Abstract
Krylov subspace methods are roughly classified into three groups: ones for Hermitian linear systems, for complex symmetric linear systems, and for non-Hermitian linear systems. Non-Hermitian linear systems include complex symmetric linear systems since a complex symmetric matrix is non-Hermitian and symmetric. Krylov subspace methods for complex symmetric linear systems use the symmetry of the coefficient matrix, leading to more efficient Krylov subspace methods than ones for non-Hermitian linear systems. This chapter also presents preconditioning techniques to boost the speed of convergence of Krylov subspace methods.
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Notes
- 1.
U and V are unitary matrices, and \(\Sigma \) is a diagonal matrix whose diagonal elements are nonnegative.
- 2.
The method was mentioned in the unpublished paper [38] by Chronopulous and Ma in 1989, and Dr. Chronopulous emailed the author about this fact on Apr. 6th, 2020.
- 3.
Codes: GNU Octave version 6.2.0, OS: macOS Moterey version 12.2.1, CPU: Apple M1 pro.
- 4.
This means that \(\mathcal {S} \cap \mathcal {G}_0\) does not contain any eigenvector of A, and the trivial invariant subspace is \(\{\textbf{0}\}\).
- 5.
Given two subspaces \(W_1\) and \(W_2\), the sum of \(W_1\) and \(W_2\) is defined by \(W_1+W_2:=\{\boldsymbol{w}_1+\boldsymbol{w}_2 \, : \, \boldsymbol{w}_1 \in W_1, \boldsymbol{w}_2 \in W_2 \}\).
- 6.
In practice, \(R_0^*\) is usually set to \(R_0^*=R_0\) or a random matrix.
- 7.
This means that \(\mathcal {S} \cap \mathcal {G}_0\) does not contain any eigenvector of A, and the trivial invariant subspace is \(\{\textbf{0}\}\).
- 8.
For \(R=(r_{i,j}) \in \mathbb {C}^{N\times m}\), the symbol \(\Vert R\Vert _\textrm{F} := (\sum _{i=1}^N \sum _{j=1}^m \bar{r}_{i,j} r_{i,j})^{1/2}\) is called the Frobenius norm of R, and for a square matrix \(A=(a_{i,j})\in \mathbb {C}^{N\times N}\), the symbol Tr\((A):=\sum _{i=1}^N a_{i,i}\) is called the trace of A.
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Sogabe, T. (2022). Classification and Theory of Krylov Subspace Methods. In: Krylov Subspace Methods for Linear Systems. Springer Series in Computational Mathematics, vol 60. Springer, Singapore. https://doi.org/10.1007/978-981-19-8532-4_3
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DOI: https://doi.org/10.1007/978-981-19-8532-4_3
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