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Sparse matrices


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Sparse matrices

  2. 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. 3. Sparse Matricesin Data StructuresSparse matrix is a two-dimensional array in which most ofthe elements have null value or zero “0”. In large numberof applications sparse matrices are used. It is wastage ofmemory and processing time if we store null values of amatrix in array. To avoid such circumstances differenttechniques are used such as linked list. In simple wordssparse matrices are matrices that allow specialtechniques to take advantage of the large number ofnull elements and the structure.
  4. 4. Symmetric classification of SparseMatrix:  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. 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. 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. 7. Importance of SparseMatricesSparse matrices occur in manyapplications including solving partialdifferential equations (PDEs), text-document matrices used for latentsemantic indexing (LSI), linear andnonlinear optimization, andmanipulating network and graphmodels.