eigen/doc/QuickReference.dox

namespace Eigen {

/** \eigenManualPage QuickRefPage Quick reference guide

\eigenAutoToc

<hr>

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\section QuickRef_Headers Modules and Header files

The Eigen library is divided in a Core module and several additional modules. Each module has a corresponding header file which has to be included in order to use the module. The \c %Dense and \c Eigen header files are provided to conveniently gain access to several modules at once.

<table class="manual">
<tr><th>Module</th><th>Header file</th><th>Contents</th></tr>
<tr            ><td>\link Core_Module Core \endlink</td><td>\code#include <Eigen/Core>\endcode</td><td>Matrix and Array classes, basic linear algebra (including triangular and selfadjoint products), array manipulation</td></tr>
<tr class="alt"><td>\link Geometry_Module Geometry \endlink</td><td>\code#include <Eigen/Geometry>\endcode</td><td>Transform, Translation, Scaling, Rotation2D and 3D rotations (Quaternion, AngleAxis)</td></tr>
<tr            ><td>\link LU_Module LU \endlink</td><td>\code#include <Eigen/LU>\endcode</td><td>Inverse, determinant, LU decompositions with solver (FullPivLU, PartialPivLU)</td></tr>
<tr class="alt"><td>\link Cholesky_Module Cholesky \endlink</td><td>\code#include <Eigen/Cholesky>\endcode</td><td>LLT and LDLT Cholesky factorization with solver</td></tr>
<tr            ><td>\link Householder_Module Householder \endlink</td><td>\code#include <Eigen/Householder>\endcode</td><td>Householder transformations; this module is used by several linear algebra modules</td></tr>
<tr class="alt"><td>\link SVD_Module SVD \endlink</td><td>\code#include <Eigen/SVD>\endcode</td><td>SVD decompositions with least-squares solver (JacobiSVD, BDCSVD)</td></tr>
<tr            ><td>\link QR_Module QR \endlink</td><td>\code#include <Eigen/QR>\endcode</td><td>QR decomposition with solver (HouseholderQR, ColPivHouseholderQR, FullPivHouseholderQR)</td></tr>
<tr class="alt"><td>\link Eigenvalues_Module Eigenvalues \endlink</td><td>\code#include <Eigen/Eigenvalues>\endcode</td><td>Eigenvalue, eigenvector decompositions (EigenSolver, SelfAdjointEigenSolver, ComplexEigenSolver)</td></tr>
<tr            ><td>\link Sparse_Module Sparse \endlink</td><td>\code#include <Eigen/Sparse>\endcode</td><td>%Sparse matrix storage and related basic linear algebra (SparseMatrix, SparseVector) \n (see \ref SparseQuickRefPage for details on sparse modules)</td></tr>
<tr class="alt"><td></td><td>\code#include <Eigen/Dense>\endcode</td><td>Includes Core, Geometry, LU, Cholesky, SVD, QR, and Eigenvalues header files</td></tr>
<tr            ><td></td><td>\code#include <Eigen/Eigen>\endcode</td><td>Includes %Dense and %Sparse header files (the whole Eigen library)</td></tr>
</table>

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\section QuickRef_Types Array, matrix and vector types


\b Recall: Eigen provides two kinds of dense objects: mathematical matrices and vectors which are both represented by the template class Matrix, and general 1D and 2D arrays represented by the template class Array:
\code
typedef Matrix<Scalar, RowsAtCompileTime, ColsAtCompileTime, Options> MyMatrixType;
typedef Array<Scalar, RowsAtCompileTime, ColsAtCompileTime, Options> MyArrayType;
\endcode

\li \c Scalar is the scalar type of the coefficients (e.g., \c float, \c double, \c bool, \c int, etc.).
\li \c RowsAtCompileTime and \c ColsAtCompileTime are the number of rows and columns of the matrix as known at compile-time or \c Dynamic.
\li \c Options can be \c ColMajor or \c RowMajor, default is \c ColMajor. (see class Matrix for more options)

All combinations are allowed: you can have a matrix with a fixed number of rows and a dynamic number of columns, etc. The following are all valid:
\code
Matrix<double, 6, Dynamic>                  // Dynamic number of columns (heap allocation)
Matrix<double, Dynamic, 2>                  // Dynamic number of rows (heap allocation)
Matrix<double, Dynamic, Dynamic, RowMajor>  // Fully dynamic, row major (heap allocation)
Matrix<double, 13, 3>                       // Fully fixed (usually allocated on stack)
\endcode

