(a) Explain what happens to the GMRES algorithm if the right-hand side 𝐛 of the linear system A

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(a) Explain what happens to the GMRES algorithm if the...

(a) Explain what happens to the GMRES algorithm if the right-hand side $\mathbf{b}$ of the linear system $A \mathbf{x}=\mathbf{b}$ is an eigenvector of $A$. (b) More generally, prove that if the Krylov subspaces generated by $\mathbf{b}$ stabilize at order $m$, then the solution ot the linear system lies in $V^{(m)}$ and so the GMRES algorithm converges to the solution at order $m$.

Key Concepts

GMRES Algorithm

GMRES, or the Generalized Minimal Residual algorithm, is an iterative method used to solve systems of linear equations, particularly those involving non-symmetric matrices. It works by building a sequence of approximations from Krylov subspaces, minimizing the residual norm over each subspace until convergence is achieved. Its framework relies on successively solving a least-squares problem within an expanding subspace, thereby ensuring that the best possible approximation is used at each step.

Eigenvector

An eigenvector is a non-zero vector that changes only in scale (and not direction) when a linear transformation (via a matrix) is applied to it; the scaling factor is known as the eigenvalue. When the right-hand side vector b is an eigenvector of the matrix A, the action of A on b has a very simple form, which can lead to rapid or even immediate convergence in iterative methods. In the context of GMRES, this simplicity results in the first Krylov subspace containing the solution when b is an eigenvector.

Krylov Subspaces

Krylov subspaces are linear spaces spanned by the set of vectors {b, Ab, A²b, ..., A^(m-1)b} for a given matrix A and starting vector b. These subspaces form the basis for many iterative methods including GMRES, providing a structured way to approximate solutions of linear systems. Their properties, such as dimensionality and behavior under the transformation A, directly influence the convergence and computational efficiency of the algorithms based on them.

Subspace Stabilization

Subspace stabilization occurs when successive Krylov subspaces stop expanding and a fixed subspace is reached, meaning that adding another power of A does not yield a new direction. When the Krylov subspace stabilizes at order m, it implies that all subsequent iterations are contained within this m-dimensional space. Therefore, if the solution to the system lies in this space, iterative methods like GMRES will converge in m steps, as no new information is provided by further matrix-vector products.

Convergence of Iterative Methods

The convergence of iterative methods is often intimately linked with the properties of the space in which the solution is sought. In the context of GMRES, if the Krylov subspace generated by the right-hand side vector b stabilizes at a finite order, the solution, if it lies within this subspace, will be exactly represented in that finite-dimensional setting. This guarantees that GMRES will converge to the solution in a number of iterations that does not exceed the dimension of the stabilized Krylov subspace.

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