/** Helper function for the balancing algorithm.
* \returns true if the row and the column, having colNorm and rowNorm
* as norms, are balanced, false otherwise.
* colB and rowB are repectively the multipliers for
* the column and the row in order to balance them.
* */
bool balanced( Scalar colNorm, Scalar rowNorm,
bool& isBalanced, Scalar& colB, Scalar& rowB );
/** Helper function for the balancing algorithm.
* \returns true if the row and the column, having colNorm and rowNorm
* as norms, are balanced, false otherwise.
* colB and rowB are repectively the multipliers for
* the column and the row in order to balance them.
* */
bool balancedR( Scalar colNorm, Scalar rowNorm,
bool& isBalanced, Scalar& colB, Scalar& rowB );
public:
/**
* Balancing algorithm from B. N. PARLETT and C. REINSCH (1969)
* "Balancing a matrix for calculation of eigenvalues and eigenvectors"
* adapted to the case of companion matrices.
* A matrix with non zero row and non zero column is balanced
* for a certain norm if the i-th row and the i-th column
* have same norm for all i.
*/
void balance();
protected:
RightColumn m_monic;
BottomLeftDiagonal m_bl_diag;
};
template< typename _Scalar, int _Deg >
inline
bool ei_companion<_Scalar,_Deg>::balanced( Scalar colNorm, Scalar rowNorm,
bool& isBalanced, Scalar& colB, Scalar& rowB )
{
if( Scalar(0) == colNorm || Scalar(0) == rowNorm ){ return true; }
else
{
//To find the balancing coefficients, if the radix is 2,
//one finds \f$ \sigma \f$ such that
// \f$ 2^{2\sigma-1} < rowNorm / colNorm \le 2^{2\sigma+1} \f$
// then the balancing coefficient for the row is \f$ 1/2^{\sigma} \f$
// and the balancing coefficient for the column is \f$ 2^{\sigma} \f$
rowB = rowNorm / ei_radix<Scalar>();
colB = Scalar(1);
const Scalar s = colNorm + rowNorm;
while (colNorm < rowB)
{
colB *= ei_radix<Scalar>();
colNorm *= ei_radix2<Scalar>();
}
rowB = rowNorm * ei_radix<Scalar>();
while (colNorm >= rowB)
{
colB /= ei_radix<Scalar>();
colNorm /= ei_radix2<Scalar>();
}
//This line is used to avoid insubstantial balancing
if ((rowNorm + colNorm) < Scalar(0.95) * s * colB)
{
isBalanced = false;
rowB = Scalar(1) / colB;
return false;
}
else{
return true; }
}
}
template< typename _Scalar, int _Deg >
inline
bool ei_companion<_Scalar,_Deg>::balancedR( Scalar colNorm, Scalar rowNorm,
bool& isBalanced, Scalar& colB, Scalar& rowB )
{
if( Scalar(0) == colNorm || Scalar(0) == rowNorm ){ return true; }
else
{
/**
* Set the norm of the column and the row to the geometric mean
* of the row and column norm
*/
const _Scalar q = colNorm/rowNorm;
if( !ei_isApprox( q, _Scalar(1) ) )
{
rowB = ei_sqrt( colNorm/rowNorm );
colB = Scalar(1)/rowB;
isBalanced = false;
return false;
}
else{
return true; }
}
}
template< typename _Scalar, int _Deg >
void ei_companion<_Scalar,_Deg>::balance()
{
EIGEN_STATIC_ASSERT( 1 < Deg, YOU_MADE_A_PROGRAMMING_MISTAKE );
const int deg = m_monic.size();
const int deg_1 = deg-1;
bool hasConverged=false;
while( !hasConverged )
{
hasConverged = true;
Scalar colNorm,rowNorm;
Scalar colB,rowB;
//First row, first column excluding the diagonal
//==============================================
colNorm = ei_abs(m_bl_diag[0]);
rowNorm = ei_abs(m_monic[0]);
//Compute balancing of the row and the column
if( !balanced( colNorm, rowNorm, hasConverged, colB, rowB ) )
{
m_bl_diag[0] *= colB;
m_monic[0] *= rowB;
}
//Middle rows and columns excluding the diagonal
//==============================================
for( int i=1; i<deg_1; ++i )
{
// column norm, excluding the diagonal
colNorm = ei_abs(m_bl_diag[i]);
// row norm, excluding the diagonal
rowNorm = ei_abs(m_bl_diag[i-1]) + ei_abs(m_monic[i]);
//Compute balancing of the row and the column
if( !balanced( colNorm, rowNorm, hasConverged, colB, rowB ) )
{
m_bl_diag[i] *= colB;
m_bl_diag[i-1] *= rowB;
m_monic[i] *= rowB;
}
}
//Last row, last column excluding the diagonal
//============================================
const int ebl = m_bl_diag.size()-1;
VectorBlock<RightColumn,Deg_1> headMonic( m_monic, 0, deg_1 );
colNorm = headMonic.array().abs().sum();
rowNorm = ei_abs( m_bl_diag[ebl] );
//Compute balancing of the row and the column
if( !balanced( colNorm, rowNorm, hasConverged, colB, rowB ) )
{
headMonic *= colB;
m_bl_diag[ebl] *= rowB;
}
}
}
#endif // EIGEN_COMPANION_H

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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.