In most cases, you can simply use one of the convenience typedefs for \ref matrixtypedefs "matrices" and \ref arraytypedefs "arrays". Some examples:
<table class="example">
<tr><th>Matrices</th><th>Arrays</th></tr>
<tr><td>\code
Matrix<float,Dynamic,Dynamic>   <=>   MatrixXf
Matrix<double,Dynamic,1>        <=>   VectorXd
Matrix<int,1,Dynamic>           <=>   RowVectorXi
Matrix<float,3,3>               <=>   Matrix3f
Matrix<float,4,1>               <=>   Vector4f
\endcode</td><td>\code
Array<float,Dynamic,Dynamic>    <=>   ArrayXXf
Array<double,Dynamic,1>         <=>   ArrayXd
Array<int,1,Dynamic>            <=>   RowArrayXi
Array<float,3,3>                <=>   Array33f
Array<float,4,1>                <=>   Array4f
\endcode</td></tr>
</table>

Conversion between the matrix and array worlds:
\code
Array44f a1, a1;
Matrix4f m1, m2;
m1 = a1 * a2;                     // coeffwise product, implicit conversion from array to matrix.
a1 = m1 * m2;                     // matrix product, implicit conversion from matrix to array.
a2 = a1 + m1.array();             // mixing array and matrix is forbidden
m2 = a1.matrix() + m1;            // and explicit conversion is required.
ArrayWrapper<Matrix4f> m1a(m1);   // m1a is an alias for m1.array(), they share the same coefficients
MatrixWrapper<Array44f> a1m(a1);
\endcode

In the rest of this document we will use the following symbols to emphasize the features which are specifics to a given kind of object:
\li <a name="matrixonly"></a>\matrixworld linear algebra matrix and vector only
\li <a name="arrayonly"></a>\arrayworld array objects only

\subsection QuickRef_Basics Basic matrix manipulation

<table class="manual">
<tr><th></th><th>1D objects</th><th>2D objects</th><th>Notes</th></tr>
<tr><td>Constructors</td>
<td>\code
Vector4d  v4;
Vector2f  v1(x, y);
Array3i   v2(x, y, z);
Vector4d  v3(x, y, z, w);

VectorXf  v5; // empty object
ArrayXf   v6(size);
\endcode</td><td>\code
Matrix4f  m1;


MatrixXf  m5; // empty object
MatrixXf  m6(nb_rows, nb_columns);
\endcode</td><td class="note">
By default, the coefficients \n are left uninitialized</td></tr>
<tr class="alt"><td>Comma initializer</td>
<td>\code
Vector3f  v1;     v1 << x, y, z;
ArrayXf   v2(4);  v2 << 1, 2, 3, 4;

\endcode</td><td>\code
Matrix3f  m1;   m1 << 1, 2, 3,
                      4, 5, 6,
                      7, 8, 9;
\endcode</td><td></td></tr>

<tr><td>Comma initializer (bis)</td>
<td colspan="2">
\include Tutorial_commainit_02.cpp
</td>
<td>
output:
\verbinclude Tutorial_commainit_02.out
</td>
</tr>

<tr class="alt"><td>Runtime info</td>
<td>\code
vector.size();

vector.innerStride();
vector.data();
\endcode</td><td>\code
matrix.rows();          matrix.cols();
matrix.innerSize();     matrix.outerSize();
matrix.innerStride();   matrix.outerStride();
matrix.data();
\endcode</td><td class="note">Inner/Outer* are storage order dependent</td></tr>
<tr><td>Compile-time info</td>
<td colspan="2">\code
ObjectType::Scalar              ObjectType::RowsAtCompileTime
ObjectType::RealScalar          ObjectType::ColsAtCompileTime
ObjectType::Index               ObjectType::SizeAtCompileTime
\endcode</td><td></td></tr>
<tr class="alt"><td>Resizing</td>
<td>\code
vector.resize(size);


vector.resizeLike(other_vector);
vector.conservativeResize(size);
\endcode</td><td>\code
matrix.resize(nb_rows, nb_cols);
matrix.resize(Eigen::NoChange, nb_cols);
matrix.resize(nb_rows, Eigen::NoChange);
matrix.resizeLike(other_matrix);
matrix.conservativeResize(nb_rows, nb_cols);
\endcode</td><td class="note">no-op if the new sizes match,<br/>otherwise data are lost<br/><br/>resizing with data preservation</td></tr>

