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727 lines
26 KiB
727 lines
26 KiB
namespace Eigen {
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/** \eigenManualPage QuickRefPage Quick reference guide
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\eigenAutoToc
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<hr>
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<a href="#" class="top">top</a>
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\section QuickRef_Headers Modules and Header files
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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.
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<table class="manual">
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<tr><th>Module</th><th>Header file</th><th>Contents</th></tr>
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<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>
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<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>
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<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>
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<tr><td>\link Cholesky_Module Cholesky \endlink</td><td>\code#include <Eigen/Cholesky>\endcode</td><td>LLT and LDLT Cholesky factorization with solver</td></tr>
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<tr class="alt"><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>
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<tr><td>\link SVD_Module SVD \endlink</td><td>\code#include <Eigen/SVD>\endcode</td><td>SVD decomposition with least-squares solver (JacobiSVD)</td></tr>
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<tr class="alt"><td>\link QR_Module QR \endlink</td><td>\code#include <Eigen/QR>\endcode</td><td>QR decomposition with solver (HouseholderQR, ColPivHouseholderQR, FullPivHouseholderQR)</td></tr>
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<tr><td>\link Eigenvalues_Module Eigenvalues \endlink</td><td>\code#include <Eigen/Eigenvalues>\endcode</td><td>Eigenvalue, eigenvector decompositions (EigenSolver, SelfAdjointEigenSolver, ComplexEigenSolver)</td></tr>
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<tr class="alt"><td>\link Sparse_modules Sparse \endlink</td><td>\code#include <Eigen/Sparse>\endcode</td><td>%Sparse matrix storage and related basic linear algebra (SparseMatrix, DynamicSparseMatrix, SparseVector)</td></tr>
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<tr><td></td><td>\code#include <Eigen/Dense>\endcode</td><td>Includes Core, Geometry, LU, Cholesky, SVD, QR, and Eigenvalues header files</td></tr>
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<tr class="alt"><td></td><td>\code#include <Eigen/Eigen>\endcode</td><td>Includes %Dense and %Sparse header files (the whole Eigen library)</td></tr>
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</table>
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<a href="#" class="top">top</a>
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\section QuickRef_Types Array, matrix and vector types
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\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:
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\code
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typedef Matrix<Scalar, RowsAtCompileTime, ColsAtCompileTime, Options> MyMatrixType;
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typedef Array<Scalar, RowsAtCompileTime, ColsAtCompileTime, Options> MyArrayType;
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\endcode
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\li \c Scalar is the scalar type of the coefficients (e.g., \c float, \c double, \c bool, \c int, etc.).
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\li \c RowsAtCompileTime and \c ColsAtCompileTime are the number of rows and columns of the matrix as known at compile-time or \c Dynamic.
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\li \c Options can be \c ColMajor or \c RowMajor, default is \c ColMajor. (see class Matrix for more options)
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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:
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\code
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Matrix<double, 6, Dynamic> // Dynamic number of columns (heap allocation)
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Matrix<double, Dynamic, 2> // Dynamic number of rows (heap allocation)
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Matrix<double, Dynamic, Dynamic, RowMajor> // Fully dynamic, row major (heap allocation)
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Matrix<double, 13, 3> // Fully fixed (usually allocated on stack)
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\endcode
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In most cases, you can simply use one of the convenience typedefs for \ref matrixtypedefs "matrices" and \ref arraytypedefs "arrays". Some examples:
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<table class="example">
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<tr><th>Matrices</th><th>Arrays</th></tr>
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<tr><td>\code
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Matrix<float,Dynamic,Dynamic> <=> MatrixXf
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Matrix<double,Dynamic,1> <=> VectorXd
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Matrix<int,1,Dynamic> <=> RowVectorXi
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Matrix<float,3,3> <=> Matrix3f
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Matrix<float,4,1> <=> Vector4f
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\endcode</td><td>\code
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Array<float,Dynamic,Dynamic> <=> ArrayXXf
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Array<double,Dynamic,1> <=> ArrayXd
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Array<int,1,Dynamic> <=> RowArrayXi
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Array<float,3,3> <=> Array33f
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Array<float,4,1> <=> Array4f
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\endcode</td></tr>
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</table>
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Conversion between the matrix and array worlds:
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\code
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Array44f a1, a1;
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Matrix4f m1, m2;
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m1 = a1 * a2; // coeffwise product, implicit conversion from array to matrix.
