Initialization time for the implicit method can be fairly large as the sparse matrices
The book follows the structures of the matrices, from tri-diagonal matrices resulting from one-dimensional mesh-based methods, through multi-diagonal or block-diagonal matrices, and ending with general sparse matrices
Our experiments on a set of 20 sparse matrices
show that IPCSR outperforms PCSR for all test cases.
For sparse matrices
, a factorization may create excessive fill-ins of the zero entries, which results in significant memory and operation costs.
In sparse matrices
only non-zero elements are stored in memory.
Real problems such as fluid dynamics in irregular and 3D-domains or finite elements calculations are more complex and the solutions involve sparse matrices
algebra, factorization or inversion of large linear systems, submatrices, complicated indexing, etc.
The reason is that the required memory in the conventional FE-BI-MLFMA only needs to store sparse matrices
, whereas that the FE-BI-MLFMA of  requires large memory during the inverse of the FEM sparse matrix solved by a sparse direct solver based on the multifrontal approach .
Most of the research on iterative methods deals with iterative methods for solving linear systems of equalities and inequalities for sparse matrices
, the most important method being GMRES method.
To do this, the cross section of the waveguide was extracted from the array shown in Figure 8(a) and placed along the diagonal of the sparse matrices
According to properties of multiscale kernels, appropriate data structures and numerical techniques should be implemented to search nonzero elements from the sparse matrix and to solve the linear system effectively since the linear systems compose sparse matrices
2) Development of compression schemes for sparse matrices
in the form of EKMR and TMR.
A simple example refers to the concept of sparse matrices