<tr><td>Coeff access with \n range checking</td>
<td>\code
vector(i)     vector.x()
vector[i]     vector.y()
              vector.z()
              vector.w()
\endcode</td><td>\code
matrix(i,j)
\endcode</td><td class="note">Range checking is disabled if \n NDEBUG or EIGEN_NO_DEBUG is defined</td></tr>

<tr class="alt"><td>Coeff access without \n range checking</td>
<td>\code
vector.coeff(i)
vector.coeffRef(i)
\endcode</td><td>\code
matrix.coeff(i,j)
matrix.coeffRef(i,j)
\endcode</td><td></td></tr>

<tr><td>Assignment/copy</td>
<td colspan="2">\code
object = expression;
object_of_float = expression_of_double.cast<float>();
\endcode</td><td class="note">the destination is automatically resized (if possible)</td></tr>

</table>

\subsection QuickRef_PredefMat Predefined Matrices

<table class="manual">
<tr>
  <th>Fixed-size matrix or vector</th>
  <th>Dynamic-size matrix</th>
  <th>Dynamic-size vector</th>
</tr>
<tr style="border-bottom-style: none;">
  <td>
\code
typedef {Matrix3f|Array33f} FixedXD;
FixedXD x;

x = FixedXD::Zero();
x = FixedXD::Ones();
x = FixedXD::Constant(value);
x = FixedXD::Random();
x = FixedXD::LinSpaced(size, low, high);

x.setZero();
x.setOnes();
x.setConstant(value);
x.setRandom();
x.setLinSpaced(size, low, high);
\endcode
  </td>
  <td>
\code
typedef {MatrixXf|ArrayXXf} Dynamic2D;
Dynamic2D x;

x = Dynamic2D::Zero(rows, cols);
x = Dynamic2D::Ones(rows, cols);
x = Dynamic2D::Constant(rows, cols, value);
x = Dynamic2D::Random(rows, cols);
N/A

x.setZero(rows, cols);
x.setOnes(rows, cols);
x.setConstant(rows, cols, value);
x.setRandom(rows, cols);
N/A
\endcode
  </td>
  <td>
\code
typedef {VectorXf|ArrayXf} Dynamic1D;
Dynamic1D x;

x = Dynamic1D::Zero(size);
x = Dynamic1D::Ones(size);
x = Dynamic1D::Constant(size, value);
x = Dynamic1D::Random(size);
x = Dynamic1D::LinSpaced(size, low, high);

x.setZero(size);
x.setOnes(size);
x.setConstant(size, value);
x.setRandom(size);
x.setLinSpaced(size, low, high);
\endcode
  </td>
</tr>

<tr><td colspan="3">Identity and \link MatrixBase::Unit basis vectors \endlink \matrixworld</td></tr>
<tr style="border-bottom-style: none;">
  <td>
\code
x = FixedXD::Identity();
x.setIdentity();

Vector3f::UnitX() // 1 0 0
Vector3f::UnitY() // 0 1 0
Vector3f::UnitZ() // 0 0 1
\endcode
  </td>
  <td>
\code
x = Dynamic2D::Identity(rows, cols);
x.setIdentity(rows, cols);


N/A
\endcode
  </td>
  <td>\code
N/A


VectorXf::Unit(size,i)
VectorXf::Unit(4,1) == Vector4f(0,1,0,0)
                    == Vector4f::UnitY()
\endcode
  </td>
</tr>
</table>


\subsection QuickRef_Map Mapping external arrays

<table class="manual">
<tr>
<td>Contiguous \n memory</td>
<td>\code
float data[] = {1,2,3,4};
Map<Vector3f> v1(data);       // uses v1 as a Vector3f object
Map<ArrayXf>  v2(data,3);     // uses v2 as a ArrayXf object
Map<Array22f> m1(data);       // uses m1 as a Array22f object
Map<MatrixXf> m2(data,2,2);   // uses m2 as a MatrixXf object
\endcode</td>
</tr>
<tr>
<td>Typical usage \n of strides</td>
<td>\code
float data[] = {1,2,3,4,5,6,7,8,9};
Map<VectorXf,0,InnerStride<2> >  v1(data,3);                      // = [1,3,5]
Map<VectorXf,0,InnerStride<> >   v2(data,3,InnerStride<>(3));     // = [1,4,7]
Map<MatrixXf,0,OuterStride<3> >  m2(data,2,3);                    // both lines     |1,4,7|
Map<MatrixXf,0,OuterStride<> >   m1(data,2,3,OuterStride<>(3));   // are equal to:  |2,5,8|
\endcode</td>
</tr>
</table>


