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RealSchur.h
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_REAL_SCHUR_H
#define EIGEN_REAL_SCHUR_H
#include "./HessenbergDecomposition.h"
namespace
Eigen
{
/** \eigenvalues_module \ingroup Eigenvalues_Module
*
*
* \class RealSchur
*
* \brief Performs a real Schur decomposition of a square matrix
*
* \tparam _MatrixType the type of the matrix of which we are computing the
* real Schur decomposition; this is expected to be an instantiation of the
* Matrix class template.
*
* Given a real square matrix A, this class computes the real Schur
* decomposition: \f$ A = U T U^T \f$ where U is a real orthogonal matrix and
* T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose
* inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
* matrix is a block-triangular matrix whose diagonal consists of 1-by-1
* blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the
* blocks on the diagonal of T are the same as the eigenvalues of the matrix
* A, and thus the real Schur decomposition is used in EigenSolver to compute
* the eigendecomposition of a matrix.
*
* Call the function compute() to compute the real Schur decomposition of a
* given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool)
* constructor which computes the real Schur decomposition at construction
* time. Once the decomposition is computed, you can use the matrixU() and
* matrixT() functions to retrieve the matrices U and T in the decomposition.
*
* The documentation of RealSchur(const MatrixType&, bool) contains an example
* of the typical use of this class.
*
* \note The implementation is adapted from
* <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
* Their code is based on EISPACK.
*
* \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver
*/
template
<
typename
_MatrixType
>
class
RealSchur
{
public:
typedef
_MatrixType
MatrixType
;
enum
{
RowsAtCompileTime
=
MatrixType
::
RowsAtCompileTime
,
ColsAtCompileTime
=
MatrixType
::
ColsAtCompileTime
,
Options
=
MatrixType
::
Options
,
MaxRowsAtCompileTime
=
MatrixType
::
MaxRowsAtCompileTime
,
MaxColsAtCompileTime
=
MatrixType
::
MaxColsAtCompileTime
};
typedef
typename
MatrixType
::
Scalar
Scalar
;
typedef
std
::
complex
<
typename
NumTraits
<
Scalar
>::
Real
>
ComplexScalar
;
typedef
typename
MatrixType
::
Index
Index
;
typedef
Matrix
<
ComplexScalar
,
ColsAtCompileTime
,
1
,
Options
&
~
RowMajor
,
MaxColsAtCompileTime
,
1
>
EigenvalueType
;
typedef
Matrix
<
Scalar
,
ColsAtCompileTime
,
1
,
Options
&
~
RowMajor
,
MaxColsAtCompileTime
,
1
>
ColumnVectorType
;
/** \brief Default constructor.
*
* \param [in] size Positive integer, size of the matrix whose Schur decomposition will be computed.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via compute(). The \p size parameter is only
* used as a hint. It is not an error to give a wrong \p size, but it may
* impair performance.
*
* \sa compute() for an example.
*/
RealSchur
(
Index
size
=
RowsAtCompileTime
==
Dynamic
?
1
:
RowsAtCompileTime
)
:
m_matT
(
size
,
size
),
m_matU
(
size
,
size
),
m_workspaceVector
(
size
),
m_hess
(
size
),
m_isInitialized
(
false
),
m_matUisUptodate
(
false
),
m_maxIters
(
-
1
)
{
}
/** \brief Constructor; computes real Schur decomposition of given matrix.
*
* \param[in] matrix Square matrix whose Schur decomposition is to be computed.
* \param[in] computeU If true, both T and U are computed; if false, only T is computed.
*
* This constructor calls compute() to compute the Schur decomposition.
*
* Example: \include RealSchur_RealSchur_MatrixType.cpp
* Output: \verbinclude RealSchur_RealSchur_MatrixType.out
*/
RealSchur
(
const
MatrixType
&
matrix
,
bool
computeU
=
true
)
:
m_matT
(
matrix
.
rows
(),
matrix
.
cols
()),
m_matU
(
matrix
.
rows
(),
matrix
.
cols
()),
m_workspaceVector
(
matrix
.
rows
()),
m_hess
(
matrix
.
rows
()),
m_isInitialized
(
false
),
m_matUisUptodate
(
false
),
m_maxIters
(
-
1
)
{
compute
(
matrix
,
computeU
);
}
/** \brief Returns the orthogonal matrix in the Schur decomposition.
