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162 lines
6.7 KiB
162 lines
6.7 KiB
namespace Eigen {
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/** \eigenManualPage TutorialAdvancedInitialization Advanced initialization
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This page discusses several advanced methods for initializing matrices. It gives more details on the
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comma-initializer, which was introduced before. It also explains how to get special matrices such as the
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identity matrix and the zero matrix.
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\eigenAutoToc
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\section TutorialAdvancedInitializationCommaInitializer The comma initializer
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Eigen offers a comma initializer syntax which allows the user to easily set all the coefficients of a matrix,
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vector or array. Simply list the coefficients, starting at the top-left corner and moving from left to right
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and from the top to the bottom. The size of the object needs to be specified beforehand. If you list too few
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or too many coefficients, Eigen will complain.
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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 Tutorial_commainit_01.cpp
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</td>
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<td>
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\verbinclude Tutorial_commainit_01.out
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</td></tr></table>
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Moreover, the elements of the initialization list may themselves be vectors or matrices. A common use is
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to join vectors or matrices together. For example, here is how to join two row vectors together. Remember
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that you have to set the size before you can use the comma initializer.
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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 Tutorial_AdvancedInitialization_Join.cpp
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</td>
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<td>
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\verbinclude Tutorial_AdvancedInitialization_Join.out
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</td></tr></table>
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We can use the same technique to initialize matrices with a block structure.
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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 Tutorial_AdvancedInitialization_Block.cpp
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</td>
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<td>
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\verbinclude Tutorial_AdvancedInitialization_Block.out
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</td></tr></table>
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The comma initializer can also be used to fill block expressions such as <tt>m.row(i)</tt>. Here is a more
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complicated way to get the same result as in the first example above:
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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 Tutorial_commainit_01b.cpp
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</td>
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<td>
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\verbinclude Tutorial_commainit_01b.out
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</td></tr></table>
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\section TutorialAdvancedInitializationSpecialMatrices Special matrices and arrays
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The Matrix and Array classes have static methods like \link DenseBase::Zero() Zero()\endlink, which can be
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used to initialize all coefficients to zero. There are three variants. The first variant takes no arguments
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and can only be used for fixed-size objects. If you want to initialize a dynamic-size object to zero, you need
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to specify the size. Thus, the second variant requires one argument and can be used for one-dimensional
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dynamic-size objects, while the third variant requires two arguments and can be used for two-dimensional
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objects. All three variants are illustrated in the following 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><td>
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\include Tutorial_AdvancedInitialization_Zero.cpp
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</td>
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<td>
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\verbinclude Tutorial_AdvancedInitialization_Zero.out
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</td></tr></table>
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Similarly, the static method \link DenseBase::Constant() Constant\endlink(value) sets all coefficients to \c value.
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If the size of the object needs to be specified, the additional arguments go before the \c value
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argument, as in <tt>MatrixXd::Constant(rows, cols, value)</tt>. The method \link DenseBase::Random() Random()
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\endlink fills the matrix or array with random coefficients. The identity matrix can be obtained by calling
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\link MatrixBase::Identity() Identity()\endlink; this method is only available for Matrix, not for Array,
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because "identity matrix" is a linear algebra concept. The method
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\link DenseBase::LinSpaced LinSpaced\endlink(size, low, high) is only available for vectors and
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one-dimensional arrays; it yields a vector of the specified size whose coefficients are equally spaced between
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\c low and \c high. The method \c LinSpaced() is illustrated in the following example, which prints a table
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with angles in degrees, the corresponding angle in radians, and their sine and cosine.
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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 Tutorial_AdvancedInitialization_LinSpaced.cpp
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</td>
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<td>
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\verbinclude Tutorial_AdvancedInitialization_LinSpaced.out
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</td></tr></table>
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This example shows that objects like the ones returned by LinSpaced() can be assigned to variables (and
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expressions). Eigen defines utility functions like \link DenseBase::setZero() setZero()\endlink,
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\link MatrixBase::setIdentity() \endlink and \link DenseBase::setLinSpaced() \endlink to do this
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conveniently. The following example contrasts three ways to construct the matrix
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\f$ J = \bigl[ \begin{smallmatrix} O & I \\ I & O \end{smallmatrix} \bigr] \f$: using static methods and
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assignment, using static methods and the comma-initializer, or using the setXxx() methods.
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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 Tutorial_AdvancedInitialization_ThreeWays.cpp
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</td>
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<td>
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\verbinclude Tutorial_AdvancedInitialization_ThreeWays.out
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</td></tr></table>
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A summary of all pre-defined matrix, vector and array objects can be found in the \ref QuickRefPage.
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\section TutorialAdvancedInitializationTemporaryObjects Usage as temporary objects
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As shown above, static methods as Zero() and Constant() can be used to initialize variables at the time of
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declaration or at the right-hand side of an assignment operator. You can think of these methods as returning a
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matrix or array; in fact, they return so-called \ref TopicEigenExpressionTemplates "expression objects" which
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evaluate to a matrix or array when needed, so that this syntax does not incur any overhead.
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These expressions can also be used as a temporary object. The second example in
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the \ref GettingStarted guide, which we reproduce here, already illustrates this.
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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 QuickStart_example2_dynamic.cpp
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</td>
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<td>
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\verbinclude QuickStart_example2_dynamic.out
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</td></tr></table>
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The expression <tt>m + MatrixXf::Constant(3,3,1.2)</tt> constructs the 3-by-3 matrix expression with all its coefficients
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equal to 1.2 plus the corresponding coefficient of \a m.
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The comma-initializer, too, can also be used to construct temporary objects. The following example constructs a random
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matrix of size 2-by-3, and then multiplies this matrix on the left with
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\f$ \bigl[ \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \bigr] \f$.
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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 Tutorial_AdvancedInitialization_CommaTemporary.cpp
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</td>
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<td>
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\verbinclude Tutorial_AdvancedInitialization_CommaTemporary.out
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</td></tr></table>
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The \link CommaInitializer::finished() finished() \endlink method is necessary here to get the actual matrix
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object once the comma initialization of our temporary submatrix is done.
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*/
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}
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