265 lines
12 KiB
Plaintext
265 lines
12 KiB
Plaintext
namespace Eigen {
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/** \eigenManualPage TutorialMatrixClass The Matrix class
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\eigenAutoToc
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In Eigen, all matrices and vectors are objects of the Matrix template class.
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Vectors are just a special case of matrices, with either 1 row or 1 column.
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\section TutorialMatrixFirst3Params The first three template parameters of Matrix
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The Matrix class takes six template parameters, but for now it's enough to
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learn about the first three first parameters. The three remaining parameters have default
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values, which for now we will leave untouched, and which we
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\ref TutorialMatrixOptTemplParams "discuss below".
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The three mandatory template parameters of Matrix are:
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\code
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Matrix<typename Scalar, int RowsAtCompileTime, int ColsAtCompileTime>
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\endcode
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\li \c Scalar is the scalar type, i.e. the type of the coefficients.
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That is, if you want a matrix of floats, choose \c float here.
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See \ref TopicScalarTypes "Scalar types" for a list of all supported
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scalar types and for how to extend support to new types.
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\li \c RowsAtCompileTime and \c ColsAtCompileTime are the number of rows
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and columns of the matrix as known at compile time (see
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\ref TutorialMatrixDynamic "below" for what to do if the number is not
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known at compile time).
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We offer a lot of convenience typedefs to cover the usual cases. For example, \c Matrix4f is
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a 4x4 matrix of floats. Here is how it is defined by Eigen:
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\code
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typedef Matrix<float, 4, 4> Matrix4f;
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\endcode
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We discuss \ref TutorialMatrixTypedefs "below" these convenience typedefs.
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\section TutorialMatrixVectors Vectors
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As mentioned above, in Eigen, vectors are just a special case of
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matrices, with either 1 row or 1 column. The case where they have 1 column is the most common;
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such vectors are called column-vectors, often abbreviated as just vectors. In the other case
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where they have 1 row, they are called row-vectors.
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For example, the convenience typedef \c Vector3f is a (column) vector of 3 floats. It is defined as follows by Eigen:
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\code
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typedef Matrix<float, 3, 1> Vector3f;
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\endcode
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We also offer convenience typedefs for row-vectors, for example:
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\code
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typedef Matrix<int, 1, 2> RowVector2i;
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\endcode
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\section TutorialMatrixDynamic The special value Dynamic
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Of course, Eigen is not limited to matrices whose dimensions are known at compile time.
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The \c RowsAtCompileTime and \c ColsAtCompileTime template parameters can take the special
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value \c Dynamic which indicates that the size is unknown at compile time, so must
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be handled as a run-time variable. In Eigen terminology, such a size is referred to as a
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\em dynamic \em size; while a size that is known at compile time is called a
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\em fixed \em size. For example, the convenience typedef \c MatrixXd, meaning
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a matrix of doubles with dynamic size, is defined as follows:
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\code
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typedef Matrix<double, Dynamic, Dynamic> MatrixXd;
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\endcode
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And similarly, we define a self-explanatory typedef \c VectorXi as follows:
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\code
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typedef Matrix<int, Dynamic, 1> VectorXi;
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\endcode
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You can perfectly have e.g. a fixed number of rows with a dynamic number of columns, as in:
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\code
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Matrix<float, 3, Dynamic>
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\endcode
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\section TutorialMatrixConstructors Constructors
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A default constructor is always available, never performs any dynamic memory allocation, and never initializes the matrix coefficients. You can do:
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\code
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Matrix3f a;
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MatrixXf b;
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\endcode
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Here,
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\li \c a is a 3x3 matrix, with a static float[9] array of uninitialized coefficients,
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\li \c b is a dynamic-size matrix whose size is currently 0x0, and whose array of
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coefficients hasn't yet been allocated at all.
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Constructors taking sizes are also available. For matrices, the number of rows is always passed first.