#ifndef EIGEN_POLYNOMIAL_SOLVER_H
#define EIGEN_POLYNOMIAL_SOLVER_H
/** \ingroup Polynomials_Module
* \class PolynomialSolverBase.
*
* \brief Defined to be inherited by polynomial solvers: it provides
* convenient methods such as
* - real roots,
* - greatest, smallest complex roots,
* - real roots with greatest, smallest absolute real value,
* - greatest, smallest real roots.
*
* It stores the set of roots as a vector of complexes.
*
*/
template< typename _Scalar, int _Deg >
class PolynomialSolverBase
{
public:
EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_Deg==Dynamic ? Dynamic : _Deg)
typedef _Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef std::complex<RealScalar> RootType;
typedef Matrix<RootType,_Deg,1> RootsType;
protected:
template< typename OtherPolynomial >
inline void setPolynomial( const OtherPolynomial& poly ){
m_roots.resize(poly.size()); }
public:
template< typename OtherPolynomial >
inline PolynomialSolverBase( const OtherPolynomial& poly ){
setPolynomial( poly() ); }
inline PolynomialSolverBase(){}
public:
/** \returns the complex roots of the polynomial */
inline const RootsType& roots() const { return m_roots; }
public:
/** Clear and fills the back insertion sequence with the real roots of the polynomial
* i.e. the real part of the complex roots that have an imaginary part which
* absolute value is smaller than absImaginaryThreshold.
* absImaginaryThreshold takes the dummy_precision associated
* with the _Scalar template parameter of the PolynomialSolver class as the default value.
*
* \param[out] bi_seq : the back insertion sequence (stl concept)
* \param[in] absImaginaryThreshold : the maximum bound of the imaginary part of a complex
* number that is considered as real.
* */
template<typename Stl_back_insertion_sequence>
inline void realRoots( Stl_back_insertion_sequence& bi_seq,
const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
{
bi_seq.clear();
for( int i=0; i<m_roots.size(); ++i )
{
if( ei_abs( m_roots[i].imag() ) < absImaginaryThreshold ){
bi_seq.push_back( m_roots[i].real() ); }
}
}
protected:
template<typename squaredNormBinaryPredicate>
inline const RootType& selectComplexRoot_withRespectToNorm( squaredNormBinaryPredicate& pred ) const
{
int res=0;
RealScalar norm2 = ei_abs2( m_roots[0] );
for( int i=1; i<m_roots.size(); ++i )
{
const RealScalar currNorm2 = ei_abs2( m_roots[i] );
if( pred( currNorm2, norm2 ) ){
res=i; norm2=currNorm2; }
}
return m_roots[res];
}
public:
/**
* \returns the complex root with greatest norm.
*/
inline const RootType& greatestRoot() const
{
std::greater<Scalar> greater;
return selectComplexRoot_withRespectToNorm( greater );
}
/**
* \returns the complex root with smallest norm.
*/
inline const RootType& smallestRoot() const
{
std::less<Scalar> less;
return selectComplexRoot_withRespectToNorm( less );
}
protected:
template<typename squaredRealPartBinaryPredicate>
inline const RealScalar& selectRealRoot_withRespectToAbsRealPart(
squaredRealPartBinaryPredicate& pred,
bool& hasArealRoot,
const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
{
hasArealRoot = false;
int res=0;
RealScalar abs2;
for( int i=0; i<m_roots.size(); ++i )
{
if( ei_abs( m_roots[i].imag() ) < absImaginaryThreshold )
{
if( !hasArealRoot )
{
hasArealRoot = true;
res = i;
abs2 = m_roots[i].real() * m_roots[i].real();
}
else
{
const RealScalar currAbs2 = m_roots[i].real() * m_roots[i].real();
if( pred( currAbs2, abs2 ) )
{
abs2 = currAbs2;
res = i;
}
}
}
else
{
if( ei_abs( m_roots[i].imag() ) < ei_abs( m_roots[res].imag() ) ){
res = i; }
}
}
return m_roots[res].real();
}
template<typename RealPartBinaryPredicate>
inline const RealScalar& selectRealRoot_withRespectToRealPart(
RealPartBinaryPredicate& pred,
bool& hasArealRoot,
const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
{
hasArealRoot = false;
int res=0;
RealScalar val;
for( int i=0; i<m_roots.size(); ++i )
{
if( ei_abs( m_roots[i].imag() ) < absImaginaryThreshold )
{
if( !hasArealRoot )
{
hasArealRoot = true;
res = i;
val = m_roots[i].real();
}
else
{
const RealScalar curr = m_roots[i].real();
if( pred( curr, val ) )
{
val = curr;
res = i;
}
}
}
else
{
if( ei_abs( m_roots[i].imag() ) < ei_abs( m_roots[res].imag() ) ){
res = i; }
}
}
return m_roots[res].real();
}
public:
/**
* \returns a real root with greatest absolute magnitude.
