Eigen sparse matrix operation

In many applications （ for example , The finite element method ）, Usually deal with very large matrices , Among them, only a few coefficients are not zero . under these circumstances , You can reduce memory consumption and improve performance by using a dedicated representation that stores only non-zero coefficients . Such a matrix is called sparse matrix .

1. Sparse matrix module

modular	The header file	Content
SparseCore	#include <Eigen/SparseCore>	SparseMatrix and SparseVector class 、 matrix assemble 、 Basic sparse linear algebra （ Including sparse triangle solver ）
SparseCholesky	#include <Eigen/SparseCholesky>	Directly use sparse LLT and LDLT Cholesky Decomposition to solve the sparse self adjoint positive definite problem
SparseLU	#include<Eigen/SparseLU>	sparse LU Decomposition to solve the general square sparse system
SparseQR	#include<Eigen/SparseQR	Sparse for solving sparse linear least squares problem QR decompose
IterativeLinearSolvers	#include <Eigen/IterativeLinearSolvers>	Iterative solver for solving large general linear square problems （ Including the self adjoint positive definite problem ）
Sparse	#include <Eigen/Sparse>	Including all the above modules

2. Sparse matrix format

SparseMatrix Class is Eigen The main sparse matrix representation of the sparse module ; It provides high performance and low memory usage . It implements widely used compressed columns （ Or yes ） A more general variant of the storage scheme . It consists of four compact arrays ：

Values: Store non-zero coefficient values .
InnerIndices: Store non-zero rows （ Or column ） Indexes .
OuterStarts: For each column （ Or yes ） Store the first non-zero index in the first two arrays .
InnerNNZs: Store each column （ Corresponding line ） Nonzero number of . inner It refers to an internal vector , It is the column of the column master matrix , Or row of the row master matrix . The word external refers to the other direction .

at present , The elements of a given internal vector are always sorted by adding an internal index . “_” Represents the available space to quickly insert new elements . Assume no reallocation , The insertion of random elements is therefore in O(nnz_j) in , among nnz_j Is the nonzero number of the corresponding internal vector . On the other hand , It is more efficient to insert elements with increased internal indexes into a given internal vector , Because this only needs to increase the corresponding InnerNNZs entry , namely O(1) operation .

The case of no available space is a special case , Called compression mode . It corresponds to the widely used compressed column （ Or yes ） Storage plan （CCS or CRS）. By calling SparseMatrix::makeCompressed() function , Any SparseMatrix Convert to this form . under these circumstances , It can be noted that InnerNNZs An array with the OuterStarts It's redundant , Because our equation ：InnerNNZs[j] = OuterStarts[j+1]-OuterStarts[j]. therefore , Actually call SparseMatrix::makeCompressed() This buffer will be released .

reference

Eigen On sparse matrix _ The blog of rainbow cat who loves Mathematics -CSDN Blog _eigen sparsematrix

Eigen course 3 - Sparse matrix operation _xuezhisdc The blog of -CSDN Blog _eigen sparse matrix

Eigen course 5 - Solve sparse linear equations _xuezhisdc The blog of -CSDN Blog _eigen System of linear equations

Eigen: Solving Sparse Linear Systems 

Eigen: Quick reference guide for sparse matrices