<a href="#" class="top">top</a>
\section QuickRef_ArithmeticOperators Arithmetic Operators

<table class="manual">
<tr><td>
add \n subtract</td><td>\code
mat3 = mat1 + mat2;           mat3 += mat1;
mat3 = mat1 - mat2;           mat3 -= mat1;\endcode
</td></tr>
<tr class="alt"><td>
scalar product</td><td>\code
mat3 = mat1 * s1;             mat3 *= s1;           mat3 = s1 * mat1;
mat3 = mat1 / s1;             mat3 /= s1;\endcode
</td></tr>
<tr><td>
matrix/vector \n products \matrixworld</td><td>\code
col2 = mat1 * col1;
row2 = row1 * mat1;           row1 *= mat1;
mat3 = mat1 * mat2;           mat3 *= mat1; \endcode
</td></tr>
<tr class="alt"><td>
transposition \n adjoint \matrixworld</td><td>\code
mat1 = mat2.transpose();      mat1.transposeInPlace();
mat1 = mat2.adjoint();        mat1.adjointInPlace();
\endcode
</td></tr>
<tr><td>
\link MatrixBase::dot dot \endlink product \n inner product \matrixworld</td><td>\code
scalar = vec1.dot(vec2);
scalar = col1.adjoint() * col2;
scalar = (col1.adjoint() * col2).value();\endcode
</td></tr>
<tr class="alt"><td>
outer product \matrixworld</td><td>\code
mat = col1 * col2.transpose();\endcode
</td></tr>

<tr><td>
\link MatrixBase::norm() norm \endlink \n \link MatrixBase::normalized() normalization \endlink \matrixworld</td><td>\code
scalar = vec1.norm();         scalar = vec1.squaredNorm()
vec2 = vec1.normalized();     vec1.normalize(); // inplace \endcode
</td></tr>

<tr class="alt"><td>
\link MatrixBase::cross() cross product \endlink \matrixworld</td><td>\code
#include <Eigen/Geometry>
vec3 = vec1.cross(vec2);\endcode</td></tr>
</table>

<a href="#" class="top">top</a>
\section QuickRef_Coeffwise Coefficient-wise \& Array operators

In addition to the aforementioned operators, Eigen supports numerous coefficient-wise operator and functions.
Most of them unambiguously makes sense in array-world\arrayworld. The following operators are readily available for arrays,
or available through .array() for vectors and matrices:

<table class="manual">
<tr><td>Arithmetic operators</td><td>\code
array1 * array2     array1 / array2     array1 *= array2    array1 /= array2
array1 + scalar     array1 - scalar     array1 += scalar    array1 -= scalar
\endcode</td></tr>
<tr><td>Comparisons</td><td>\code
array1 < array2     array1 > array2     array1 < scalar     array1 > scalar
array1 <= array2    array1 >= array2    array1 <= scalar    array1 >= scalar
array1 == array2    array1 != array2    array1 == scalar    array1 != scalar
array1.min(array2)  array1.max(array2)  array1.min(scalar)  array1.max(scalar)
\endcode</td></tr>
<tr><td>Trigo, power, and \n misc functions \n and the STL-like variants</td><td>\code
array1.abs2()
array1.abs()                  abs(array1)
array1.sqrt()                 sqrt(array1)
array1.log()                  log(array1)
array1.log10()                log10(array1)
array1.exp()                  exp(array1)
array1.pow(array2)            pow(array1,array2)
array1.pow(scalar)            pow(array1,scalar)
                              pow(scalar,array2)
array1.square()
array1.cube()
array1.inverse()

array1.sin()                  sin(array1)
array1.cos()                  cos(array1)
array1.tan()                  tan(array1)
array1.asin()                 asin(array1)
array1.acos()                 acos(array1)
array1.atan()                 atan(array1)
array1.sinh()                 sinh(array1)
array1.cosh()                 cosh(array1)
array1.tanh()                 tanh(array1)
array1.arg()                  arg(array1)

array1.floor()                floor(array1)
array1.ceil()                 ceil(array1)
array1.round()                round(aray1)

array1.isFinite()             isfinite(array1)
array1.isInf()                isinf(array1)
array1.isNaN()                isnan(array1)
\endcode
</td></tr>
</table>