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For vectors, just pass the vector size. They allocate the array of coefficients
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with the given size, but don't initialize the coefficients themselves:
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\code
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MatrixXf a(10,15);
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VectorXf b(30);
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\endcode
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Here,
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\li \c a is a 10x15 dynamic-size matrix, with allocated but currently uninitialized coefficients.
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\li \c b is a dynamic-size vector of size 30, with allocated but currently uninitialized coefficients.
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In order to offer a uniform API across fixed-size and dynamic-size matrices, it is legal to use these
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constructors on fixed-size matrices, even if passing the sizes is useless in this case. So this is legal:
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\code
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Matrix3f a(3,3);
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\endcode
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and is a no-operation.
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Finally, we also offer some constructors to initialize the coefficients of small fixed-size vectors up to size 4:
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\code
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Vector2d a(5.0, 6.0);
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Vector3d b(5.0, 6.0, 7.0);
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Vector4d c(5.0, 6.0, 7.0, 8.0);
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\endcode
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\section TutorialMatrixCoeffAccessors Coefficient accessors
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The primary coefficient accessors and mutators in Eigen are the overloaded parenthesis operators.
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For matrices, the row index is always passed first. For vectors, just pass one index.
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The numbering starts at 0. This example is self-explanatory:
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<table class="example">
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<tr><th>Example:</th><th>Output:</th></tr>
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<tr><td>
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\include tut_matrix_coefficient_accessors.cpp
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</td>
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<td>
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\verbinclude tut_matrix_coefficient_accessors.out
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</td></tr></table>
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Note that the syntax <tt> m(index) </tt>
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is not restricted to vectors, it is also available for general matrices, meaning index-based access
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in the array of coefficients. This however depends on the matrix's storage order. All Eigen matrices default to
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column-major storage order, but this can be changed to row-major, see \ref TopicStorageOrders "Storage orders".
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The operator[] is also overloaded for index-based access in vectors, but keep in mind that C++ doesn't allow operator[] to
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take more than one argument. We restrict operator[] to vectors, because an awkwardness in the C++ language
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would make matrix[i,j] compile to the same thing as matrix[j] !
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\section TutorialMatrixCommaInitializer Comma-initialization
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%Matrix and vector coefficients can be conveniently set using the so-called \em comma-initializer syntax.
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For now, it is enough to know this example:
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<table class="example">
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<tr><th>Example:</th><th>Output:</th></tr>
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<tr>
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<td>\include Tutorial_commainit_01.cpp </td>
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<td>\verbinclude Tutorial_commainit_01.out </td>
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</tr></table>
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The right-hand side can also contain matrix expressions as discussed in \ref TutorialAdvancedInitialization "this page".
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\section TutorialMatrixSizesResizing Resizing
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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.
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<table class="example">
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<tr><th>Example:</th><th>Output:</th></tr>
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<tr>
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<td>\include tut_matrix_resize.cpp </td>
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<td>\verbinclude tut_matrix_resize.out </td>
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</tr></table>
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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.
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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.
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All these methods are still available on fixed-size matrices, for the sake of API uniformity. Of course, you can't actually
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resize a fixed-size matrix. Trying to change a fixed size to an actually different value will trigger an assertion failure;
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but the following code is legal:
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<table class="example">
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<tr><th>Example:</th><th>Output:</th></tr>
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<tr>
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<td>\include tut_matrix_resize_fixed_size.cpp </td>
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<td>\verbinclude tut_matrix_resize_fixed_size.out </td>
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</tr></table>
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\section TutorialMatrixAssignment Assignment and resizing
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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:
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<table class="example">
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<tr><th>Example:</th><th>Output:</th></tr>
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<tr>
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<td>\include tut_matrix_assignment_resizing.cpp </td>
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<td>\verbinclude tut_matrix_assignment_resizing.out </td>
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</tr></table>
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Of course, if the left-hand side is of fixed size, resizing it is not allowed.
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If you do not want this automatic resizing to happen (for example for debugging purposes), you can disable it, see
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\ref TopicResizing "this page".
