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 namespace Eigen { /** \eigenManualPage TutorialMatrixClass The Matrix class \eigenAutoToc In Eigen, all matrices and vectors are objects of the Matrix template class. Vectors are just a special case of matrices, with either 1 row or 1 column. \section TutorialMatrixFirst3Params The first three template parameters of Matrix The Matrix class takes six template parameters, but for now it's enough to learn about the first three first parameters. The three remaining parameters have default values, which for now we will leave untouched, and which we \ref TutorialMatrixOptTemplParams "discuss below". The three mandatory template parameters of Matrix are: \code Matrix \endcode \li \c Scalar is the scalar type, i.e. the type of the coefficients. That is, if you want a matrix of floats, choose \c float here. See \ref TopicScalarTypes "Scalar types" for a list of all supported scalar types and for how to extend support to new types. \li \c RowsAtCompileTime and \c ColsAtCompileTime are the number of rows and columns of the matrix as known at compile time (see \ref TutorialMatrixDynamic "below" for what to do if the number is not known at compile time). We offer a lot of convenience typedefs to cover the usual cases. For example, \c Matrix4f is a 4x4 matrix of floats. Here is how it is defined by Eigen: \code typedef Matrix Matrix4f; \endcode We discuss \ref TutorialMatrixTypedefs "below" these convenience typedefs. \section TutorialMatrixVectors Vectors As mentioned above, in Eigen, vectors are just a special case of matrices, with either 1 row or 1 column. The case where they have 1 column is the most common; such vectors are called column-vectors, often abbreviated as just vectors. In the other case where they have 1 row, they are called row-vectors. For example, the convenience typedef \c Vector3f is a (column) vector of 3 floats. It is defined as follows by Eigen: \code typedef Matrix Vector3f; \endcode We also offer convenience typedefs for row-vectors, for example: \code typedef Matrix RowVector2i; \endcode \section TutorialMatrixDynamic The special value Dynamic Of course, Eigen is not limited to matrices whose dimensions are known at compile time. The \c RowsAtCompileTime and \c ColsAtCompileTime template parameters can take the special value \c Dynamic which indicates that the size is unknown at compile time, so must be handled as a run-time variable. In Eigen terminology, such a size is referred to as a \em dynamic \em size; while a size that is known at compile time is called a \em fixed \em size. For example, the convenience typedef \c MatrixXd, meaning a matrix of doubles with dynamic size, is defined as follows: \code typedef Matrix MatrixXd; \endcode And similarly, we define a self-explanatory typedef \c VectorXi as follows: \code typedef Matrix VectorXi; \endcode You can perfectly have e.g. a fixed number of rows with a dynamic number of columns, as in: \code Matrix \endcode \section TutorialMatrixConstructors Constructors A default constructor is always available, never performs any dynamic memory allocation, and never initializes the matrix coefficients. You can do: \code Matrix3f a; MatrixXf b; \endcode Here, \li \c a is a 3-by-3 matrix, with a plain float array of uninitialized coefficients, \li \c b is a dynamic-size matrix whose size is currently 0-by-0, and whose array of coefficients hasn't yet been allocated at all. Constructors taking sizes are also available. For matrices, the number of rows is always passed first. For vectors, just pass the vector size. They allocate the array of coefficients with the given size, but don't initialize the coefficients themselves: \code MatrixXf a(10,15); VectorXf b(30); \endcode Here, \li \c a is a 10x15 dynamic-size matrix, with allocated but currently uninitialized coefficients. \li \c b is a dynamic-size vector of size 30, with allocated but currently uninitialized coefficients. In order to offer a uniform API across fixed-size and dynamic-size matrices, it is legal to use these constructors on fixed-size matrices, even if passing the sizes is useless in this case. So this is legal: \code Matrix3f a(3,3); \endcode and is a no-operation. Finally, we also offer some constructors to initialize the coefficients of small fixed-size vectors up to size 4: \code Vector2d a(5.0, 6.0); Vector3d b(5.0, 6.0, 7.0); Vector4d c(5.0, 6.0, 7.0, 8.0); \endcode \section TutorialMatrixCoeffAccessors Coefficient accessors The primary coefficient accessors and mutators in Eigen are the overloaded parenthesis operators. For matrices, the row index is always passed first. For vectors, just pass one index. The numbering starts at 0. This example is self-explanatory:
Example:Output:
\include tut_matrix_coefficient_accessors.cpp \verbinclude tut_matrix_coefficient_accessors.out
Note that the syntax m(index) is not restricted to vectors, it is also available for general matrices, meaning index-based access in the array of coefficients. This however depends on the matrix's storage order. All Eigen matrices default to column-major storage order, but this can be changed to row-major, see \ref TopicStorageOrders "Storage orders". The operator[] is also overloaded for index-based access in vectors, but keep in mind that C++ doesn't allow operator[] to take more than one argument. We restrict operator[] to vectors, because an awkwardness in the C++ language would make matrix[i,j] compile to the same thing as matrix[j] ! \section TutorialMatrixCommaInitializer Comma-initialization %Matrix and vector coefficients can be conveniently set using the so-called \em comma-initializer syntax. For now, it is enough to know this example:
Example:Output:
\include Tutorial_commainit_01.cpp \verbinclude Tutorial_commainit_01.out
The right-hand side can also contain matrix expressions as discussed in \ref TutorialAdvancedInitialization "this page". \section TutorialMatrixSizesResizing Resizing The current size of a matrix can be retrieved by \link EigenBase::rows() rows()\endlink, \link EigenBase::cols() cols() \endlink and \link EigenBase::size() size()\endlink. These methods return the number of rows, the number of columns and the number of coefficients, respectively. Resizing a dynamic-size matrix is done by the \link PlainObjectBase::resize(Index,Index) resize() \endlink method.