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a1 = m1 * m2; // matrix product, implicit conversion from matrix to array.
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a2 = a1 + m1.array(); // mixing array and matrix is forbidden
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m2 = a1.matrix() + m1; // and explicit conversion is required.
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ArrayWrapper<Matrix4f> m1a(m1); // m1a is an alias for m1.array(), they share the same coefficients
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MatrixWrapper<Array44f> a1m(a1);
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\endcode
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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:
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\li <a name="matrixonly"></a>\matrixworld linear algebra matrix and vector only
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\li <a name="arrayonly"></a>\arrayworld array objects only
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\subsection QuickRef_Basics Basic matrix manipulation
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<table class="manual">
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<tr><th></th><th>1D objects</th><th>2D objects</th><th>Notes</th></tr>
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<tr><td>Constructors</td>
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<td>\code
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Vector4d v4;
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Vector2f v1(x, y);
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Array3i v2(x, y, z);
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Vector4d v3(x, y, z, w);
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VectorXf v5; // empty object
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ArrayXf v6(size);
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\endcode</td><td>\code
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Matrix4f m1;
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MatrixXf m5; // empty object
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MatrixXf m6(nb_rows, nb_columns);
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\endcode</td><td class="note">
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By default, the coefficients \n are left uninitialized</td></tr>
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<tr class="alt"><td>Comma initializer</td>
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<td>\code
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Vector3f v1; v1 << x, y, z;
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ArrayXf v2(4); v2 << 1, 2, 3, 4;
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\endcode</td><td>\code
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Matrix3f m1; m1 << 1, 2, 3,
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4, 5, 6,
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7, 8, 9;
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\endcode</td><td></td></tr>
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<tr><td>Comma initializer (bis)</td>
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<td colspan="2">
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\include Tutorial_commainit_02.cpp
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</td>
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<td>
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output:
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\verbinclude Tutorial_commainit_02.out
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</td>
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</tr>
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<tr class="alt"><td>Runtime info</td>
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<td>\code
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vector.size();
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vector.innerStride();
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vector.data();
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\endcode</td><td>\code
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matrix.rows(); matrix.cols();
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matrix.innerSize(); matrix.outerSize();
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matrix.innerStride(); matrix.outerStride();
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matrix.data();
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\endcode</td><td class="note">Inner/Outer* are storage order dependent</td></tr>
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<tr><td>Compile-time info</td>
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<td colspan="2">\code
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ObjectType::Scalar ObjectType::RowsAtCompileTime
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ObjectType::RealScalar ObjectType::ColsAtCompileTime
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ObjectType::Index ObjectType::SizeAtCompileTime
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\endcode</td><td></td></tr>
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<tr class="alt"><td>Resizing</td>
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<td>\code
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vector.resize(size);
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vector.resizeLike(other_vector);
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vector.conservativeResize(size);
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\endcode</td><td>\code
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matrix.resize(nb_rows, nb_cols);
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matrix.resize(Eigen::NoChange, nb_cols);
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matrix.resize(nb_rows, Eigen::NoChange);
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matrix.resizeLike(other_matrix);
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matrix.conservativeResize(nb_rows, nb_cols);
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\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>
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<tr><td>Coeff access with \n range checking</td>
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<td>\code
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vector(i) vector.x()
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vector[i] vector.y()
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vector.z()
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vector.w()
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\endcode</td><td>\code
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matrix(i,j)
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\endcode</td><td class="note">Range checking is disabled if \n NDEBUG or EIGEN_NO_DEBUG is defined</td></tr>
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<tr class="alt"><td>Coeff access without \n range checking</td>
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<td>\code
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vector.coeff(i)
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vector.coeffRef(i)
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\endcode</td><td>\code
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matrix.coeff(i,j)
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matrix.coeffRef(i,j)
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\endcode</td><td></td></tr>
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<tr><td>Assignment/copy</td>
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<td colspan="2">\code
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object = expression;
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object_of_float = expression_of_double.cast<float>();
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\endcode</td><td class="note">the destination is automatically resized (if possible)</td></tr>
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</table>
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\subsection QuickRef_PredefMat Predefined Matrices
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<table class="manual">
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<tr>
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<th>Fixed-size matrix or vector</th>
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<th>Dynamic-size matrix</th>
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<th>Dynamic-size vector</th>
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</tr>
