SPARSE MATRICESPRESENTATION BY ZAIN ZAFAR

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 |

└ ┘

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.

Symmetric classification of Sparse

Matrix:

Triangular Matrices:

Triangular matrices have the same

number of rows as they have

columns; that is, they have n rows

and n columns. In triangular matrix

both main and lower diagonals

are filled with non-zero values or

main diagonal and upper storing

diagonals are filled with non-zero

values.

Band Matrices:

An important special type of

sparse matrices is band

matrix, defined as follows. The

lower bandwidth of a matrix A is

the smallest number p such that

the entry aij vanishes whenever i > j

+ p.

Types of Triangular Matrices:

Upper triangular matrix: Lower triangular matrix:

A matrix A is a lower triangular

matrix if its nonzero elements are

found only in the lower triangle of

the matrix, including the main

diagonal;

A matrix A is an upper triangular

matrix if its nonzero elements are

found only in the upper triangle of

the matrix, including the main

diagonal;

Types of Band Matrices:

Diagonal matrix

Let A be a square matrix (with

entries in any field). If all off-

diagonal entries of A are zero,

then A is a diagonal matrix.

Tri-diagonal matrix

A tri-diagonal matrix is a matrix

that has nonzero elements only in

the main diagonal, the first

diagonal below this, and the first

diagonal above the main

diagonal.

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.