Table of contents
Declaring sparse matrices and vectors
The SparseMatrix class is the main sparse matrix representation of the Eigen's sparse module which offers high performance, low memory usage, and compatibility with most of sparse linear algebra packages. Because of its limited flexibility, we also provide a DynamicSparseMatrix variante taillored for low-level sparse matrix assembly. Both of them can be either row major or column major:
#include <Eigen/Sparse> SparseMatrix<std::complex<float> > m1(1000,2000); // declare a 1000x2000 col-major compressed sparse matrix of complex<float> SparseMatrix<double,RowMajor> m2(1000,2000); // declare a 1000x2000 row-major compressed sparse matrix of double DynamicSparseMatrix<std::complex<float> > m1(1000,2000); // declare a 1000x2000 col-major dynamic sparse matrix of complex<float> DynamicSparseMatrix<double,RowMajor> m2(1000,2000); // declare a 1000x2000 row-major dynamic sparse matrix of double
Although a sparse matrix could also be used to represent a sparse vector, for that purpose it is better to use the specialized SparseVector class:
SparseVector<std::complex<float> > v1(1000); // declare a column sparse vector of complex<float> of size 1000 SparseVector<double,RowMajor> v2(1000); // declare a row sparse vector of double of size 1000
Overview of the internal sparse storage
In order to get the best of the Eigen's sparse objects, it is important to have a rough idea of the way they are internally stored. The SparseMatrix class implements the common and generic Compressed Column/Row Storage scheme. It consists of three compact arrays storing the values with their respective inner coordinates, and pointer indices to the begining of each outer vector. For instance, let m
be a column-major sparse matrix. Then its nonzero coefficients are sequentially stored in memory in a column-major order (values). A second array of integer stores the respective row index of each coefficient (inner 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 (outer indices).
Here is an example, with the matrix:
0 | 3 | 0 | 0 | 0 |
22 | 0 | 0 | 0 | 17 |
7 | 5 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 14 | 0 | 8 |
and its internal representation using the Compressed Column Storage format:
Values: | 22 | 7 | 3 | 5 | 14 | 1 | 17 | 8 |
Inner indices: | 1 | 2 | 0 | 2 | 4 | 2 | 1 | 4 |
0 | 2 | 4 | 5 | 6 | 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 inner and outer dimensions. For instance, it is easy to loop over the coefficients of an inner vector (e.g., a column of a column-major matrix), but completely inefficient to do the same for an outer vector (e.g., a row of a col-major matrix).
The SparseVector class implements the same compressed storage scheme but, of course, without any outer index buffer.
Since all nonzero coefficients of such a matrix are sequentially stored in memory, random insertion of new nonzeros can be extremely costly. To overcome this limitation, Eigen's sparse module provides a DynamicSparseMatrix class which is basically implemented as an array of SparseVector. In other words, a DynamicSparseMatrix is a SparseMatrix where the values and inner-indices arrays have been splitted into multiple small and resizable arrays. Assuming the number of nonzeros per inner vector is relatively low, this slight modification allow for very fast random insertion at the cost of a slight memory overhead and a lost of compatibility with other sparse libraries used by some of our highlevel solvers. Note that the major memory overhead comes from the extra memory preallocated by each inner vector to avoid an expensive memory reallocation at every insertion.
To summarize, it is recommanded to use a SparseMatrix whenever this is possible, and reserve the use of DynamicSparseMatrix for matrix assembly purpose when a SparseMatrix is not flexible enough. The respective pro/cons of both representations are summarized in the following table:
SparseMatrix | DynamicSparseMatrix | |
memory usage | *** | ** |
sorted insertion | *** | *** |
random insertion in sorted inner vector | ** | ** |
sorted insertion in random inner vector | - | *** |
random insertion | - | ** |
coeff wise unary operators | *** | *** |
coeff wise binary operators | *** | *** |
matrix products | *** | **(*) |
transpose | ** | *** |
redux | *** | ** |
*= scalar | *** | ** |
Compatibility with highlevel solvers (TAUCS, Cholmod, SuperLU, UmfPack) | *** | - |
Matrix and vector properties
Here mat and vec represents any sparse-matrix and sparse-vector types respectively.