*
* \returns A const reference to the matrix U.
*
* \pre Either the constructor RealSchur(const MatrixType&, bool) or the
* member function compute(const MatrixType&, bool) has been called before
* to compute the Schur decomposition of a matrix, and \p computeU was set
* to true (the default value).
*
* \sa RealSchur(const MatrixType&, bool) for an example
*/
const
MatrixType
&
matrixU
()
const
{
eigen_assert
(
m_isInitialized
&&
"RealSchur is not initialized."
);
eigen_assert
(
m_matUisUptodate
&&
"The matrix U has not been computed during the RealSchur decomposition."
);
return
m_matU
;
}
/** \brief Returns the quasi-triangular matrix in the Schur decomposition.
*
* \returns A const reference to the matrix T.
*
* \pre Either the constructor RealSchur(const MatrixType&, bool) or the
* member function compute(const MatrixType&, bool) has been called before
* to compute the Schur decomposition of a matrix.
*
* \sa RealSchur(const MatrixType&, bool) for an example
*/
const
MatrixType
&
matrixT
()
const
{
eigen_assert
(
m_isInitialized
&&
"RealSchur is not initialized."
);
return
m_matT
;
}
/** \brief Computes Schur decomposition of given matrix.
*
* \param[in] matrix Square matrix whose Schur decomposition is to be computed.
* \param[in] computeU If true, both T and U are computed; if false, only T is computed.
* \returns Reference to \c *this
*
* The Schur decomposition is computed by first reducing the matrix to
* Hessenberg form using the class HessenbergDecomposition. The Hessenberg
* matrix is then reduced to triangular form by performing Francis QR
* iterations with implicit double shift. The cost of computing the Schur
* decomposition depends on the number of iterations; as a rough guide, it
* may be taken to be \f$25n^3\f$ flops if \a computeU is true and
* \f$10n^3\f$ flops if \a computeU is false.
*
* Example: \include RealSchur_compute.cpp
* Output: \verbinclude RealSchur_compute.out
*
* \sa compute(const MatrixType&, bool, Index)
*/
RealSchur
&
compute
(
const
MatrixType
&
matrix
,
bool
computeU
=
true
);
/** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T
* \param[in] matrixH Matrix in Hessenberg form H
* \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
* \param computeU Computes the matriX U of the Schur vectors
* \return Reference to \c *this
*
* This routine assumes that the matrix is already reduced in Hessenberg form matrixH
* using either the class HessenbergDecomposition or another mean.
* It computes the upper quasi-triangular matrix T of the Schur decomposition of H
* When computeU is true, this routine computes the matrix U such that
* A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
*
* NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
* is not available, the user should give an identity matrix (Q.setIdentity())
*
* \sa compute(const MatrixType&, bool)
*/
template
<
typename
HessMatrixType
,
typename
OrthMatrixType
>
RealSchur
&
computeFromHessenberg
(
const
HessMatrixType
&
matrixH
,
const
OrthMatrixType
&
matrixQ
,
bool
computeU
);
/** \brief Reports whether previous computation was successful.
*
* \returns \c Success if computation was succesful, \c NoConvergence otherwise.
*/
ComputationInfo
info
()
const
{
eigen_assert
(
m_isInitialized
&&
"RealSchur is not initialized."
);
return
m_info
;
}
/** \brief Sets the maximum number of iterations allowed.
*
* If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
* of the matrix.
*/
RealSchur
&
setMaxIterations
(
Index
maxIters
)
{
m_maxIters
=
maxIters
;
return
*
this
;
}
/** \brief Returns the maximum number of iterations. */
Index
getMaxIterations
()
{
return
m_maxIters
;
}
/** \brief Maximum number of iterations per row.
*
* If not otherwise specified, the maximum number of iterations is this number times the size of the
* matrix. It is currently set to 40.