Example:Output:
\include tut_matrix_resize.cpp \verbinclude tut_matrix_resize.out
The resize() method is a no-operation if the actual matrix size doesn't change; otherwise it is destructive: the values of the coefficients may change. If you want a conservative variant of resize() which does not change the coefficients, use \link PlainObjectBase::conservativeResize() conservativeResize()\endlink, see \ref TopicResizing "this page" for more details. All these methods are still available on fixed-size matrices, for the sake of API uniformity. Of course, you can't actually resize a fixed-size matrix. Trying to change a fixed size to an actually different value will trigger an assertion failure; but the following code is legal:
Example:Output:
\include tut_matrix_resize_fixed_size.cpp \verbinclude tut_matrix_resize_fixed_size.out
\section TutorialMatrixAssignment Assignment and resizing Assignment is the action of copying a matrix into another, using \c operator=. Eigen resizes the matrix on the left-hand side automatically so that it matches the size of the matrix on the right-hand size. For example:
Example:Output:
\include tut_matrix_assignment_resizing.cpp \verbinclude tut_matrix_assignment_resizing.out
Of course, if the left-hand side is of fixed size, resizing it is not allowed. If you do not want this automatic resizing to happen (for example for debugging purposes), you can disable it, see \ref TopicResizing "this page". \section TutorialMatrixFixedVsDynamic Fixed vs. Dynamic size When should one use fixed sizes (e.g. \c Matrix4f), and when should one prefer dynamic sizes (e.g. \c MatrixXf)? The simple answer is: use fixed sizes for very small sizes where you can, and use dynamic sizes for larger sizes or where you have to. For small sizes, especially for sizes smaller than (roughly) 16, using fixed sizes is hugely beneficial to performance, as it allows Eigen to avoid dynamic memory allocation and to unroll loops. Internally, a fixed-size Eigen matrix is just a plain array, i.e. doing \code Matrix4f mymatrix; \endcode really amounts to just doing \code float mymatrix; \endcode so this really has zero runtime cost. By contrast, the array of a dynamic-size matrix is always allocated on the heap, so doing \code MatrixXf mymatrix(rows,columns); \endcode amounts to doing \code float *mymatrix = new float[rows*columns]; \endcode and in addition to that, the MatrixXf object stores its number of rows and columns as member variables. The limitation of using fixed sizes, of course, is that this is only possible when you know the sizes at compile time. Also, for large enough sizes, say for sizes greater than (roughly) 32, the performance benefit of using fixed sizes becomes negligible. Worse, trying to create a very large matrix using fixed sizes inside a function could result in a stack overflow, since Eigen will try to allocate the array automatically as a local variable, and this is normally done on the stack. Finally, depending on circumstances, Eigen can also be more aggressive trying to vectorize (use SIMD instructions) when dynamic sizes are used, see \ref TopicVectorization "Vectorization". \section TutorialMatrixOptTemplParams Optional template parameters We mentioned at the beginning of this page that the Matrix class takes six template parameters, but so far we only discussed the first three. The remaining three parameters are optional. Here is the complete list of template parameters: \code Matrix \endcode \li \c Options is a bit field. Here, we discuss only one bit: \c RowMajor. It specifies that the matrices of this type use row-major storage order; by default, the storage order is column-major. See the page on \ref TopicStorageOrders "storage orders". For example, this type means row-major 3x3 matrices: \code Matrix \endcode \li \c MaxRowsAtCompileTime and \c MaxColsAtCompileTime are useful when you want to specify that, even though the exact sizes of your matrices are not known at compile time, a fixed upper bound is known at compile time. The biggest reason why you might want to do that is to avoid dynamic memory allocation. For example the following matrix type uses a plain array of 12 floats, without dynamic memory allocation: \code Matrix \endcode \section TutorialMatrixTypedefs Convenience typedefs Eigen defines the following Matrix typedefs: \li MatrixNt for Matrix. For example, MatrixXi for Matrix. \li VectorNt for Matrix. For example, Vector2f for Matrix. \li RowVectorNt for Matrix. For example, RowVector3d for Matrix. Where: \li N can be any one of \c 2, \c 3, \c 4, or \c X (meaning \c Dynamic). \li t can be any one of \c i (meaning int), \c f (meaning float), \c d (meaning double), \c cf (meaning complex), or \c cd (meaning complex). The fact that typedefs are only defined for these five types doesn't mean that they are the only supported scalar types. For example, all standard integer types are supported, see \ref TopicScalarTypes "Scalar types". */ }