1.4 The Arnoldi Iterations and Decomposition

The Arnoldi decomposition for matrix A is to achieve Q ^T AQ = H or AQ = QH, where Q is orthogonal and H is upper Hessenberg

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There are three methods of realizing this decomposition and we shall use the third method quite often.

? Method 1 Givens rotation. The Givens matrix has four main nonzeros that are in turn determined by an angle parameter ?

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Note P ^T = P = P ^?1.

We shall use Givens rotations to zero out these positions of A: ( j, i) = (3, 1), , ( n, 1), (4, 2), , ( n, n ?2). Each of these ( j, i) pairs defines a Givens rotation P( i + 1, j) and the crucial observation is that the target ( j, i) position of matrices P( i + 1, j) A and P( i + 1, j) AP( i + 1, j) are identical (this would be false if matrix P( i, j) is used). Therefore, to specify P( i + 1, j)= P( j, i + 1), we only need to consider P( i + 1, j) A whose ( j