*/
static
const
int
m_maxIterationsPerRow
=
40
;
private:
MatrixType
m_matT
;
MatrixType
m_matU
;
ColumnVectorType
m_workspaceVector
;
HessenbergDecomposition
<
MatrixType
>
m_hess
;
ComputationInfo
m_info
;
bool
m_isInitialized
;
bool
m_matUisUptodate
;
Index
m_maxIters
;
typedef
Matrix
<
Scalar
,
3
,
1
>
Vector3s
;
Scalar
computeNormOfT
();
Index
findSmallSubdiagEntry
(
Index
iu
);
void
splitOffTwoRows
(
Index
iu
,
bool
computeU
,
const
Scalar
&
exshift
);
void
computeShift
(
Index
iu
,
Index
iter
,
Scalar
&
exshift
,
Vector3s
&
shiftInfo
);
void
initFrancisQRStep
(
Index
il
,
Index
iu
,
const
Vector3s
&
shiftInfo
,
Index
&
im
,
Vector3s
&
firstHouseholderVector
);
void
performFrancisQRStep
(
Index
il
,
Index
im
,
Index
iu
,
bool
computeU
,
const
Vector3s
&
firstHouseholderVector
,
Scalar
*
workspace
);
};
template
<
typename
MatrixType
>
RealSchur
<
MatrixType
>&
RealSchur
<
MatrixType
>::
compute
(
const
MatrixType
&
matrix
,
bool
computeU
)
{
eigen_assert
(
matrix
.
cols
()
==
matrix
.
rows
());
Index
maxIters
=
m_maxIters
;
if
(
maxIters
==
-
1
)
maxIters
=
m_maxIterationsPerRow
*
matrix
.
rows
();
// Step 1. Reduce to Hessenberg form
m_hess
.
compute
(
matrix
);
// Step 2. Reduce to real Schur form
computeFromHessenberg
(
m_hess
.
matrixH
(),
m_hess
.
matrixQ
(),
computeU
);
return
*
this
;
}
template
<
typename
MatrixType
>
template
<
typename
HessMatrixType
,
typename
OrthMatrixType
>
RealSchur
<
MatrixType
>&
RealSchur
<
MatrixType
>::
computeFromHessenberg
(
const
HessMatrixType
&
matrixH
,
const
OrthMatrixType
&
matrixQ
,
bool
computeU
)
{
m_matT
=
matrixH
;
if
(
computeU
)
m_matU
=
matrixQ
;
Index
maxIters
=
m_maxIters
;
if
(
maxIters
==
-
1
)
maxIters
=
m_maxIterationsPerRow
*
matrixH
.
rows
();
m_workspaceVector
.
resize
(
m_matT
.
cols
());
Scalar
*
workspace
=
&
m_workspaceVector
.
coeffRef
(
0
);
// The matrix m_matT is divided in three parts.
// Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
// Rows il,...,iu is the part we are working on (the active window).
// Rows iu+1,...,end are already brought in triangular form.
Index
iu
=
m_matT
.
cols
()
-
1
;
Index
iter
=
0
;
// iteration count for current eigenvalue
Index
totalIter
=
0
;
// iteration count for whole matrix
Scalar
exshift
(
0
);
// sum of exceptional shifts
Scalar
norm
=
computeNormOfT
();
if
(
norm
!=
0
)
{
while
(
iu
>=
0
)
{
Index
il
=
findSmallSubdiagEntry
(
iu
);
// Check for convergence
if
(
il
==
iu
)
// One root found
{
m_matT
.
coeffRef
(
iu
,
iu
)
=
m_matT
.
coeff
(
iu
,
iu
)
+
exshift
;
if
(
iu
>
0
)
m_matT
.