* A real root is defined as the real part of a complex root with absolute imaginary
* part smallest than absImaginaryThreshold.
* absImaginaryThreshold takes the dummy_precision associated
* with the _Scalar template parameter of the PolynomialSolver class as the default value.
* If no real root is found the boolean hasArealRoot is set to false and the real part of
* the root with smallest absolute imaginary part is returned instead.
*
* \param[out] hasArealRoot : boolean true if a real root is found according to the
* absImaginaryThreshold criterion, false otherwise.
* \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
* whether or not a root is real.
*/
inline const RealScalar& absGreatestRealRoot(
bool& hasArealRoot,
const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
{
std::greater<Scalar> greater;
return selectRealRoot_withRespectToAbsRealPart( greater, hasArealRoot, absImaginaryThreshold );
}
/**
* \returns a real root with smallest absolute magnitude.
* A real root is defined as the real part of a complex root with absolute imaginary
* part smallest than absImaginaryThreshold.
* absImaginaryThreshold takes the dummy_precision associated
* with the _Scalar template parameter of the PolynomialSolver class as the default value.
* If no real root is found the boolean hasArealRoot is set to false and the real part of
* the root with smallest absolute imaginary part is returned instead.
*
* \param[out] hasArealRoot : boolean true if a real root is found according to the
* absImaginaryThreshold criterion, false otherwise.
* \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
* whether or not a root is real.
*/
inline const RealScalar& absSmallestRealRoot(
bool& hasArealRoot,
const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
{
std::less<Scalar> less;
return selectRealRoot_withRespectToAbsRealPart( less, hasArealRoot, absImaginaryThreshold );
}
/**
* \returns the real root with greatest value.
* A real root is defined as the real part of a complex root with absolute imaginary
* part smallest than absImaginaryThreshold.
* absImaginaryThreshold takes the dummy_precision associated
* with the _Scalar template parameter of the PolynomialSolver class as the default value.
* If no real root is found the boolean hasArealRoot is set to false and the real part of
* the root with smallest absolute imaginary part is returned instead.
*
* \param[out] hasArealRoot : boolean true if a real root is found according to the
* absImaginaryThreshold criterion, false otherwise.
* \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
* whether or not a root is real.
*/
inline const RealScalar& greatestRealRoot(
bool& hasArealRoot,
const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
{
std::greater<Scalar> greater;
return selectRealRoot_withRespectToRealPart( greater, hasArealRoot, absImaginaryThreshold );
}
/**
* \returns the real root with smallest value.
* A real root is defined as the real part of a complex root with absolute imaginary
* part smallest than absImaginaryThreshold.
* absImaginaryThreshold takes the dummy_precision associated
* with the _Scalar template parameter of the PolynomialSolver class as the default value.
* If no real root is found the boolean hasArealRoot is set to false and the real part of
* the root with smallest absolute imaginary part is returned instead.
*
* \param[out] hasArealRoot : boolean true if a real root is found according to the
* absImaginaryThreshold criterion, false otherwise.
* \param[in] absImaginaryThreshold : threshold on the absolute imaginary part to decide
* whether or not a root is real.
*/
inline const RealScalar& smallestRealRoot(
bool& hasArealRoot,
const RealScalar& absImaginaryThreshold = NumTraits<Scalar>::dummy_precision() ) const
{
std::less<Scalar> less;
return selectRealRoot_withRespectToRealPart( less, hasArealRoot, absImaginaryThreshold );
}
protected:
RootsType m_roots;
};
#define EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES( BASE ) \
typedef typename BASE::Scalar Scalar; \
typedef typename BASE::RealScalar RealScalar; \
typedef typename BASE::RootType RootType; \
typedef typename BASE::RootsType RootsType;
/** \ingroup Polynomials_Module
*
* \class PolynomialSolver
*
* \brief A polynomial solver
*
* Computes the complex roots of a real polynomial.
*
* \param _Scalar the scalar type, i.e., the type of the polynomial coefficients
* \param _Deg the degree of the polynomial, can be a compile time value or Dynamic.
* Notice that the number of polynomial coefficients is _Deg+1.
*
* This class implements a polynomial solver and provides convenient methods such as
* - real roots,
* - greatest, smallest complex roots,
* - real roots with greatest, smallest absolute real value.
* - greatest, smallest real roots.
*
* WARNING: this polynomial solver is experimental, part of the unsuported Eigen modules.