The following coefficient-wise operators are available for all kind of expressions (matrices, vectors, and arrays), and for both real or complex scalar types:

<table class="manual">
<tr><th>Eigen's API</th><th>STL-like APIs\arrayworld </th><th>Comments</th></tr>
<tr><td>\code
mat1.real()
mat1.imag()
mat1.conjugate()
\endcode
</td><td>\code
real(array1)
imag(array1)
conj(array1)
\endcode
</td><td>
\code
 // read-write, no-op for real expressions
 // read-only for real, read-write for complexes
 // no-op for real expressions
\endcode
</td></tr>
</table>

Some coefficient-wise operators are readily available for for matrices and vectors through the following cwise* methods:
<table class="manual">
<tr><th>Matrix API \matrixworld</th><th>Via Array conversions</th></tr>
<tr><td>\code
mat1.cwiseMin(mat2)         mat1.cwiseMin(scalar)
mat1.cwiseMax(mat2)         mat1.cwiseMax(scalar)
mat1.cwiseAbs2()
mat1.cwiseAbs()
mat1.cwiseSqrt()
mat1.cwiseInverse()
mat1.cwiseProduct(mat2)
mat1.cwiseQuotient(mat2)
mat1.cwiseEqual(mat2)       mat1.cwiseEqual(scalar)
mat1.cwiseNotEqual(mat2)
\endcode
</td><td>\code
mat1.array().min(mat2.array())    mat1.array().min(scalar)
mat1.array().max(mat2.array())    mat1.array().max(scalar)
mat1.array().abs2()
mat1.array().abs()
mat1.array().sqrt()
mat1.array().inverse()
mat1.array() * mat2.array()
mat1.array() / mat2.array()
mat1.array() == mat2.array()      mat1.array() == scalar
mat1.array() != mat2.array()
\endcode</td></tr>
</table>
The main difference between the two API is that the one based on cwise* methods returns an expression in the matrix world,
while the second one (based on .array()) returns an array expression.
Recall that .array() has no cost, it only changes the available API and interpretation of the data.

It is also very simple to apply any user defined function \c foo using DenseBase::unaryExpr together with <a href="http://en.cppreference.com/w/cpp/utility/functional/ptr_fun">std::ptr_fun</a> (c++03), <a href="http://en.cppreference.com/w/cpp/utility/functional/ref">std::ref</a> (c++11), or <a href="http://en.cppreference.com/w/cpp/language/lambda">lambdas</a> (c++11):
\code
mat1.unaryExpr(std::ptr_fun(foo));
mat1.unaryExpr(std::ref(foo));
mat1.unaryExpr([](double x) { return foo(x); });
\endcode


<a href="#" class="top">top</a>
\section QuickRef_Reductions Reductions

Eigen provides several reduction methods such as:
\link DenseBase::minCoeff() minCoeff() \endlink, \link DenseBase::maxCoeff() maxCoeff() \endlink,
\link DenseBase::sum() sum() \endlink, \link DenseBase::prod() prod() \endlink,
\link MatrixBase::trace() trace() \endlink \matrixworld,
\link MatrixBase::norm() norm() \endlink \matrixworld, \link MatrixBase::squaredNorm() squaredNorm() \endlink \matrixworld,
\link DenseBase::all() all() \endlink, and \link DenseBase::any() any() \endlink.
All reduction operations can be done matrix-wise,
\link DenseBase::colwise() column-wise \endlink or
\link DenseBase::rowwise() row-wise \endlink. Usage example:
<table class="manual">
<tr><td rowspan="3" style="border-right-style:dashed;vertical-align:middle">\code
      5 3 1
mat = 2 7 8
      9 4 6 \endcode
</td> <td>\code mat.minCoeff(); \endcode</td><td>\code 1 \endcode</td></tr>
<tr class="alt"><td>\code mat.colwise().minCoeff(); \endcode</td><td>\code 2 3 1 \endcode</td></tr>
<tr style="vertical-align:middle"><td>\code mat.rowwise().minCoeff(); \endcode</td><td>\code
1
2
4
\endcode</td></tr>
</table>