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\section TutorialMatrixFixedVsDynamic Fixed vs. Dynamic size
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When should one use fixed sizes (e.g. \c Matrix4f), and when should one prefer dynamic sizes (e.g. \c MatrixXf)?
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The simple answer is: use fixed
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sizes for very small sizes where you can, and use dynamic sizes for larger sizes or where you have to. For small sizes,
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especially for sizes smaller than (roughly) 16, using fixed sizes is hugely beneficial
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to performance, as it allows Eigen to avoid dynamic memory allocation and to unroll
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loops. Internally, a fixed-size Eigen matrix is just a plain static array, i.e. doing
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\code Matrix4f mymatrix; \endcode
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really amounts to just doing
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\code float mymatrix[16]; \endcode
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so this really has zero runtime cost. By contrast, the array of a dynamic-size matrix
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is always allocated on the heap, so doing
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\code MatrixXf mymatrix(rows,columns); \endcode
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amounts to doing
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\code float *mymatrix = new float[rows*columns]; \endcode
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and in addition to that, the MatrixXf object stores its number of rows and columns as
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member variables.
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The limitation of using fixed sizes, of course, is that this is only possible
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when you know the sizes at compile time. Also, for large enough sizes, say for sizes
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greater than (roughly) 32, the performance benefit of using fixed sizes becomes negligible.
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Worse, trying to create a very large matrix using fixed sizes could result in a stack overflow,
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since Eigen will try to allocate the array as a static array, which by default goes on the stack.
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Finally, depending on circumstances, Eigen can also be more aggressive trying to vectorize
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(use SIMD instructions) when dynamic sizes are used, see \ref TopicVectorization "Vectorization".
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\section TutorialMatrixOptTemplParams Optional template parameters
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We mentioned at the beginning of this page that the Matrix class takes six template parameters,
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but so far we only discussed the first three. The remaining three parameters are optional. Here is
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the complete list of template parameters:
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\code
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Matrix<typename Scalar,
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int RowsAtCompileTime,
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int ColsAtCompileTime,
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int Options = 0,
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int MaxRowsAtCompileTime = RowsAtCompileTime,
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int MaxColsAtCompileTime = ColsAtCompileTime>
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\endcode
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\li \c Options is a bit field. Here, we discuss only one bit: \c RowMajor. It specifies that the matrices
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of this type use row-major storage order; by default, the storage order is column-major. See the page on
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\ref TopicStorageOrders "storage orders". For example, this type means row-major 3x3 matrices:
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\code
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Matrix<float, 3, 3, RowMajor>
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\endcode
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\li \c MaxRowsAtCompileTime and \c MaxColsAtCompileTime are useful when you want to specify that, even though
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the exact sizes of your matrices are not known at compile time, a fixed upper bound is known at
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compile time. The biggest reason why you might want to do that is to avoid dynamic memory allocation.
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For example the following matrix type uses a static array of 12 floats, without dynamic memory allocation:
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\code
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Matrix<float, Dynamic, Dynamic, 0, 3, 4>
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\endcode
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\section TutorialMatrixTypedefs Convenience typedefs
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Eigen defines the following Matrix typedefs:
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\li MatrixNt for Matrix<type, N, N>. For example, MatrixXi for Matrix<int, Dynamic, Dynamic>.
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\li VectorNt for Matrix<type, N, 1>. For example, Vector2f for Matrix<float, 2, 1>.
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\li RowVectorNt for Matrix<type, 1, N>. For example, RowVector3d for Matrix<double, 1, 3>.
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Where:
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\li N can be any one of \c 2, \c 3, \c 4, or \c X (meaning \c Dynamic).
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\li t can be any one of \c i (meaning int), \c f (meaning float), \c d (meaning double),
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\c cf (meaning complex<float>), or \c cd (meaning complex<double>). The fact that typedefs are only
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defined for these five types doesn't mean that they are the only supported scalar types. For example,
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all standard integer types are supported, see \ref TopicScalarTypes "Scalar types".
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*/
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}
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