*
*
* Currently a QR algorithm is used to compute the eigenvalues of the companion matrix of
* the polynomial to compute its roots.
* This supposes that the complex moduli of the roots are all distinct: e.g. there should
* be no multiple roots or conjugate roots for instance.
* With 32bit (float) floating types this problem shows up frequently.
* However, almost always, correct accuracy is reached even in these cases for 64bit
* (double) floating types and small polynomial degree (<20).
*/
template< typename _Scalar, int _Deg >
class PolynomialSolver : public PolynomialSolverBase<_Scalar,_Deg>
{
public:
EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,_Deg==Dynamic ? Dynamic : _Deg)
typedef PolynomialSolverBase<_Scalar,_Deg> PS_Base;
EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES( PS_Base )
typedef Matrix<Scalar,_Deg,_Deg> CompanionMatrixType;
typedef EigenSolver<CompanionMatrixType> EigenSolverType;
public:
/** Computes the complex roots of a new polynomial. */
template< typename OtherPolynomial >
void compute( const OtherPolynomial& poly )
{
assert( Scalar(0) != poly[poly.size()-1] );
ei_companion<Scalar,_Deg> companion( poly );
companion.balance();
m_eigenSolver.compute( companion.denseMatrix() );
m_roots = m_eigenSolver.eigenvalues();
}
public:
template< typename OtherPolynomial >
inline PolynomialSolver( const OtherPolynomial& poly ){
compute( poly ); }
inline PolynomialSolver(){}
protected:
using PS_Base::m_roots;
EigenSolverType m_eigenSolver;
};
template< typename _Scalar >
class PolynomialSolver<_Scalar,1> : public PolynomialSolverBase<_Scalar,1>
{
public:
typedef PolynomialSolverBase<_Scalar,1> PS_Base;
EIGEN_POLYNOMIAL_SOLVER_BASE_INHERITED_TYPES( PS_Base )
public:
/** Computes the complex roots of a new polynomial. */
template< typename OtherPolynomial >
void compute( const OtherPolynomial& poly )
{
assert( Scalar(0) != poly[poly.size()-1] );
m_roots[0] = -poly[0]/poly[poly.size()-1];
}
protected:
using PS_Base::m_roots;
};
#endif // EIGEN_POLYNOMIAL_SOLVER_H

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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.
#ifndef EIGEN_POLYNOMIAL_UTILS_H
#define EIGEN_POLYNOMIAL_UTILS_H
/** \ingroup Polynomials_Module
* \returns the evaluation of the polynomial at x using Horner algorithm.
*
* \param[in] poly : the vector of coefficients of the polynomial ordered
* by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
* e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$.
* \param[in] x : the value to evaluate the polynomial at.
*
* <i><b>Note for stability:</b></i>
* <dd> \f$ |x| \le 1 \f$ </dd>
*/
template <typename Polynomials, typename T>
inline
T poly_eval_horner( const Polynomials& poly, const T& x )
{
T val=poly[poly.size()-1];
for( int i=poly.size()-2; i>=0; --i ){
val = val*x + poly[i]; }
return val;
}
/** \ingroup Polynomials_Module
* \returns the evaluation of the polynomial at x using stabilized Horner algorithm.
*
* \param[in] poly : the vector of coefficients of the polynomial ordered
* by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
* e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$.
* \param[in] x : the value to evaluate the polynomial at.
*/
template <typename Polynomials, typename T>
inline
T poly_eval( const Polynomials& poly, const T& x )
{
typedef typename NumTraits<T>::Real Real;
if( ei_abs2( x ) <= Real(1) ){
return poly_eval_horner( poly, x ); }
else
{
T val=poly[0];
T inv_x = T(1)/x;
for( int i=1; i<poly.size(); ++i ){
val = val*inv_x + poly[i]; }
return std::pow(x,(T)(poly.size()-1)) * val;
}
}
/** \ingroup Polynomials_Module
* \returns a maximum bound for the absolute value of any root of the polynomial.
*
* \param[in] poly : the vector of coefficients of the polynomial ordered
* by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
* e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$.
*
* <i><b>Precondition:</b></i>
* <dd> the leading coefficient of the input polynomial poly must be non zero </dd>
*/
template <typename Polynomial>
inline
typename NumTraits<typename Polynomial::Scalar>::Real cauchy_max_bound( const Polynomial& poly )
{
typedef typename Polynomial::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real Real;
assert( Scalar(0) != poly[poly.size()-1] );
const Scalar inv_leading_coeff = Scalar(1)/poly[poly.size()-1];
Real cb(0);
for( int i=0; i<poly.size()-1; ++i ){
cb += ei_abs(poly[i]*inv_leading_coeff); }
return cb + Real(1);
}
/** \ingroup Polynomials_Module
* \returns a minimum bound for the absolute value of any non zero root of the polynomial.
* \param[in] poly : the vector of coefficients of the polynomial ordered
* by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
* e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$.