Standard dimensions | mat.rows() mat.cols() | vec.size() |
Sizes along the inner/outer dimensions | mat.innerSize() mat.outerSize() | |
Number of non zero coefficiens | mat.nonZeros() | vec.nonZeros() |
Iterating over the nonzero coefficients
Iterating over the coefficients of a sparse matrix can be done only in the same order than the storage order. Here is an example:
SparseMatrixType mat(rows,cols); for (int k=0; k\<m1.outerSize(); ++k) for (SparseMatrixType::InnerIterator it(mat,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() } | SparseVector<double> vec(size); for (SparseVector<double>::InnerIterator it(vec); it; ++it) { it.value(); // == vec[ it.index() ] it.index(); } |
DynamicSparseMatrix<float> aux(1000,1000); for (...) for each i for each j interacting with i aux.coeffRef(i,j) += foo(o1,o2); SparseMatrix<float> mat(aux); // convert the DynamicSparseMatrix to a SparseMatrix
Sometimes, however, we simply want to set all the coefficients of a matrix before using it through standard matrix operations (addition, product, etc.). In that case it faster to use the low-level startFill()/fill()/fillrand()/endFill() interface. Even though this interface is availabe for both sparse matrix types, their respective restrictions slightly differ from one representation to the other. In all case, a call to startFill() set the matrix to zero, and the fill*() functions will fail if the coefficient already exist.
As a first difference, for SparseMatrix, the fill*() functions can only be called inside a startFill()/endFill() pair, and no other member functions are allowed during the filling process, i.e., until endFill() has been called. On the other hand, a DynamicSparseMatrix is always in a stable state, and the startFill()/endFill() functions are only for compatibility purpose.
Another difference is that the fill*() functions must be called with increasing outer indices for a SparseMatrix, while they can be random for a DynamicSparseMatrix.
Finally, the fill() function assumes the coefficient are inserted in a sorted order per inner vector, while the fillrand() variante allows random insertions (the outer indices must still be sorted for SparseMatrix).
Some examples:
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:
SparseMatrix<double,RowMajor> m(rows,cols); m.startFill(rows*cols*percent_of_non_zero); // estimate of the number of nonzeros (optional) for (int i=0; i\<rows; ++i) for (int j=0; j\<cols; ++j) if (rand()\<percent_of_non_zero) m.fill(i,j) = rand(); m.endFill();
2 - If you can set each outer vector in a consistent order, but do not have sorted data for each inner vector, then you can use fillrand() instead of fill():
SparseMatrix<double,RowMajor> m(rows,cols); m.startFill(rows*cols*percent_of_non_zero); // estimate of the number of nonzeros (optional) for (int i=0; i\<rows; ++i) for (int k=0; k\<cols*percent_of_non_zero; ++k) m.fillrand(i,rand(0,cols)) = rand(); m.endFill();
3 - Eventually, if none of the above solution is practicable for you, then you have to use a RandomSetter which temporarily wraps the matrix into a more flexible hash map allowing complete random accesses:
SparseMatrix<double,RowMajor> m(rows,cols); { RandomSetter<SparseMatrix<double,RowMajor> > setter(m); for (int k=0; k\<cols*rows*percent_of_non_zero; ++k) setter(rand(0,rows), rand(0,cols)) = rand(); }
m
is set at the destruction of the setter, hence the use of a nested block. This imposed syntax has the advantage to emphasize the critical section where m is not valid and cannot be used.SparseMatrixType sm1, sm2, sm3; sm3 = sm1.transpose() + sm2; // invalid sm3 = SparseMatrixType(sm1.transpose()) + sm2; // correct
Here are some examples of the supported operations:
s_1 *= 0.5; sm4 = sm1 + sm2 + sm3; // only if s_1, s_2 and s_3 have the same storage order sm3 = sm1 * sm2; dv3 = sm1 * dv2; dm3 = sm1 * dm2; dm3 = dm2 * sm1; sm3 = sm1.cwise() * sm2; // only if s_1 and s_2 have the same storage order dv2 = sm1.marked<UpperTriangular>().solveTriangular(dv2);
The product of a sparse matrix A time a dense matrix/vector dv with A symmetric can be optimized by telling that to Eigen:
res = A.marked<SeflAdjoint>() * dv; // if all coefficients of A are stored res = A.marked<SeflAdjoint|UpperTriangular>() * dv; // if only the upper part of A is stored res = A.marked<SeflAdjoint|LowerTriangular>() * dv; // if only the lower part of A is stored