(32) where for are sub-matrices and, for are sub-vectors. Such a system of equations can be solved by solving the following sub-problems

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Module 4 : Solving Linear Algebraic Equations Section 4 : Direct Methods for Solving Sparse Linear Systems 4 Direct Methods for Solving Sparse Linear Systems A system of Linear equations --------(32)

1 Module 4 : Solving Linear Algebraic Equations Section 4 : Direct Methods for Solving Sparse Linear Systems 4 Direct Methods for Solving Sparse Linear Systems A system of Linear equations (32) is called sparse if only a relatively small number of its matrix elements ( patterns that frequently occur are ) are nonzero. The sparse Tridiagonal Band diagonal with band width M Block diagonal matrices Lower / upper triangular and block lower / upper triangular matrices We have encountered numerous situations in the module on Problem Discretization using Approximation Theory where such matrices arise. It is wasteful to apply general direct methods on these problems. Special methods are evolved for solving such sparse systems, which can achieve considerable reduction in the computation time and memory space requirements. In this section, some of the sparse matrix algorithms are discussed in detail. This is meant to be only a brief introduction to sparse matrix computations and the treatment of the topic is, by no means, exhaustive. 4.1 Block Diagonal Matrices In some applications, such as solving ODE-BVP / PDE using orthogonal collocations on finite elements, we encounter equations with a block diagonal matrices, i.e. where for are sub-matrices and, for are sub-vectors. Such a system of equations can be solved by solving the following sub-problems (33) where each equation is solved using, say, Gaussian elimination. Each Gaussian elimination sub-problem would require and Defining dimension of vector as the number of multiplications and divisions in the conventional Gaussian elimination equals It can be easily shown that

-------(34) where matrix is a tridiagonal matrix.

2 i.e. 4.2 Thomas Algorithm for Tridiagonal and Block Tridiagonal Matrices [2] Consider system of equation given by following equation (34) where matrix is a tridiagonal matrix. Thomas algorithm is the Gaussian elimination algorithm tailored to solve this type of sparse system. Step 1:Triangularization: Forward sweep with normalization (35) (36) (37) (38) This sequence of operations finally results in the following system of equations Step 2: Backward sweep leads to solution vector (39, 40)

Total no of multiplications and divisions in Thomas algorithm is which is significantly smaller than the operations (approximately) necessary for carrying out the Gaussian elimination and backward

3 Total no of multiplications and divisions in Thomas algorithm is which is significantly smaller than the operations (approximately) necessary for carrying out the Gaussian elimination and backward sweep for a dense matrix. The Thomas algorithm can be easily extended to solve a system of equations that involves a block tridiagonal matrix. Consider block tridiagonal system of the form (41) where and are matrices and (, represent vectors of appropriate dimensions. Step 1:Block Triangularization (42) (43) Step 2: Backward sweep (44) (45, 46) 4.3 Triangular and Block Triangular Matrices A triangular matrix is a sparse matrix with zero-valued elements above or below the diagonal. For example, a lower triangular matrix can be represented as follows To solve a system, the following algorithm is used (47)

4 (48) The operational count i.e., the number of multiplications and divisions, for this elimination process is (49) which is considerably smaller than the Gaussian elimination for a dense matrix. In some applications we encounter equations with a block triangular matrices. For example, where are sub-matrices while and are sub-vectors for. The solution of this type of systems is completely analogous to that of lower triangular systems,except that sub-matrices and sub-vectors are used in place of scalars. Thus, the equivalent algorithm for blocktriangular matrix can be stated as follows (50) (51) The above form does not imply that the inverse can find each by Gaussian elimination. should be computed explicitly. For example, we 4.4 Solution of a Large System By Partitioning If matrix is equation (LAE) is very large, then we can partition matrix and vector as where is a square matrix. this results in two equations (52) which can be solved sequentially as follows (53) (54) (55) (56)

5 It is also possible to work with higher number of partitions equal to say 9, and solve a given large dimensional problem.

Transactions on Modelling and Simulation vol 18, 1997 WIT Press, ISSN X

Transactions on Modelling and Simulation vol 18, 1997 WIT Press, ISSN X Boundary element analysis of masonry structures Y. F. Rashed*, M. H. Abdalla and M. A. R. Youssef Department of Structural Engineering, Cairo University, Giza, *Present address: Wessex nstitute of Technology,