Special versions of \link DenseBase::minCoeff(IndexType*,IndexType*) const minCoeff \endlink and \link DenseBase::maxCoeff(IndexType*,IndexType*) const maxCoeff \endlink:
\code
int i, j;
s = vector.minCoeff(&i);        // s == vector[i]
s = matrix.maxCoeff(&i, &j);    // s == matrix(i,j)
\endcode
Typical use cases of all() and any():
\code
if((array1 > 0).all()) ...      // if all coefficients of array1 are greater than 0 ...
if((array1 < array2).any()) ... // if there exist a pair i,j such that array1(i,j) < array2(i,j) ...
\endcode


<a href="#" class="top">top</a>\section QuickRef_Blocks Sub-matrices

Read-write access to a \link DenseBase::col(Index) column \endlink
or a \link DenseBase::row(Index) row \endlink of a matrix (or array):
\code
mat1.row(i) = mat2.col(j);
mat1.col(j1).swap(mat1.col(j2));
\endcode

Read-write access to sub-vectors:
<table class="manual">
<tr>
<th>Default versions</th>
<th>Optimized versions when the size \n is known at compile time</th></tr>
<th></th>

<tr><td>\code vec1.head(n)\endcode</td><td>\code vec1.head<n>()\endcode</td><td>the first \c n coeffs </td></tr>
<tr><td>\code vec1.tail(n)\endcode</td><td>\code vec1.tail<n>()\endcode</td><td>the last \c n coeffs </td></tr>
<tr><td>\code vec1.segment(pos,n)\endcode</td><td>\code vec1.segment<n>(pos)\endcode</td>
    <td>the \c n coeffs in the \n range [\c pos : \c pos + \c n - 1]</td></tr>
<tr class="alt"><td colspan="3">

Read-write access to sub-matrices:</td></tr>
<tr>
  <td>\code mat1.block(i,j,rows,cols)\endcode
      \link DenseBase::block(Index,Index,Index,Index) (more) \endlink</td>
  <td>\code mat1.block<rows,cols>(i,j)\endcode
      \link DenseBase::block(Index,Index) (more) \endlink</td>
  <td>the \c rows x \c cols sub-matrix \n starting from position (\c i,\c j)</td></tr>
<tr><td>\code
 mat1.topLeftCorner(rows,cols)
 mat1.topRightCorner(rows,cols)
 mat1.bottomLeftCorner(rows,cols)
 mat1.bottomRightCorner(rows,cols)\endcode
 <td>\code
 mat1.topLeftCorner<rows,cols>()
 mat1.topRightCorner<rows,cols>()
 mat1.bottomLeftCorner<rows,cols>()
 mat1.bottomRightCorner<rows,cols>()\endcode
 <td>the \c rows x \c cols sub-matrix \n taken in one of the four corners</td></tr>
 <tr><td>\code
 mat1.topRows(rows)
 mat1.bottomRows(rows)
 mat1.leftCols(cols)
 mat1.rightCols(cols)\endcode
 <td>\code
 mat1.topRows<rows>()
 mat1.bottomRows<rows>()
 mat1.leftCols<cols>()
 mat1.rightCols<cols>()\endcode
 <td>specialized versions of block() \n when the block fit two corners</td></tr>
</table>


<a href="#" class="top">top</a>\section QuickRef_Misc Miscellaneous operations

\subsection QuickRef_Reverse Reverse
Vectors, rows, and/or columns of a matrix can be reversed (see DenseBase::reverse(), DenseBase::reverseInPlace(), VectorwiseOp::reverse()).
\code
vec.reverse()           mat.colwise().reverse()   mat.rowwise().reverse()
vec.reverseInPlace()
\endcode

\subsection QuickRef_Replicate Replicate
Vectors, matrices, rows, and/or columns can be replicated in any direction (see DenseBase::replicate(), VectorwiseOp::replicate())
\code
vec.replicate(times)                                          vec.replicate<Times>
mat.replicate(vertical_times, horizontal_times)               mat.replicate<VerticalTimes, HorizontalTimes>()
mat.colwise().replicate(vertical_times, horizontal_times)     mat.colwise().replicate<VerticalTimes, HorizontalTimes>()
mat.rowwise().replicate(vertical_times, horizontal_times)     mat.rowwise().replicate<VerticalTimes, HorizontalTimes>()
\endcode


<a href="#" class="top">top</a>\section QuickRef_DiagTriSymm Diagonal, Triangular, and Self-adjoint matrices
(matrix world \matrixworld)