*/
template <typename Polynomial>
inline
typename NumTraits<typename Polynomial::Scalar>::Real cauchy_min_bound( const Polynomial& poly )
{
typedef typename Polynomial::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real Real;
int i=0;
while( i<poly.size()-1 && Scalar(0) == poly(i) ){ ++i; }
if( poly.size()-1 == i ){
return Real(1); }
const Scalar inv_min_coeff = Scalar(1)/poly[i];
Real cb(1);
for( int j=i+1; j<poly.size(); ++j ){
cb += ei_abs(poly[j]*inv_min_coeff); }
return Real(1)/cb;
}
/** \ingroup Polynomials_Module
* Given the roots of a polynomial compute the coefficients in the
* monomial basis of the monic polynomial with same roots and minimal degree.
* If RootVector is a vector of complexes, Polynomial should also be a vector
* of complexes.
* \param[in] rv : a vector containing the roots of a polynomial.
* \param[out] poly : the vector of coefficients of the polynomial ordered
* by degrees i.e. poly[i] is the coefficient of degree i of the polynomial
* e.g. \f$ 3 + x^2 \f$ is stored as a vector \f$ [ 3, 0, 1 ] \f$.
*/
template <typename RootVector, typename Polynomial>
void roots_to_monicPolynomial( const RootVector& rv, Polynomial& poly )
{
typedef typename Polynomial::Scalar Scalar;
poly.setZero( rv.size()+1 );
poly[0] = -rv[0]; poly[1] = Scalar(1);
for( int i=1; i<(int)rv.size(); ++i )
{
for( int j=i+1; j>0; --j ){ poly[j] = poly[j-1] - rv[i]*poly[j]; }
poly[0] = -rv[i]*poly[0];
}
}
#endif // EIGEN_POLYNOMIAL_UTILS_H

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#include <unsupported/Eigen/Polynomials>
#include <vector>
#include <iostream>
using namespace Eigen;
using namespace std;
int main()
{
typedef Matrix<double,5,1> Vector5d;
Vector5d roots = Vector5d::Random();
cout << "Roots: " << roots.transpose() << endl;
Eigen::Matrix<double,6,1> polynomial;
roots_to_monicPolynomial( roots, polynomial );
PolynomialSolver<double,5> psolve( polynomial );
cout << "Complex roots: " << psolve.roots().transpose() << endl;
std::vector<double> realRoots;
psolve.realRoots( realRoots );
Map<Vector5d> mapRR( &realRoots[0] );
cout << "Real roots: " << mapRR.transpose() << endl;
cout << endl;
cout << "Illustration of the convergence problem with the QR algorithm: " << endl;
cout << "---------------------------------------------------------------" << endl;
Eigen::Matrix<float,7,1> hardCase_polynomial;
hardCase_polynomial <<
-0.957, 0.9219, 0.3516, 0.9453, -0.4023, -0.5508, -0.03125;
cout << "Hard case polynomial defined by floats: " << hardCase_polynomial.transpose() << endl;
PolynomialSolver<float,6> psolvef( hardCase_polynomial );
cout << "Complex roots: " << psolvef.roots().transpose() << endl;
Eigen::Matrix<float,6,1> evals;
for( int i=0; i<6; ++i ){ evals[i] = std::abs( poly_eval( hardCase_polynomial, psolvef.roots()[i] ) ); }
cout << "Norms of the evaluations of the polynomial at the roots: " << evals.transpose() << endl << endl;
cout << "Using double's almost always solves the problem for small degrees: " << endl;
cout << "-------------------------------------------------------------------" << endl;
PolynomialSolver<double,6> psolve6d( hardCase_polynomial.cast<double>() );
cout << "Complex roots: " << psolve6d.roots().transpose() << endl;
for( int i=0; i<6; ++i )
{
std::complex<float> castedRoot( psolve6d.roots()[i].real(), psolve6d.roots()[i].imag() );
evals[i] = std::abs( poly_eval( hardCase_polynomial, castedRoot ) );
}
cout << "Norms of the evaluations of the polynomial at the roots: " << evals.transpose() << endl << endl;
cout.precision(10);
cout << "The last root in float then in double: " << psolvef.roots()[5] << "\t" << psolve6d.roots()[5] << endl;
std::complex<float> castedRoot( psolve6d.roots()[5].real(), psolve6d.roots()[5].imag() );
cout << "Norm of the difference: " << ei_abs( psolvef.roots()[5] - castedRoot ) << endl;
}

20
unsupported/doc/examples/PolynomialUtils1.cpp Normal file
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@@ -0,0 +1,20 @@