coeffRef
(
iu
,
iu
-
1
)
=
Scalar
(
0
);
iu
--
;
iter
=
0
;
}
else
if
(
il
==
iu
-
1
)
// Two roots found
{
splitOffTwoRows
(
iu
,
computeU
,
exshift
);
iu
-=
2
;
iter
=
0
;
}
else
// No convergence yet
{
// The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )
Vector3s
firstHouseholderVector
(
0
,
0
,
0
),
shiftInfo
;
computeShift
(
iu
,
iter
,
exshift
,
shiftInfo
);
iter
=
iter
+
1
;
totalIter
=
totalIter
+
1
;
if
(
totalIter
>
maxIters
)
break
;
Index
im
;
initFrancisQRStep
(
il
,
iu
,
shiftInfo
,
im
,
firstHouseholderVector
);
performFrancisQRStep
(
il
,
im
,
iu
,
computeU
,
firstHouseholderVector
,
workspace
);
}
}
}
if
(
totalIter
<=
maxIters
)
m_info
=
Success
;
else
m_info
=
NoConvergence
;
m_isInitialized
=
true
;
m_matUisUptodate
=
computeU
;
return
*
this
;
}
/** \internal Computes and returns vector L1 norm of T */
template
<
typename
MatrixType
>
inline
typename
MatrixType
::
Scalar
RealSchur
<
MatrixType
>::
computeNormOfT
()
{
const
Index
size
=
m_matT
.
cols
();
// FIXME to be efficient the following would requires a triangular reduxion code
// Scalar norm = m_matT.upper().cwiseAbs().sum()
// + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
Scalar
norm
(
0
);
for
(
Index
j
=
0
;
j
<
size
;
++
j
)
norm
+=
m_matT
.
col
(
j
).
segment
(
0
,
(
std
::
min
)(
size
,
j
+
2
)).
cwiseAbs
().
sum
();
return
norm
;
}
/** \internal Look for single small sub-diagonal element and returns its index */
template
<
typename
MatrixType
>
inline
typename
MatrixType
::
Index
RealSchur
<
MatrixType
>::
findSmallSubdiagEntry
(
Index
iu
)
{
using
std
::
abs
;
Index
res
=
iu
;
while
(
res
>
0
)
{
Scalar
s
=
abs
(
m_matT
.
coeff
(
res
-
1
,
res
-
1
))
+
abs
(
m_matT
.
coeff
(
res
,
res
));
if
(
abs
(
m_matT
.
coeff
(
res
,
res
-
1
))
<=
NumTraits
<
Scalar
>::
epsilon
()
*
s
)
break
;
res
--
;
}
return
res
;
}
/** \internal Update T given that rows iu-1 and iu decouple from the rest. */
template
<
typename
MatrixType
>
inline
void
RealSchur
<
MatrixType
>::
splitOffTwoRows
(
Index
iu
,
bool
computeU
,
const
Scalar
&
exshift
)
{
using
std
::
sqrt
;
using
std
::
abs
;
const
Index
size
=
m_matT
.
cols
();
// The eigenvalues of the 2x2 matrix [a b; c d] are
// trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
Scalar
p
=
Scalar
(
0.5
)
*
(
m_matT
.
coeff
(
iu
-
1
,
iu
-
1
)
-
m_matT
.
coeff
(
iu
,
iu
));
Scalar
q
=
p
*
p
+
m_matT
.
coeff
(
iu
,
iu
-
1
)
*
m_matT
.
coeff
(
iu
-
1
,
iu
);
// q = tr^2 / 4 - det = discr/4
m_matT
.
coeffRef
(
iu
,
iu
)
+=
exshift
;
m_matT
.
coeffRef
(
iu
-
1
,
iu
-
1
)
+=
exshift
;
if
(
q
>=
Scalar
(
0
))
// Two real eigenvalues
{
Scalar
z
=
sqrt
(
abs
(
q
));
JacobiRotation
<
Scalar
>
rot
;
if
(
p
>=
Scalar
(
0
))
rot
.
makeGivens
(
p
+
z
,
m_matT
.
coeff
(
iu
,
iu
-
1
));
else
rot
.
makeGivens
(
p
-
z
,
m_matT
.
coeff
(
iu
,
iu
-
1
));
m_matT
.
rightCols
(
size
-
iu
+
1
).
applyOnTheLeft
(
iu
-
1
,
iu
,
rot
.
adjoint
());
m_matT
.
topRows
(
iu
+
1
).
applyOnTheRight
(
iu
-
1
,
iu
,
rot
);
m_matT
.
coeffRef
(
iu
,
iu
-
1
)
=
Scalar
(
0
);
if
(
computeU
)
m_matU
.
applyOnTheRight
(
iu
-
1
,
iu
,
rot
);
}
if
(
iu
>
1
)
m_matT
.