\subsection QuickRef_Diagonal Diagonal matrices

<table class="example">
<tr><th>Operation</th><th>Code</th></tr>
<tr><td>
view a vector \link MatrixBase::asDiagonal() as a diagonal matrix \endlink \n </td><td>\code
mat1 = vec1.asDiagonal();\endcode
</td></tr>
<tr><td>
Declare a diagonal matrix</td><td>\code
DiagonalMatrix<Scalar,SizeAtCompileTime> diag1(size);
diag1.diagonal() = vector;\endcode
</td></tr>
<tr><td>Access the \link MatrixBase::diagonal() diagonal \endlink and \link MatrixBase::diagonal(Index) super/sub diagonals \endlink of a matrix as a vector (read/write)</td>
 <td>\code
vec1 = mat1.diagonal();        mat1.diagonal() = vec1;      // main diagonal
vec1 = mat1.diagonal(+n);      mat1.diagonal(+n) = vec1;    // n-th super diagonal
vec1 = mat1.diagonal(-n);      mat1.diagonal(-n) = vec1;    // n-th sub diagonal
vec1 = mat1.diagonal<1>();     mat1.diagonal<1>() = vec1;   // first super diagonal
vec1 = mat1.diagonal<-2>();    mat1.diagonal<-2>() = vec1;  // second sub diagonal
\endcode</td>
</tr>

<tr><td>Optimized products and inverse</td>
 <td>\code
mat3  = scalar * diag1 * mat1;
mat3 += scalar * mat1 * vec1.asDiagonal();
mat3 = vec1.asDiagonal().inverse() * mat1
mat3 = mat1 * diag1.inverse()
\endcode</td>
</tr>

</table>

\subsection QuickRef_TriangularView Triangular views

TriangularView gives a view on a triangular part of a dense matrix and allows to perform optimized operations on it. The opposite triangular part is never referenced and can be used to store other information.

\note The .triangularView() template member function requires the \c template keyword if it is used on an
object of a type that depends on a template parameter; see \ref TopicTemplateKeyword for details.

<table class="example">
<tr><th>Operation</th><th>Code</th></tr>
<tr><td>
Reference to a triangular with optional \n
unit or null diagonal (read/write):
</td><td>\code
m.triangularView<Xxx>()
\endcode \n
\c Xxx = ::Upper, ::Lower, ::StrictlyUpper, ::StrictlyLower, ::UnitUpper, ::UnitLower
</td></tr>
<tr><td>
Writing to a specific triangular part:\n (only the referenced triangular part is evaluated)
</td><td>\code
m1.triangularView<Eigen::Lower>() = m2 + m3 \endcode
</td></tr>
<tr><td>
Conversion to a dense matrix setting the opposite triangular part to zero:
</td><td>\code
m2 = m1.triangularView<Eigen::UnitUpper>()\endcode
</td></tr>
<tr><td>
Products:
</td><td>\code
m3 += s1 * m1.adjoint().triangularView<Eigen::UnitUpper>() * m2
m3 -= s1 * m2.conjugate() * m1.adjoint().triangularView<Eigen::Lower>() \endcode
</td></tr>
<tr><td>
Solving linear equations:\n
\f$ M_2 := L_1^{-1} M_2 \f$ \n
\f$ M_3 := {L_1^*}^{-1} M_3 \f$ \n
\f$ M_4 := M_4 U_1^{-1} \f$
</td><td>\n \code
L1.triangularView<Eigen::UnitLower>().solveInPlace(M2)
L1.triangularView<Eigen::Lower>().adjoint().solveInPlace(M3)
U1.triangularView<Eigen::Upper>().solveInPlace<OnTheRight>(M4)\endcode
</td></tr>
</table>

\subsection QuickRef_SelfadjointMatrix Symmetric/selfadjoint views

Just as for triangular matrix, you can reference any triangular part of a square matrix to see it as a selfadjoint
matrix and perform special and optimized operations. Again the opposite triangular part is never referenced and can be
used to store other information.

\note The .selfadjointView() template member function requires the \c template keyword if it is used on an
object of a type that depends on a template parameter; see \ref TopicTemplateKeyword for details.