#include <unsupported/Eigen/Polynomials>
#include <iostream>
using namespace Eigen;
using namespace std;
int main()
{
Vector4d roots = Vector4d::Random();
cout << "Roots: " << roots.transpose() << endl;
Eigen::Matrix<double,5,1> polynomial;
roots_to_monicPolynomial( roots, polynomial );
cout << "Polynomial: ";
for( int i=0; i<4; ++i ){ cout << polynomial[i] << ".x^" << i << "+ "; }
cout << polynomial[4] << ".x^4" << endl;
Vector4d evaluation;
for( int i=0; i<4; ++i ){
evaluation[i] = poly_eval( polynomial, roots[i] ); }
cout << "Evaluation of the polynomial at the roots: " << evaluation.transpose();
}

16
unsupported/test/CMakeLists.txt
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@@ -24,3 +24,19 @@ if(FFTW_FOUND)
ei_add_test(FFTW "-DEIGEN_FFTW_DEFAULT " "-lfftw3 -lfftw3f -lfftw3l" )
endif(FFTW_FOUND)
find_package(GSL)
if(GSL_FOUND AND GSL_VERSION_MINOR LESS 9)
set(GSL_FOUND "")
endif(GSL_FOUND AND GSL_VERSION_MINOR LESS 9)
if(GSL_FOUND)
add_definitions("-DHAS_GSL" ${GSL_DEFINITIONS})
include_directories(${GSL_INCLUDE_DIR})
ei_add_property(EIGEN_TESTED_BACKENDS "GSL, ")
else(GSL_FOUND)
ei_add_property(EIGEN_MISSING_BACKENDS "GSL, ")
set(GSL_LIBRARIES " ")
endif(GSL_FOUND)
ei_add_test(polynomialutils)
ei_add_test(polynomialsolver " " "${GSL_LIBRARIES}" )

263
unsupported/test/polynomialsolver.cpp Normal file
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@@ -0,0 +1,263 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.
#include "main.h"
#include <unsupported/Eigen/Polynomials>
#include <iostream>
#include <algorithm>
#ifdef HAS_GSL
#include "gsl_helper.h"
#endif
using namespace std;
template<int Size>
struct ei_increment_if_fixed_size
{
enum {
ret = (Size == Dynamic) ? Dynamic : Size+1
};
};
template<int Deg, typename POLYNOMIAL, typename SOLVER>
bool aux_evalSolver( const POLYNOMIAL& pols, SOLVER& psolve )
{
typedef typename POLYNOMIAL::Scalar Scalar;
typedef typename SOLVER::RootsType RootsType;
typedef Matrix<Scalar,Deg,1> EvalRootsType;
const int deg = pols.size()-1;
psolve.compute( pols );
const RootsType& roots( psolve.roots() );
EvalRootsType evr( deg );
for( int i=0; i<roots.size(); ++i ){
evr[i] = std::abs( poly_eval( pols, roots[i] ) ); }
bool evalToZero = evr.isZero( test_precision<Scalar>() );
if( !evalToZero )
{
cerr << "WRONG root: " << endl;
cerr << "Polynomial: " << pols.transpose() << endl;
cerr << "Roots found: " << roots.transpose() << endl;
cerr << "Abs value of the polynomial at the roots: " << evr.transpose() << endl;
cerr << endl;
}
#ifdef HAS_GSL
if (ei_is_same_type< Scalar, double>::ret)
{
typedef GslTraits<Scalar> Gsl;
RootsType gslRoots(deg);
Gsl::eigen_poly_solve( pols, gslRoots );
EvalRootsType gslEvr( deg );
for( int i=0; i<gslRoots.size(); ++i )
{
gslEvr[i] = std::abs( poly_eval( pols, gslRoots[i] ) );
}
bool gslEvalToZero = gslEvr.isZero( test_precision<Scalar>() );
if( !evalToZero )
{
if( !gslEvalToZero ){
cerr << "GSL also failed" << endl; }
else{
cerr << "GSL did NOT failed" << endl; }
cerr << "GSL roots found: " << gslRoots.transpose() << endl;
cerr << "Abs value of the polynomial at the GSL roots: " << gslEvr.transpose() << endl;
cerr << endl;
}
}
#endif //< HAS_GSL
std::vector<Scalar> rootModuli( roots.size() );
Map< EvalRootsType > aux( &rootModuli[0], roots.size() );
aux = roots.array().abs();
std::sort( rootModuli.begin(), rootModuli.end() );
bool distinctModuli=true;
for( size_t i=1; i<rootModuli.size() && distinctModuli; ++i )
{
if( ei_isApprox( rootModuli[i], rootModuli[i-1] ) ){
distinctModuli = false; }
}
VERIFY( evalToZero || !distinctModuli );
return distinctModuli;
}
template<int Deg, typename POLYNOMIAL>
void evalSolver( const POLYNOMIAL& pols )
{
typedef typename POLYNOMIAL::Scalar Scalar;
typedef PolynomialSolver<Scalar, Deg > PolynomialSolverType;
PolynomialSolverType psolve;
aux_evalSolver<Deg, POLYNOMIAL, PolynomialSolverType>( pols, psolve );
}
template< int Deg, typename POLYNOMIAL, typename ROOTS, typename REAL_ROOTS >
