Initialization time for the implicit method can be fairly large as the

sparse matrices are allocated.

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 [26], whereas that the FE-BI-MLFMA of [10] requires large memory during the inverse of the FEM sparse matrix solved by a sparse direct solver based on the multifrontal approach [27].

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 [[mu]'.

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 mentioned before.