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<tr style="border-bottom-style: none;">
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<td>
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\code
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typedef {Matrix3f|Array33f} FixedXD;
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FixedXD x;
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x = FixedXD::Zero();
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x = FixedXD::Ones();
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x = FixedXD::Constant(value);
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x = FixedXD::Random();
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x = FixedXD::LinSpaced(size, low, high);
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x.setZero();
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x.setOnes();
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x.setConstant(value);
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x.setRandom();
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x.setLinSpaced(size, low, high);
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\endcode
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</td>
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<td>
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\code
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typedef {MatrixXf|ArrayXXf} Dynamic2D;
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Dynamic2D x;
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x = Dynamic2D::Zero(rows, cols);
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x = Dynamic2D::Ones(rows, cols);
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x = Dynamic2D::Constant(rows, cols, value);
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x = Dynamic2D::Random(rows, cols);
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N/A
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x.setZero(rows, cols);
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x.setOnes(rows, cols);
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x.setConstant(rows, cols, value);
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x.setRandom(rows, cols);
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N/A
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\endcode
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</td>
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<td>
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\code
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typedef {VectorXf|ArrayXf} Dynamic1D;
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Dynamic1D x;
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x = Dynamic1D::Zero(size);
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x = Dynamic1D::Ones(size);
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x = Dynamic1D::Constant(size, value);
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x = Dynamic1D::Random(size);
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x = Dynamic1D::LinSpaced(size, low, high);
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x.setZero(size);
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x.setOnes(size);
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x.setConstant(size, value);
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x.setRandom(size);
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x.setLinSpaced(size, low, high);
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\endcode
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</td>
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</tr>
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<tr><td colspan="3">Identity and \link MatrixBase::Unit basis vectors \endlink \matrixworld</td></tr>
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<tr style="border-bottom-style: none;">
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<td>
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\code
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x = FixedXD::Identity();
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x.setIdentity();
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Vector3f::UnitX() // 1 0 0
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Vector3f::UnitY() // 0 1 0
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Vector3f::UnitZ() // 0 0 1
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\endcode
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</td>
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<td>
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\code
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x = Dynamic2D::Identity(rows, cols);
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x.setIdentity(rows, cols);
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N/A
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\endcode
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</td>
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<td>\code
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N/A
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VectorXf::Unit(size,i)
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VectorXf::Unit(4,1) == Vector4f(0,1,0,0)
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== Vector4f::UnitY()
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\endcode
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</td>
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</tr>
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</table>
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\subsection QuickRef_Map Mapping external arrays
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<table class="manual">
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<tr>
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<td>Contiguous \n memory</td>
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<td>\code
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float data[] = {1,2,3,4};
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Map<Vector3f> v1(data); // uses v1 as a Vector3f object
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Map<ArrayXf> v2(data,3); // uses v2 as a ArrayXf object
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Map<Array22f> m1(data); // uses m1 as a Array22f object
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Map<MatrixXf> m2(data,2,2); // uses m2 as a MatrixXf object
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\endcode</td>
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</tr>
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<tr>
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<td>Typical usage \n of strides</td>
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<td>\code
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float data[] = {1,2,3,4,5,6,7,8,9};
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Map<VectorXf,0,InnerStride<2> > v1(data,3); // = [1,3,5]
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Map<VectorXf,0,InnerStride<> > v2(data,3,InnerStride<>(3)); // = [1,4,7]
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Map<MatrixXf,0,OuterStride<3> > m2(data,2,3); // both lines |1,4,7|
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Map<MatrixXf,0,OuterStride<> > m1(data,2,3,OuterStride<>(3)); // are equal to: |2,5,8|
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\endcode</td>
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</tr>
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</table>
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<a href="#" class="top">top</a>
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\section QuickRef_ArithmeticOperators Arithmetic Operators
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<table class="manual">
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<tr><td>
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add \n subtract</td><td>\code
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mat3 = mat1 + mat2; mat3 += mat1;
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mat3 = mat1 - mat2; mat3 -= mat1;\endcode
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</td></tr>
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<tr class="alt"><td>
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scalar product</td><td>\code
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mat3 = mat1 * s1; mat3 *= s1; mat3 = s1 * mat1;
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mat3 = mat1 / s1; mat3 /= s1;\endcode