void evalSolverSugarFunction( const POLYNOMIAL& pols, const ROOTS& roots, const REAL_ROOTS& real_roots )
{
typedef typename POLYNOMIAL::Scalar Scalar;
typedef PolynomialSolver<Scalar, Deg > PolynomialSolverType;
PolynomialSolverType psolve;
if( aux_evalSolver<Deg, POLYNOMIAL, PolynomialSolverType>( pols, psolve ) )
{
//It is supposed that
// 1) the roots found are correct
// 2) the roots have distinct moduli
typedef typename POLYNOMIAL::Scalar Scalar;
typedef typename REAL_ROOTS::Scalar Real;
typedef PolynomialSolver<Scalar, Deg > PolynomialSolverType;
typedef typename PolynomialSolverType::RootsType RootsType;
typedef Matrix<Scalar,Deg,1> EvalRootsType;
//Test realRoots
std::vector< Real > calc_realRoots;
psolve.realRoots( calc_realRoots );
VERIFY( calc_realRoots.size() == (size_t)real_roots.size() );
const Scalar psPrec = ei_sqrt( test_precision<Scalar>() );
for( size_t i=0; i<calc_realRoots.size(); ++i )
{
bool found = false;
for( size_t j=0; j<calc_realRoots.size()&& !found; ++j )
{
if( ei_isApprox( calc_realRoots[i], real_roots[j] ), psPrec ){
found = true; }
}
VERIFY( found );
}
//Test greatestRoot
VERIFY( ei_isApprox( roots.array().abs().maxCoeff(),
ei_abs( psolve.greatestRoot() ), psPrec ) );
//Test smallestRoot
VERIFY( ei_isApprox( roots.array().abs().minCoeff(),
ei_abs( psolve.smallestRoot() ), psPrec ) );
bool hasRealRoot;
//Test absGreatestRealRoot
Real r = psolve.absGreatestRealRoot( hasRealRoot );
VERIFY( hasRealRoot == (real_roots.size() > 0 ) );
if( hasRealRoot ){
VERIFY( ei_isApprox( real_roots.array().abs().maxCoeff(), ei_abs(r), psPrec ) ); }
//Test absSmallestRealRoot
r = psolve.absSmallestRealRoot( hasRealRoot );
VERIFY( hasRealRoot == (real_roots.size() > 0 ) );
if( hasRealRoot ){
VERIFY( ei_isApprox( real_roots.array().abs().minCoeff(), ei_abs( r ), psPrec ) ); }
//Test greatestRealRoot
r = psolve.greatestRealRoot( hasRealRoot );
VERIFY( hasRealRoot == (real_roots.size() > 0 ) );
if( hasRealRoot ){
VERIFY( ei_isApprox( real_roots.array().maxCoeff(), r, psPrec ) ); }
//Test smallestRealRoot
r = psolve.smallestRealRoot( hasRealRoot );
VERIFY( hasRealRoot == (real_roots.size() > 0 ) );
if( hasRealRoot ){
VERIFY( ei_isApprox( real_roots.array().minCoeff(), r, psPrec ) ); }
}
}
template<typename _Scalar, int _Deg>
void polynomialsolver(int deg)
{
typedef ei_increment_if_fixed_size<_Deg> Dim;
typedef Matrix<_Scalar,Dim::ret,1> PolynomialType;
typedef Matrix<_Scalar,_Deg,1> EvalRootsType;
cout << "Standard cases" << endl;
PolynomialType pols = PolynomialType::Random(deg+1);
evalSolver<_Deg,PolynomialType>( pols );
cout << "Hard cases" << endl;
_Scalar multipleRoot = ei_random<_Scalar>();
EvalRootsType allRoots = EvalRootsType::Constant(deg,multipleRoot);
roots_to_monicPolynomial( allRoots, pols );
evalSolver<_Deg,PolynomialType>( pols );
cout << "Test sugar" << endl;
EvalRootsType realRoots = EvalRootsType::Random(deg);
roots_to_monicPolynomial( realRoots, pols );
evalSolverSugarFunction<_Deg>(
pols,
realRoots.template cast <
std::complex<
typename NumTraits<_Scalar>::Real
>
>(),
realRoots );
}
template<typename _Scalar> void polynomialsolver_scalar()
{
CALL_SUBTEST_1( (polynomialsolver<_Scalar,1>(1)) );
CALL_SUBTEST_2( (polynomialsolver<_Scalar,2>(2)) );
CALL_SUBTEST_3( (polynomialsolver<_Scalar,3>(3)) );
CALL_SUBTEST_4( (polynomialsolver<_Scalar,4>(4)) );
CALL_SUBTEST_5( (polynomialsolver<_Scalar,5>(5)) );
CALL_SUBTEST_6( (polynomialsolver<_Scalar,6>(6)) );
CALL_SUBTEST_7( (polynomialsolver<_Scalar,7>(7)) );
CALL_SUBTEST_8( (polynomialsolver<_Scalar,8>(8)) );
CALL_SUBTEST_9( (polynomialsolver<_Scalar,Dynamic>(
ei_random<int>(9,45)
)) );
}
void test_polynomialsolver()
{
for(int i = 0; i < g_repeat; i++)
{
polynomialsolver_scalar<double>();
polynomialsolver_scalar<float>();
}
}

124
unsupported/test/polynomialutils.cpp Normal file
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@@ -0,0 +1,124 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.
