Arnoldi Iteration - Properties of The Arnoldi Iteration

Properties of The Arnoldi Iteration

Let Qn denote the m-by-n matrix formed by the first n Arnoldi vectors q1, q2, …, qn, and let Hn be the (upper Hessenberg) matrix formed by the numbers hj,k computed by the algorithm:

H_n = \begin{bmatrix} h_{1,1} & h_{1,2} & h_{1,3} & \cdots & h_{1,n} \\ h_{2,1} & h_{2,2} & h_{2,3} & \cdots & h_{2,n} \\ 0 & h_{3,2} & h_{3,3} & \cdots & h_{3,n} \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & h_{n,n-1} & h_{n,n} \end{bmatrix}.

We then have

This yields an alternative interpretation of the Arnoldi iteration as a (partial) orthogonal reduction of A to Hessenberg form. The matrix Hn can be viewed as the representation in the basis formed by the Arnoldi vectors of the orthogonal projection of A onto the Krylov subspace .

The matrix Hn can be characterized by the following optimality condition. The characteristic polynomial of Hn minimizes ||p(A)q1||2 among all monic polynomials of degree n. This optimality problem has a unique solution if and only if the Arnoldi iteration does not break down.

The relation between the Q matrices in subsequent iterations is given by

where

\tilde{H}_n = \begin{bmatrix} h_{1,1} & h_{1,2} & h_{1,3} & \cdots & h_{1,n} \\ h_{2,1} & h_{2,2} & h_{2,3} & \cdots & h_{2,n} \\ 0 & h_{3,2} & h_{3,3} & \cdots & h_{3,n} \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & 0 & h_{n,n-1} & h_{n,n} \\ 0 & \cdots & \cdots & 0 & h_{n+1,n} \end{bmatrix}

is an (n+1)-by-n matrix formed by adding an extra row to Hn.

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