From 86a192681e465e81cf1444e46fbadb1fb5049d54 Mon Sep 17 00:00:00 2001 From: Gael Guennebaud Date: Fri, 16 Jan 2009 17:20:12 +0000 Subject: * Started a tutorial page for sparse matrix. * Updated Mainpage.dox's directives used by kde's scripts --- doc/TutorialSparse.dox | 196 +++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 196 insertions(+) create mode 100644 doc/TutorialSparse.dox (limited to 'doc/TutorialSparse.dox') diff --git a/doc/TutorialSparse.dox b/doc/TutorialSparse.dox new file mode 100644 index 000000000..2b3fdc61c --- /dev/null +++ b/doc/TutorialSparse.dox @@ -0,0 +1,196 @@ +namespace Eigen { + +/** \page TutorialSparse Tutorial - Getting started with the sparse module + \ingroup Tutorial + +
\ref index "Overview" + | \ref TutorialCore "Core features" + | \ref TutorialGeometry "Geometry" + | \ref TutorialAdvancedLinearAlgebra "Advanced linear algebra" + | \b Sparse \b matrix +
+ +\b Table \b of \b contents \n + - \ref TutorialSparseIntro + - \ref TutorialSparseFilling + - \ref TutorialSparseFeatureSet + - \ref TutorialSparseDirectSolvers +
+ +\section TutorialSparseIntro Introduction + +In many applications (e.g., finite element methods) it is common to deal with very large matrices where only a few coefficients are different than zero. Both in term of memory consumption and performance, it is fundamental to use an adequate representation storing only nonzero coefficients. Such a matrix is called a sparse matrix. + +In order to get the best of the Eigen's sparse matrix, it is important to have a rough idea of the way they are internally stored. In Eigen we chose to use the common and generic Compressed Column/Row Storage scheme. Let m be a column-major sparse matrix. Then its nonzero coefficients are sequentially stored in memory in a col-major order (\em values). A second array of integer stores the respective row index of each coefficient (\em inner \em indices). Finally, a third array of integer, having the same length than the number of columns, stores the index in the previous arrays of the first element of each column (\em outer \em indices). + +Here is an example, with the matrix: + + + + + + +
03000
2200017
75010
00000
001408
+ +and its internal representation using the Compressed Column Storage format: + + + +
Values: 22735141178
Inner indices: 1202 42 14
+Outer indices:
02456\em 7
+ +As you can guess, here the storage order is even more important than with dense matrix. We will therefore often make a clear difference between the \em inner dimension and the \em outer dimension. For instance, it is easy to loop over the coefficients of an \em inner \em vector (e.g., a column of a col-major matrix), but very inefficient to do the same for an \em outer \em vector (e.g., a row of a col-major matrix). + + +\b Eigen's \b sparse \b matrix \b and \b vector \b class \n +Before using any sparse feature you must include the Sparse header file: +\code +#include +\endcode +In Eigen, sparse matrices are handled by the SparseMatrix class: +\code +SparseMatrix > m1(1000,2000); // declare a 1000x2000 col-major sparse matrix of complex +SparseMatrix m2(1000,2000); // declare a 1000x2000 row-major sparse matrix of double +\endcode +where \c Scalar is the type of the coefficient and \c Options is a set of bit flags allowing to control the shape and storage of the matrix. A typical choice for \c Options is \c RowMajor to get a row-major matrix (the default is col-major). + +Unlike the dense path, the sparse module provides a special class to declare a sparse vector: +\code +SparseVector > v1(1000); // declare a column sparse vector of complex of size 1000 +SparseVector v2(1000); // declare a row sparse vector of double of size 1000 +\endcode +Note that here the size of a vector denotes its dimension and not the number of nonzero coefficients which is initially zero. + +\b Matrix \b and \b vector \b properties \n + + + + + + + + + + +
Standard \n dimensions\code +mat.rows() +mat.cols()\endcode\code +vec.size() \endcode
Sizes along the \n inner/outer dimensions\code +mat.innerSize() +mat.outerSize()\endcode
Number of non \n zero coefficiens\code +mat.nonZeros() \endcode\code +vec.nonZeros() \endcode
+ +\b Iterating \b over \b the \b nonzero \b coefficients \n + +Iterating over the coefficients of a sparse matrix can be done only in the same order than the storage order. Here is an example: + + +
+\code +SparseMatrix m1(rows,cols); +for (int k=0; k::InnerIterator it(m1,k); it; ++it) + { + it.value(); + it.row(); // row index + it.col(); // col index (here it is equal to k) + it.index(); // inner index, here it is equal to it.row() + } +\endcode + +\code +SparseVector v1(size); +for (SparseVector::InnerIterator it(v1); it; ++it) +{ + it.value(); // == v1[ it.index() ] + it.index(); +} +\endcode +
+ + +\section TutorialSparseFilling Filling a sparse matrix + +Unlike dense matrix, filling a sparse matrix efficiently is a though problem which requires some special care from the user. Indeed, directly filling in a random order a matrix in a compressed storage format would requires a lot of searches and memory copies, and some trade of between the efficiency and the ease of use have to be found. In Eigen we currently provide three ways to set the elements of a sparse matrix: + +1 - If you can set the coefficients in exactly the same order that the storage order, then the matrix can be filled directly and very efficiently. Here is an example initializing a random, row-major sparse matrix: +\code +SparseMatrix m(rows,cols); +m.startFill(rows*cols*percent_of_non_zero); // estimate of the number of nonzeros (optional) +for (int i=0; i\ m(rows,cols); +m.startFill(rows*cols*percent_of_non_zero); // estimate of the number of nonzeros (optional) +for (int i=0; i\ m(rows,cols); +{ + RandomSetter > setter(m); + for (int k=0; k\().solveTriangular(dv2); +\endcode + +The product of a sparse matrix A time a dense matrix/vector dv with A symmetric can be optimized by telling that to Eigen: +\code +res = A.marked() * dv; // if all coefficients of A are stored +res = A.marked() * dv; // if only the upper part of A is stored +res = A.marked() * dv; // if only the lower part of A is stored +\endcode + + +\section TutorialSparseDirectSolvers Using the direct solvers + +TODO + +\subsection TutorialSparseDirectSolvers_LLT LLT +Cholmod, Taucs. + +\subsection TutorialSparseDirectSolvers_LDLT LDLT + + +\subsection TutorialSparseDirectSolvers_LU LU +SuperLU, UmfPack. + +*/ + +} -- cgit v1.2.3