By Sergio Pissanetzky

This hugely technical e-book hopes to carry sparse matrix know-how within sight of engineers, programmers, analysts, academics and scholars.

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The following algorithm requires only one inspection of a and b plus a fixed, small number of elementary operations for each nonzero of the vectors. Since the final result is a single real number, h, the algorithm does not have a symbolic section. An expanded integer array IP is used to store pointers to the positions of the nonzeros in AN. This array, which must be initialized to zero, is constructed by means of a single run over JA. 15 the result is: 1. FUNDAMENTALS position: value IP: 37 1 2 3 4 5 6 7 8 9 10 11 0 0 2 4 0 0 3 0 0 1 0 .

7. 8. 9. 10. 11. 12. 13. 14. 15. 1. 1) where A is a nonsingular square n by n real sparse matrix and b a full vector. The algorithms may be grouped into two categories: direct methods and iterative methods. Direct methods are based on Gauss elimination: the equations or the unknowns of the system are modified in successive steps until the solution is found. In the iterative methods, an initial guess is usually made for x, and this guess is then improved until sufficient accuracy is obtained. Both methods have advantages and disadvantages in each particular case and it is difficult to state general rules as to which is the most convenient.

LINEAR ALGEBRAIC EQUATIONS 49 elements of the line being equal to 0. The simplest permutation matrix is I, the identity matrix. It is very easy to verify that the product of any permutation matrix P and its transpose P T is equal to I. Thus: ρ-ι=ρτ and P is an orthogonal matrix. If a matrix with n rows is pre-multiplied by P, its rows are permuted. Similarly, the n columns of a matrix are permuted by post-multiplication with a permutation matrix. P can be stored in the computer memory as a vector of integers : the integer at position i is the column index of the unit element of row i of P.