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Tobias Wood
2023-11-29 11:12:48 +00:00
parent 9ea520fc45
commit f38e16c193
534 changed files with 103368 additions and 116934 deletions
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unsupported/Eigen/MatrixFunctions
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@@ -19,39 +19,39 @@
#include "../../Eigen/Eigenvalues"
/**
* \defgroup MatrixFunctions_Module Matrix functions module
* \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_log "MatrixBase::log()", for computing the matrix logarithm
* - \ref matrixbase_pow "MatrixBase::pow()", for computing the matrix power
* - \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
* - \ref matrixbase_sqrt "MatrixBase::sqrt()", for computing the matrix square root
*
* 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
* series
* \f[ f(0) + f'(0) x + \frac{f''(0)}{2} x^2 + \frac{f'''(0)}{3!} x^3 + \cdots \f]
* converges to \f$ f(x) \f$. In this case, we can define the matrix
* function by the same series:
* \f[ f(M) = f(0) + f'(0) M + \frac{f''(0)}{2} M^2 + \frac{f'''(0)}{3!} M^3 + \cdots \f]
*
*/
* \defgroup MatrixFunctions_Module Matrix functions module
* \brief This module aims to provide various methods for the computation of
* matrix functions.
*
* To use this module, add
* \code
* #include <unsupported/Eigen/MatrixFunctions>
* \endcode
* at the start of your source file.
*
* This module defines the following MatrixBase methods.
* - \ref matrixbase_cos "MatrixBase::cos()", for computing the matrix cosine
* - \ref matrixbase_cosh "MatrixBase::cosh()", for computing the matrix hyperbolic cosine
* - \ref matrixbase_exp "MatrixBase::exp()", for computing the matrix exponential
* - \ref matrixbase_log "MatrixBase::log()", for computing the matrix logarithm
* - \ref matrixbase_pow "MatrixBase::pow()", for computing the matrix power
* - \ref matrixbase_matrixfunction "MatrixBase::matrixFunction()", for computing general matrix functions
* - \ref matrixbase_sin "MatrixBase::sin()", for computing the matrix sine
* - \ref matrixbase_sinh "MatrixBase::sinh()", for computing the matrix hyperbolic sine
* - \ref matrixbase_sqrt "MatrixBase::sqrt()", for computing the matrix square root
*
* These methods are the main entry points to this module.
*
* %Matrix functions are defined as follows. Suppose that \f$ f \f$
* is an entire function (that is, a function on the complex plane
* that is everywhere complex differentiable). Then its Taylor
* series
* \f[ f(0) + f'(0) x + \frac{f''(0)}{2} x^2 + \frac{f'''(0)}{3!} x^3 + \cdots \f]
* converges to \f$ f(x) \f$. In this case, we can define the matrix
* function by the same series:
* \f[ f(M) = f(0) + f'(0) M + \frac{f''(0)}{2} M^2 + \frac{f'''(0)}{3!} M^3 + \cdots \f]
*
*/
#include "../../Eigen/src/Core/util/DisableStupidWarnings.h"
@@ -65,8 +65,7 @@
#include "../../Eigen/src/Core/util/ReenableStupidWarnings.h"
/**
/**
\page matrixbaseextra_page
\ingroup MatrixFunctions_Module
@@ -182,7 +181,7 @@ const MatrixLogarithmReturnValue<Derived> MatrixBase<Derived>::log() const
\param[in] M invertible matrix whose logarithm is to be computed.
\returns expression representing the matrix logarithm root of \p M.
The matrix logarithm of \f$ M \f$ is a matrix \f$ X \f$ such that
The matrix logarithm of \f$ M \f$ is a matrix \f$ X \f$ such that
\f$ \exp(X) = M \f$ where exp denotes the matrix exponential. As for
the scalar logarithm, the equation \f$ \exp(X) = M \f$ may have
multiple solutions; this function returns a matrix whose eigenvalues
@@ -209,14 +208,14 @@ Nicholas J. Higham,
SIAM 2008. ISBN 978-0-898716-46-7.
