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Move documentation of MatrixBase methods in MatrixFunctions to module page.

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I think that because MatrixFunctions is in unsupported/ and MatrixBase is
not, doxygen does not include the MatrixBase methods defined and documented
in the MatrixFunctions module with the other MatrixBase methods. This is a
kludge, but at least the documentation is not lost.
This commit is contained in:
Jitse Niesen
2010-03-22 13:58:19 +00:00
parent 525d6b655f
commit 307c428253
3 changed files with 209 additions and 150 deletions
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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