A Kronecker Product Preconditioner for Stochastic Galerkin Finite Element Discretizations
SIAM Journal on Scientific Computing
A note on shifted Hessenberg systems and frequency response computation
ACM Transactions on Mathematical Software (TOMS)
Numerical analysis of rational processes beyond Markov chains
Performance Evaluation
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A fast method for solving a linear system of the form $(A^{(p)} \otimes \cdots \otimes A^{(1)} - \lambda I) x = b$ is given where each $A^{(i)}$ is an $n_i$-by-$n_i$ matrix. The first step is to convert the problem to triangular form $(T^{(p)} \otimes \cdots \otimes T^{(1)} - \lambda I) y = c$ by computing the (complex) Schur decompositions of the $A^{(i)}$. This is followed by a recursive back-substitution process that fully exploits the Kronecker structure and requires just $O(N(n_1 + \cdots + n_p))$ flops where $N = n_1 \cdots n_p$. A similar method is employed when the real Schur decomposition is used to convert each $A^{(i)}$ to quasi-triangular form. The numerical properties of these new methods are the same as if we explicitly formed $(T^{(p)} \otimes \cdots \otimes T^{(1)}- \lambda I)$ and used conventional back-substitution to solve for $y$.