coeffRef
(
iu
-
1
,
iu
-
2
)
=
Scalar
(
0
);
}
/** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
template
<
typename
MatrixType
>
inline
void
RealSchur
<
MatrixType
>::
computeShift
(
Index
iu
,
Index
iter
,
Scalar
&
exshift
,
Vector3s
&
shiftInfo
)
{
using
std
::
sqrt
;
using
std
::
abs
;
shiftInfo
.
coeffRef
(
0
)
=
m_matT
.
coeff
(
iu
,
iu
);
shiftInfo
.
coeffRef
(
1
)
=
m_matT
.
coeff
(
iu
-
1
,
iu
-
1
);
shiftInfo
.
coeffRef
(
2
)
=
m_matT
.
coeff
(
iu
,
iu
-
1
)
*
m_matT
.
coeff
(
iu
-
1
,
iu
);
// Wilkinson's original ad hoc shift
if
(
iter
==
10
)
{
exshift
+=
shiftInfo
.
coeff
(
0
);
for
(
Index
i
=
0
;
i
<=
iu
;
++
i
)
m_matT
.
coeffRef
(
i
,
i
)
-=
shiftInfo
.
coeff
(
0
);
Scalar
s
=
abs
(
m_matT
.
coeff
(
iu
,
iu
-
1
))
+
abs
(
m_matT
.
coeff
(
iu
-
1
,
iu
-
2
));
shiftInfo
.
coeffRef
(
0
)
=
Scalar
(
0.75
)
*
s
;
shiftInfo
.
coeffRef
(
1
)
=
Scalar
(
0.75
)
*
s
;
shiftInfo
.
coeffRef
(
2
)
=
Scalar
(
-
0.4375
)
*
s
*
s
;
}
// MATLAB's new ad hoc shift
if
(
iter
==
30
)
{
Scalar
s
=
(
shiftInfo
.
coeff
(
1
)
-
shiftInfo
.
coeff
(
0
))
/
Scalar
(
2.0
);
s
=
s
*
s
+
shiftInfo
.
coeff
(
2
);
if
(
s
>
Scalar
(
0
))
{
s
=
sqrt
(
s
);
if
(
shiftInfo
.
coeff
(
1
)
<
shiftInfo
.
coeff
(
0
))
s
=
-
s
;
s
=
s
+
(
shiftInfo
.
coeff
(
1
)
-
shiftInfo
.
coeff
(
0
))
/
Scalar
(
2.0
);
s
=
shiftInfo
.
coeff
(
0
)
-
shiftInfo
.
coeff
(
2
)
/
s
;
exshift
+=
s
;
for
(
Index
i
=
0
;
i
<=
iu
;
++
i
)
m_matT
.
coeffRef
(
i
,
i
)
-=
s
;
shiftInfo
.
setConstant
(
Scalar
(
0.964
));
}
}
}
/** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
template
<
typename
MatrixType
>
inline
void
RealSchur
<
MatrixType
>::
initFrancisQRStep
(
Index
il
,
Index
iu
,
const
Vector3s
&
shiftInfo
,
Index
&
im
,
Vector3s
&
firstHouseholderVector
)
{
using
std
::
abs
;
Vector3s
&
v
=
firstHouseholderVector
;
// alias to save typing
for
(
im
=
iu
-
2
;
im
>=
il
;
--
im
)
{
const
Scalar
Tmm
=
m_matT
.
coeff
(
im
,
im
);
const
Scalar
r
=
shiftInfo
.
coeff
(
0
)
-
Tmm
;
const
Scalar
s
=
shiftInfo
.
coeff
(
1
)
-
Tmm
;
v
.
coeffRef
(
0
)
=
(
r
*
s
-
shiftInfo
.
coeff
(
2
))
/
m_matT
.
coeff
(
im
+
1
,
im
)
+
m_matT
.
coeff
(
im
,
im
+
1
);
v
.
coeffRef
(
1
)
=
m_matT
.
coeff
(
im
+
1
,
im
+
1
)
-
Tmm
-
r
-
s
;
v
.
coeffRef
(
2
)
=
m_matT
.
coeff
(
im
+
2
,
im
+
1
);
if
(
im
==
il
)
{
break
;
}
const
Scalar
lhs
=
m_matT
.
