Lecture 13 - 13 13.1 Classical Iterative Methods for the...

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13 Classical Iterative Methods for the Solution of Linear Systems 13.1 Why Iterative Methods? Virtually all methods for solving A x = b or A x = λ x require O ( m 3 ) operations. In practical applications A often has a certain structure and/or is sparse , i.e., A contains many zeros. A typical problem that arises in practice is the Poisson problem mentioned at the beginning of the class. We want to find u such that -∇ 2 u ( x, y ) = - [ u xx ( x, y ) + u yy ( x, y )] = f ( x, y ) , in Ω = [0 , 1] 2 u ( x, y ) = 0 , on Ω . One of the standard numerical algorithms is a finite difference approach. The Laplacian is discretized on a grid of ( n + 1) 2 equally spaced points ( x i , y j ) = ( ih, jh ), i, j = 0 , . . . , n with h = 1 n . This results in the discrete Laplacian 2 u ( x i , y j ) u i - 1 ,j + u i,j - 1 + u i +1 ,j + u i,j +1 - 4 u i,j h 2 , where u i,j = u ( x i , y j ). The boundary conditions of the PDE allow us to set the solution at the points along the boundary as u i, 0 = u i,n = u 0 ,j = u n,j = 0 , i, j = 0 , 1 , . . . , n. At the ( n - 1) 2 interior grid points we obtain the following system of linear equations for the values of u there 4 u i,j - u i - 1 ,j - u i,j - 1 - u i +1 ,j - u i,j +1 = f i,j n 2 , i, j = 1 , . . . , n - 1 The system matrix is of size m × m , where m = ( n - 1) 2 . Each row contains at most five nonzero entries, and therefore is very sparse. Thus, special methods are called for to take advantage of this sparsity when we solve this linear system. Obviously, a full-blown LU or Cholesky factorization will be much too costly if m is large (typical values for m are often 10 6 or even larger). 13.2 The Splitting Approach The basic iterative scheme to solve A x = b will be of the form x ( k ) = G x ( k - 1) + c , k = 1 , 2 , 3 , . . . . (38) Here we assume that A C m × m , x (0) is an initial guess for the solution, and G and c are a constant iteration matrix and vector, respectively, defining the iterative scheme. Most classical iterative methods are based on a splitting of the matrix A of the form A = M - N 100
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with a nonsingular matrix M . One then defines G = M - 1 N and c = M - 1 b . Then (38) becomes x ( k ) = M - 1 N x ( k - 1) + M - 1 b or M x ( k ) = N x ( k - 1) + b . (39) In practice we will want to choose the splitting factors so that 1. (39) is easily solved, 2. (39) converges rapidly. Theorem 13.1 If G = M - 1 N < 1 then (38) converges to a solution of A x = b for any initial guess x (0) . Proof (38) describes a fixed point iteration (i.e., is of the form x = g ( x )), and the fixed point of (38) is a solution of A x = b as can be seen from x = G x + c ⇐⇒ x = M - 1 N x + M - 1 b ⇐⇒ M x = N x + b ⇐⇒ ( M - N ) = A x = b .
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