Lecture 14 - 14 14.1 Arnoldi Iteration and GMRES Arnoldi...

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14 Arnoldi Iteration and GMRES 14.1 Arnoldi Iteration The classical iterative solvers we have discussed up to this point were of the form x ( k ) = G x ( k - 1) + c with constant G and c . Such methods are also known as stationary methods . We will now study a different class of iterative solvers based on optimization . All methods will require a “black box” implementation of a matrix-vector product. In most library implementations of such solvers the user can therefore provide a custom function which computes the matrix-vector product as efficiently as possible for the specific system matrix at hand. One of the main ingredients in all of the following methods are Krylov subspaces . Given A C m × m and b C m one generates { b , A b , A 2 b , A 3 b , . . . } , which is referred to as a Krylov sequence . Clearly, (fast) matrix-vector products play a crucial role in generating this sequence since each subsequent vector in the sequence is obtain from the previous one by multiplication by A . With the Krylov sequence at hand one defines K n = span { b , A b , A 2 b , . . . , A n - 1 b } as the n -th order Krylov subspace . The Arnoldi iteration method to be derived will be applicable to both linear systems and eigenvalue problems, and therefore we are interested in re-examining similarity transformations of the form A = QHQ * , where H is an upper Hessenberg matrix. In our earlier work we used Householder reflectors to transform A to upper Hessen- berg form. This had its advantages since the resulting algorithm is a stable one. When studying the QR factorization we also looked at the modified Gram-Schmidt algorithm. That algorithm was less stable. However, it has the advantage that one get one col- umn of the unitary matrix Q one column at a time, i.e., the modified Gram-Schmidt algorithm can be stopped at any time and yields a partial set of orthonormal column vectors. On the other hand, with Householder reflectors we always have to perform the entire QR factorization before we get (all) orthonormal vectors. Thus, Arnoldi iteration can be seen as the use of the modified Gram-Schmidt algo- rithm in the context of Hessenberg reduction. 14.2 Derivation of Arnoldi Iteration We start with the similarity transformation A = QHQ * with m × m matrices A , Q , and H . Clearly, this is equivalent to AQ = QH. 108
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Now we take n < m , so that the eigenvalue equations above can be written as A [ q 1 , q 2 , . . . , q n , q n +1 , . . . , q m ] = [ q 1 , q 2 , . . . , q n , q n +1 , . . . , q m ] × × h 11 h 12 . . . h 1 n . . . h 1 m h 21 h 22 . . . h 2 n . . . h 2 m 0 h 32 h 33 h 3 n . . . h 3 m 0 h 43 . . . 0 . . . h n - 1 ,n - 2 . . . . . . . . . . . . h n,n - 1 h nn 0 h n +1 ,n 0 . . . . . . 0 . . . 0 h m,m - 1 h mm . Next, we consider only part of this system. Namely, we let Q n = [ q 1 , q 2 , . . . , q n ] Q n +1 = [ q 1 , q 2 , . . . , q n , q n +1 ] e H n = h 11 h 12 . . . h 1 n h 21 h 22 . . . h 2 n 0 h 32 h 33 h 3 n 0 h 43 .
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