<table class="example">
<tr><th>Operation</th><th>Code</th></tr>
<tr><td>
Conversion to a dense matrix:
</td><td>\code
m2 = m.selfadjointView<Eigen::Lower>();\endcode
</td></tr>
<tr><td>
Product with another general matrix or vector:
</td><td>\code
m3  = s1 * m1.conjugate().selfadjointView<Eigen::Upper>() * m3;
m3 -= s1 * m3.adjoint() * m1.selfadjointView<Eigen::Lower>();\endcode
</td></tr>
<tr><td>
Rank 1 and rank K update: \n
\f$ upper(M_1) \mathrel{{+}{=}} s_1 M_2 M_2^* \f$ \n
\f$ lower(M_1) \mathbin{{-}{=}} M_2^* M_2 \f$
</td><td>\n \code
M1.selfadjointView<Eigen::Upper>().rankUpdate(M2,s1);
M1.selfadjointView<Eigen::Lower>().rankUpdate(M2.adjoint(),-1); \endcode
</td></tr>
<tr><td>
Rank 2 update: (\f$ M \mathrel{{+}{=}} s u v^* + s v u^* \f$)
</td><td>\code
M.selfadjointView<Eigen::Upper>().rankUpdate(u,v,s);
\endcode
</td></tr>
<tr><td>
Solving linear equations:\n(\f$ M_2 := M_1^{-1} M_2 \f$)
</td><td>\code
// via a standard Cholesky factorization
m2 = m1.selfadjointView<Eigen::Upper>().llt().solve(m2);
// via a Cholesky factorization with pivoting
m2 = m1.selfadjointView<Eigen::Lower>().ldlt().solve(m2);
\endcode
</td></tr>
</table>

*/

/*
<table class="tutorial_code">
<tr><td>
\link MatrixBase::asDiagonal() make a diagonal matrix \endlink \n from a vector </td><td>\code
mat1 = vec1.asDiagonal();\endcode
</td></tr>
<tr><td>
Declare a diagonal matrix</td><td>\code
DiagonalMatrix<Scalar,SizeAtCompileTime> diag1(size);
diag1.diagonal() = vector;\endcode
</td></tr>
<tr><td>Access \link MatrixBase::diagonal() the diagonal and super/sub diagonals of a matrix \endlink as a vector (read/write)</td>
 <td>\code
vec1 = mat1.diagonal();            mat1.diagonal() = vec1;      // main diagonal
vec1 = mat1.diagonal(+n);          mat1.diagonal(+n) = vec1;    // n-th super diagonal
vec1 = mat1.diagonal(-n);          mat1.diagonal(-n) = vec1;    // n-th sub diagonal
vec1 = mat1.diagonal<1>();         mat1.diagonal<1>() = vec1;   // first super diagonal
vec1 = mat1.diagonal<-2>();        mat1.diagonal<-2>() = vec1;  // second sub diagonal
\endcode</td>
</tr>

<tr><td>View on a triangular part of a matrix (read/write)</td>
 <td>\code
mat2 = mat1.triangularView<Xxx>();
// Xxx = Upper, Lower, StrictlyUpper, StrictlyLower, UnitUpper, UnitLower
mat1.triangularView<Upper>() = mat2 + mat3; // only the upper part is evaluated and referenced
\endcode</td></tr>

<tr><td>View a triangular part as a symmetric/self-adjoint matrix (read/write)</td>
 <td>\code
mat2 = mat1.selfadjointView<Xxx>();     // Xxx = Upper or Lower
mat1.selfadjointView<Upper>() = mat2 + mat2.adjoint();  // evaluated and write to the upper triangular part only
\endcode</td></tr>

</table>

Optimized products:
\code
mat3 += scalar * vec1.asDiagonal() * mat1
mat3 += scalar * mat1 * vec1.asDiagonal()
mat3.noalias() += scalar * mat1.triangularView<Xxx>() * mat2
mat3.noalias() += scalar * mat2 * mat1.triangularView<Xxx>()
mat3.noalias() += scalar * mat1.selfadjointView<Upper or Lower>() * mat2
mat3.noalias() += scalar * mat2 * mat1.selfadjointView<Upper or Lower>()
mat1.selfadjointView<Upper or Lower>().rankUpdate(mat2);
mat1.selfadjointView<Upper or Lower>().rankUpdate(mat2.adjoint(), scalar);
\endcode

Inverse products: (all are optimized)
\code
mat3 = vec1.asDiagonal().inverse() * mat1
mat3 = mat1 * diag1.inverse()
mat1.triangularView<Xxx>().solveInPlace(mat2)
mat1.triangularView<Xxx>().solveInPlace<OnTheRight>(mat2)
mat2 = mat1.selfadjointView<Upper or Lower>().llt().solve(mat2)
\endcode

*/
}