#include "main.h"
#include <unsupported/Eigen/Polynomials>
#include <iostream>
using namespace std;
template<int Size>
struct ei_increment_if_fixed_size
{
enum {
ret = (Size == Dynamic) ? Dynamic : Size+1
};
};
template<typename _Scalar, int _Deg>
void realRoots_to_monicPolynomial_test(int deg)
{
typedef ei_increment_if_fixed_size<_Deg> Dim;
typedef Matrix<_Scalar,Dim::ret,1> PolynomialType;
typedef Matrix<_Scalar,_Deg,1> EvalRootsType;
PolynomialType pols(deg+1);
EvalRootsType roots = EvalRootsType::Random(deg);
roots_to_monicPolynomial( roots, pols );
EvalRootsType evr( deg );
for( int i=0; i<roots.size(); ++i ){
evr[i] = std::abs( poly_eval( pols, roots[i] ) ); }
bool evalToZero = evr.isZero( test_precision<_Scalar>() );
if( !evalToZero ){
cerr << evr.transpose() << endl; }
VERIFY( evalToZero );
}
template<typename _Scalar> void realRoots_to_monicPolynomial_scalar()
{
CALL_SUBTEST_2( (realRoots_to_monicPolynomial_test<_Scalar,2>(2)) );
CALL_SUBTEST_3( (realRoots_to_monicPolynomial_test<_Scalar,3>(3)) );
CALL_SUBTEST_4( (realRoots_to_monicPolynomial_test<_Scalar,4>(4)) );
CALL_SUBTEST_5( (realRoots_to_monicPolynomial_test<_Scalar,5>(5)) );
CALL_SUBTEST_6( (realRoots_to_monicPolynomial_test<_Scalar,6>(6)) );
CALL_SUBTEST_7( (realRoots_to_monicPolynomial_test<_Scalar,7>(7)) );
CALL_SUBTEST_8( (realRoots_to_monicPolynomial_test<_Scalar,17>(17)) );
CALL_SUBTEST_9( (realRoots_to_monicPolynomial_test<_Scalar,Dynamic>(
ei_random<int>(18,26) )) );
}
template<typename _Scalar, int _Deg>
void CauchyBounds(int deg)
{
typedef ei_increment_if_fixed_size<_Deg> Dim;
typedef Matrix<_Scalar,Dim::ret,1> PolynomialType;
typedef Matrix<_Scalar,_Deg,1> EvalRootsType;
PolynomialType pols(deg+1);
EvalRootsType roots = EvalRootsType::Random(deg);
roots_to_monicPolynomial( roots, pols );
_Scalar M = cauchy_max_bound( pols );
_Scalar m = cauchy_min_bound( pols );
_Scalar Max = roots.array().abs().maxCoeff();
_Scalar min = roots.array().abs().minCoeff();
bool eval = (M >= Max) && (m <= min);
if( !eval )
{
cerr << "Roots: " << roots << endl;
cerr << "Bounds: (" << m << ", " << M << ")" << endl;
cerr << "Min,Max: (" << min << ", " << Max << ")" << endl;
}
VERIFY( eval );
}
template<typename _Scalar> void CauchyBounds_scalar()
{
CALL_SUBTEST_2( (CauchyBounds<_Scalar,2>(2)) );
CALL_SUBTEST_3( (CauchyBounds<_Scalar,3>(3)) );
CALL_SUBTEST_4( (CauchyBounds<_Scalar,4>(4)) );
CALL_SUBTEST_5( (CauchyBounds<_Scalar,5>(5)) );
CALL_SUBTEST_6( (CauchyBounds<_Scalar,6>(6)) );
CALL_SUBTEST_7( (CauchyBounds<_Scalar,7>(7)) );
CALL_SUBTEST_8( (CauchyBounds<_Scalar,17>(17)) );
CALL_SUBTEST_9( (CauchyBounds<_Scalar,Dynamic>(
ei_random<int>(18,26) )) );
}
void test_polynomialutils()
{
for(int i = 0; i < g_repeat; i++)
{
realRoots_to_monicPolynomial_scalar<double>();
realRoots_to_monicPolynomial_scalar<float>();
CauchyBounds_scalar<double>();
CauchyBounds_scalar<float>();
}
}