### Abstract

We consider the numerical solution of a c-stable linear equation in the tensor product space ℝn1x...xnd, arising from a discretized elliptic partial differential equation in ℝ^{d}. Utilizing the stability, we produce an equivalent d-stable generalized Stein-like equation, which can be solved iteratively. For large-scale problems defined by sparse and structured matrices, the methods can be modified for further efficiency, producing algorithms of O(Σini)+O(ns) computational complexity, under appropriate assumptions (with n_{s} being the flop count for solving a linear system associated with Ai-γIni). Illustrative numerical examples will be presented.

Original language | English |
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Article number | e2106 |

Journal | Numerical Linear Algebra with Applications |

Volume | 24 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2017 Dec 1 |

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### Keywords

- Cayley transform
- Kronecker product
- Stein equation
- Sylvester equation
- elliptic partial differential equation
- large-scale problem
- linear equation

### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics

### Cite this

*Numerical Linear Algebra with Applications*,

*24*(6), [e2106]. https://doi.org/10.1002/nla.2106