2. What are SPARSE MATRICES?
One of the most important developments in scientific computing is sparse
matrix technology. This technology includes the data structures to
represent the matrices, the techniques for manipulating them, the
algorithms used, and the efficient mapping of the data structures and
algorithms to high performance. A sparse matrix is a matrix having a
relatively small number of nonzero elements.
Consider the following as an example of a sparse matrix A:
┌ ┐
| 11 0 13 0 0 0 |
| 21 22 0 24 0 0 |
| 0 32 33 0 35 0 |
| 0 0 43 44 0 46 |
| 51 0 0 54 55 0 |
| 61 62 0 0 65 66 |
└ ┘
3. Sparse Matrices
in Data Structures
Sparse matrix is a two-dimensional array in which most of
the elements have null value or zero “0”. In large number
of applications sparse matrices are used. It is wastage of
memory and processing time if we store null values of a
matrix in array. To avoid such circumstances different
techniques are used such as linked list. In simple words
sparse matrices are matrices that allow special
techniques to take advantage of the large number of
null elements and the structure.
4. Symmetric classification of Sparse
Matrix:
Triangular Matrices: Band Matrices:
Triangular matrices have the same An important special type of
number of rows as they have sparse matrices is band
columns; that is, they have n rows matrix, defined as follows. The
and n columns. In triangular matrix lower bandwidth of a matrix A is
both main and lower diagonals the smallest number p such that
are filled with non-zero values or the entry aij vanishes whenever i > j
main diagonal and upper storing + p.
diagonals are filled with non-zero
values.
5. Types of Triangular Matrices:
Upper triangular matrix: Lower triangular matrix:
A matrix A is an upper triangular A matrix A is a lower triangular
matrix if its nonzero elements are matrix if its nonzero elements are
found only in the upper triangle of found only in the lower triangle of
the matrix, including the main the matrix, including the main
diagonal; diagonal;
6. Types of Band Matrices:
Diagonal matrix Tri-diagonal matrix
Let A be a square matrix (with A tri-diagonal matrix is a matrix
entries in any field). If all off- that has nonzero elements only in
diagonal entries of A are zero, the main diagonal, the first
then A is a diagonal matrix. diagonal below this, and the first
diagonal above the main
diagonal.
7. Importance of Sparse
Matrices
Sparse matrices occur in many
applications including solving partial
differential equations (PDEs), text-
document matrices used for latent
semantic indexing (LSI), linear and
nonlinear optimization, and
manipulating network and graph
models.
