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SPARSE MATRICES
PRESENTATION 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:                     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.
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;
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.
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.

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Sparse matrices

  • 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.