Example: The following program checks that
\f[ \log \left[ \begin{array}{ccc}
\f[ \log \left[ \begin{array}{ccc}
\frac12\sqrt2 & -\frac12\sqrt2 & 0 \\
\frac12\sqrt2 & \frac12\sqrt2 & 0 \\
0 & 0 & 1
\end{array} \right] = \left[ \begin{array}{ccc}
0 & \frac14\pi & 0 \\
0 & \frac14\pi & 0 \\
-\frac14\pi & 0 & 0 \\
0 & 0 & 0
0 & 0 & 0
\end{array} \right]. \f]
This corresponds to a rotation of \f$ \frac14\pi \f$ radians around
the z-axis. This is the inverse of the example used in the
@@ -228,7 +227,7 @@ Output: \verbinclude MatrixLogarithm.out
\note \p M has to be a matrix of \c float, \c double, `long
double`, \c complex<float>, \c complex<double>, or `complex<long double>`.
\sa MatrixBase::exp(), MatrixBase::matrixFunction(),
\sa MatrixBase::exp(), MatrixBase::matrixFunction(),
class MatrixLogarithmAtomic, MatrixBase::sqrt().
@@ -290,7 +289,7 @@ int main()
0, 0, 6, 7,
0, 0, 8, 9;
std::cout << A.pow(0.37) << std::endl;
// The 1 makes eigenvalue 0 non-semisimple.
A.coeffRef(0, 1) = 1;
@@ -343,18 +342,18 @@ double`, \c complex<float>, \c complex<double>, or
Compute a matrix function.
\code
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::matrixFunction(typename internal::stem_function<typename internal::traits<Derived>::Scalar>::type f) const
\endcode
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::matrixFunction(typename internal::stem_function<typename
internal::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.
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)
\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 =
@@ -362,17 +361,17 @@ real (e.g., \c float or \c double) and \c ComplexScalar =
should be \f$ f^{(n)}(x) \f$, the n-th derivative of f at x.
This routine uses the algorithm described in:
Philip Davies and Nicholas J. Higham,
"A Schur-Parlett algorithm for computing matrix functions",
Philip Davies and Nicholas J. Higham,
"A Schur-Parlett algorithm for computing matrix functions",
<em>SIAM J. %Matrix Anal. Applic.</em>, <b>25</b>:464&ndash;485, 2003.
The actual work is done by the MatrixFunction class.
Example: The following program checks that
\f[ \exp \left[ \begin{array}{ccc}
0 & \frac14\pi & 0 \\
\f[ \exp \left[ \begin{array}{ccc}
0 & \frac14\pi & 0 \\
-\frac14\pi & 0 & 0 \\
0 & 0 & 0
0 & 0 & 0
\end{array} \right] = \left[ \begin{array}{ccc}
\frac12\sqrt2 & -\frac12\sqrt2 & 0 \\
\frac12\sqrt2 & \frac12\sqrt2 & 0 \\
@@ -385,7 +384,7 @@ of \ref matrixbase_exp "exp()".
\include MatrixFunction.cpp
Output: \verbinclude MatrixFunction.out
Note that the function \c expfn is defined for complex numbers
Note that the function \c expfn is defined for complex numbers
\c x, even though the matrix \c A is over the reals. Instead of
\c expfn, we could also have used StdStemFunctions::exp:
\code
@@ -452,7 +451,7 @@ In the <b>real case</b>, the matrix \f$ M \f$ should be invertible and
it should have no eigenvalues which are real and negative (pairs of
complex conjugate eigenvalues are allowed). In that case, the matrix
has a square root which is also real, and this is the square root
computed by this function.
computed by this function.
The matrix square root is computed by first reducing the matrix to
quasi-triangular form with the real Schur decomposition. The square
@@ -484,12 +483,12 @@ square root of a matrix", <em>Linear Algebra Appl.</em>,
52/53:127&ndash;140, 1983.
Example: The following program checks that the square root of
\f[ \left[ \begin{array}{cc}
\f[ \left[ \begin{array}{cc}
\cos(\frac13\pi) & -\sin(\frac13\pi) \\
\sin(\frac13\pi) & \cos(\frac13\pi)
\end{array} \right], \f]
corresponding to a rotation over 60 degrees, is a rotation over 30 degrees:
\f[ \left[ \begin{array}{cc}
\f[ \left[ \begin{array}{cc}
\cos(\frac16\pi) & -\sin(\frac16\pi) \\
\sin(\frac16\pi) & \cos(\frac16\pi)
\end{array} \right]. \f]
@@ -502,4 +501,4 @@ Output: \verbinclude MatrixSquareRoot.out
*/
#endif // EIGEN_MATRIX_FUNCTIONS_MODULE_H
#endif // EIGEN_MATRIX_FUNCTIONS_MODULE_H