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</td></tr>
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<tr><td>
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matrix/vector \n products \matrixworld</td><td>\code
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col2 = mat1 * col1;
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row2 = row1 * mat1; row1 *= mat1;
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mat3 = mat1 * mat2; mat3 *= mat1; \endcode
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</td></tr>
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<tr class="alt"><td>
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transposition \n adjoint \matrixworld</td><td>\code
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mat1 = mat2.transpose(); mat1.transposeInPlace();
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mat1 = mat2.adjoint(); mat1.adjointInPlace();
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\endcode
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</td></tr>
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<tr><td>
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\link MatrixBase::dot() dot \endlink product \n inner product \matrixworld</td><td>\code
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scalar = vec1.dot(vec2);
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scalar = col1.adjoint() * col2;
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scalar = (col1.adjoint() * col2).value();\endcode
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</td></tr>
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<tr class="alt"><td>
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outer product \matrixworld</td><td>\code
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mat = col1 * col2.transpose();\endcode
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</td></tr>
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<tr><td>
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\link MatrixBase::norm() norm \endlink \n \link MatrixBase::normalized() normalization \endlink \matrixworld</td><td>\code
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scalar = vec1.norm(); scalar = vec1.squaredNorm()
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vec2 = vec1.normalized(); vec1.normalize(); // inplace \endcode
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</td></tr>
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<tr class="alt"><td>
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\link MatrixBase::cross() cross product \endlink \matrixworld</td><td>\code
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#include <Eigen/Geometry>
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vec3 = vec1.cross(vec2);\endcode</td></tr>
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</table>
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<a href="#" class="top">top</a>
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\section QuickRef_Coeffwise Coefficient-wise \& Array operators
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Coefficient-wise operators for matrices and vectors:
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<table class="manual">
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<tr><th>Matrix API \matrixworld</th><th>Via Array conversions</th></tr>
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<tr><td>\code
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mat1.cwiseMin(mat2)
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mat1.cwiseMax(mat2)
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mat1.cwiseAbs2()
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mat1.cwiseAbs()
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mat1.cwiseSqrt()
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mat1.cwiseProduct(mat2)
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mat1.cwiseQuotient(mat2)\endcode
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</td><td>\code
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mat1.array().min(mat2.array())
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mat1.array().max(mat2.array())
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mat1.array().abs2()
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mat1.array().abs()
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mat1.array().sqrt()
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mat1.array() * mat2.array()
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mat1.array() / mat2.array()
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\endcode</td></tr>
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</table>
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It is also very simple to apply any user defined function \c foo using DenseBase::unaryExpr together with std::ptr_fun:
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\code mat1.unaryExpr(std::ptr_fun(foo))\endcode
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Array operators:\arrayworld
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<table class="manual">
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<tr><td>Arithmetic operators</td><td>\code
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array1 * array2 array1 / array2 array1 *= array2 array1 /= array2
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array1 + scalar array1 - scalar array1 += scalar array1 -= scalar
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\endcode</td></tr>
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<tr><td>Comparisons</td><td>\code
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array1 < array2 array1 > array2 array1 < scalar array1 > scalar
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array1 <= array2 array1 >= array2 array1 <= scalar array1 >= scalar
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array1 == array2 array1 != array2 array1 == scalar array1 != scalar
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\endcode</td></tr>
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<tr><td>Trigo, power, and \n misc functions \n and the STL variants</td><td>\code
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array1.min(array2)
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array1.max(array2)
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array1.abs2()
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array1.abs() abs(array1)
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array1.sqrt() sqrt(array1)
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array1.log() log(array1)
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array1.exp() exp(array1)
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array1.pow(exponent) pow(array1,exponent)
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array1.square()
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array1.cube()
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array1.inverse()
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array1.sin() sin(array1)
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array1.cos() cos(array1)
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array1.tan() tan(array1)
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array1.asin() asin(array1)
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array1.acos() acos(array1)
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\endcode
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</td></tr>
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</table>
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<a href="#" class="top">top</a>
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\section QuickRef_Reductions Reductions
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Eigen provides several reduction methods such as:
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\link DenseBase::minCoeff() minCoeff() \endlink, \link DenseBase::maxCoeff() maxCoeff() \endlink,
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\link DenseBase::sum() sum() \endlink, \link DenseBase::prod() prod() \endlink,
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\link MatrixBase::trace() trace() \endlink \matrixworld,
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\link MatrixBase::norm() norm() \endlink \matrixworld, \link MatrixBase::squaredNorm() squaredNorm() \endlink \matrixworld,
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\link DenseBase::all() all() \endlink, and \link DenseBase::any() any() \endlink.
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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
|
|
|
|
*/
|
|
}
|