2. Basic Algorithms

Abstract

A sparse matrix is one whose entries are mostly zero. There are many ways of storing a sparse matrix. Whichever method is chosen, some form of compact data structure is required that avoids storing the numerically zero entries in the matrix. It needs to be simple and flexible so that it can be used in a wide range of matrix operations. This need is met by the primary data structure in CSparse, a compressed-column matrix. Basic matrix operations that operate on this data structure are presented below, including matrix-vector multiplication, matrix-matrix multiplication, matrix addition, and transpose.2.1 Sparse matrix data structuresThe simplest sparse matrix data structure is a list of the nonzero entries in arbitrary order. The list consists of two integer arrays i and j and one real array x of length equal to the number of entries in the matrix. For example, the matrixA=[4.503.203.12.900.901.73.003.50.401.0](2.1)is represented in zero-based triplet form below. A zero-based data structure for an m-by-n matrix contains row and column indices in the range 0 to m−1 and n−1, respectively. A one-based data structure has row and column indices that start with one. The one-based convention is used in linear algebra and is presented to the MATLAB user. Internally in MATLAB and also in CSparse, all algorithms and data structures are zero-based. Thus, both conventions are used in this book, depending on the context. In particular, all C code is zero-based. All MATLAB expressions, and all linear algebraic expressions, are one-based. All pseudocode is zero-based, since it closely relates to a corresponding C code. Graph examples are one-based, since they usually relate to an example matrix (which are also one-based).