coeff
(
im
,
im
-
1
)
*
(
abs
(
v
.
coeff
(
1
))
+
abs
(
v
.
coeff
(
2
)));
const
Scalar
rhs
=
v
.
coeff
(
0
)
*
(
abs
(
m_matT
.
coeff
(
im
-
1
,
im
-
1
))
+
abs
(
Tmm
)
+
abs
(
m_matT
.
coeff
(
im
+
1
,
im
+
1
)));
if
(
abs
(
lhs
)
<
NumTraits
<
Scalar
>::
epsilon
()
*
rhs
)
break
;
}
}
/** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
template
<
typename
MatrixType
>
inline
void
RealSchur
<
MatrixType
>::
performFrancisQRStep
(
Index
il
,
Index
im
,
Index
iu
,
bool
computeU
,
const
Vector3s
&
firstHouseholderVector
,
Scalar
*
workspace
)
{
eigen_assert
(
im
>=
il
);
eigen_assert
(
im
<=
iu
-
2
);
const
Index
size
=
m_matT
.
cols
();
for
(
Index
k
=
im
;
k
<=
iu
-
2
;
++
k
)
{
bool
firstIteration
=
(
k
==
im
);
Vector3s
v
;
if
(
firstIteration
)
v
=
firstHouseholderVector
;
else
v
=
m_matT
.
template
block
<
3
,
1
>
(
k
,
k
-
1
);
Scalar
tau
,
beta
;
Matrix
<
Scalar
,
2
,
1
>
ess
;
v
.
makeHouseholder
(
ess
,
tau
,
beta
);
if
(
beta
!=
Scalar
(
0
))
// if v is not zero
{
if
(
firstIteration
&&
k
>
il
)
m_matT
.
coeffRef
(
k
,
k
-
1
)
=
-
m_matT
.
coeff
(
k
,
k
-
1
);
else
if
(
!
firstIteration
)
m_matT
.
coeffRef
(
k
,
k
-
1
)
=
beta
;
// These Householder transformations form the O(n^3) part of the algorithm
m_matT
.
block
(
k
,
k
,
3
,
size
-
k
).
applyHouseholderOnTheLeft
(
ess
,
tau
,
workspace
);
m_matT
.
block
(
0
,
k
,
(
std
::
min
)(
iu
,
k
+
3
)
+
1
,
3
).
applyHouseholderOnTheRight
(
ess
,
tau
,
workspace
);
if
(
computeU
)
m_matU
.
block
(
0
,
k
,
size
,
3
).
applyHouseholderOnTheRight
(
ess
,
tau
,
workspace
);
}
}
Matrix
<
Scalar
,
2
,
1
>
v
=
m_matT
.
template
block
<
2
,
1
>
(
iu
-
1
,
iu
-
2
);
Scalar
tau
,
beta
;
Matrix
<
Scalar
,
1
,
1
>
ess
;
v
.
makeHouseholder
(
ess
,
tau
,
beta
);
if
(
beta
!=
Scalar
(
0
))
// if v is not zero
{
m_matT
.
coeffRef
(
iu
-
1
,
iu
-
2
)
=
beta
;
m_matT
.
block
(
iu
-
1
,
iu
-
1
,
2
,
size
-
iu
+
1
).
applyHouseholderOnTheLeft
(
ess
,
tau
,
workspace
);
m_matT
.
block
(
0
,
iu
-
1
,
iu
+
1
,
2
).
applyHouseholderOnTheRight
(
ess
,
tau
,
workspace
);
if
(
computeU
)
m_matU
.
block
(
0
,
iu
-
1
,
size
,
2
).
applyHouseholderOnTheRight
(
ess
,
tau
,
workspace
);
}
// clean up pollution due to round-off errors
for
(
Index
i
=
im
+
2
;
i
<=
iu
;
++
i
)
{
m_matT
.
coeffRef
(
i
,
i
-
2
)
=
Scalar
(
0
);
if
(
i
>
im
+
2
)
m_matT
.
coeffRef
(
i
,
i
-
3
)
=
Scalar
(
0
);
}
}
}
// end namespace Eigen
#endif
// EIGEN_REAL_SCHUR_H
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