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iterative - G1BINM Introduction to Numerical Methods 71 7...

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G1BINM Introduction to Numerical Methods 7–1 7 Iterative methods for matrix equations 7.1 The need for iterative methods We have seen that Gaussian elimination provides a method for finding the exact solution (if rounding errors can be avoided) of a system of equations Ax = b . However Gaussian elimination requires approximately n 3 / 3 operations (where n is the size of the system), which may become prohibitively time-consuming if n is very large. Another weakness is that Gaussian elimination requires us to store all the components of the matrix A . In many real applications (especially the numerical solution of differential equations), the matrix A is sparse , meaning that most of its elements are zero, in which case keeping track of the whole matrix is wasteful. In situations like these it may be preferable to adopt a method which produces an approximate rather than exact solution. We will describe three iterative methods , which start from an initial guess x 0 and produce successively better approximations x 1 , x 2 , . . . . The iteration can be halted as soon as an adequate degree of accuracy is obtained, and the hope is that this takes a significantly shorter time than the exact method of Gaussian elimination would require. 7.2 Splitting the matrix All the methods we will consider involve splitting the matrix A into the difference between two new matrices S and T : A = S - T . Thus the equation Ax = b gives Sx = T x + b , based on which we can try the iteration Sx k +1 = T x k + b . (7.1) Now if this procedure converges, say x k x as k → ∞ , then clearly x solves the original problem Ax = b , but it is not at all clear from the outset whether a scheme like (7.1) converges or not. Evidently there are many possible ways to split the matrix A . The tests of a good choice are:
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7–2 School of Mathematical Sciences University of Nottingham the new vector x k +1 should be easy to compute , that is S should be easily invertible (for example S might be diagonal or triangular); the scheme should converge as rapidly as possible towards the true solution. These two requirements are conflicting: a choice of splitting which is particularly easy to invert (see e.g. Jacobi’s method below) may not converge especially rapidly (or at all). At the other extreme we can converge exactly , in just one step, by using S = A , T = 0 ; but S = A is usually difficult to invert: that’s the whole point of splitting! It is convenient to introduce the notation A = L + D + U (= S - T ) , where L is strictly lower triangular, D is diagonal, U is strictly upper triangular. For example, if A = 1 2 3 4 5 6 7 8 9 , then L = 0 0 0 4 0 0 7 8 0 , D = 1 0 0 0 5 0 0 0 9 , U = 0 2 3 0 0 6 0 0 0 . We will consider three possible splittings. 1. Jacobi’s method S = D , T = - ( L + U ); 2. The Gauss-Seidel method S = L + D , T = - U ; 3. Successive over-relaxation (SOR) a combination of 1 and 2.
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