Large-scale polynomial eigenvalue problems can be solved by Krylov methods operating on an equivalent linear eigenproblem (linearization) of size $$d\cdot n$$ d · n , where d is the polynomial degree and n is the problem size, or by projection methods that keep the computation in the n -dimensional space. Jacobi–Davidson belongs to the latter class of methods, and, since it is a preconditioned eigensolver, it may be competitive in cases where explicitly computing a matrix factorization is exceedingly expensive. However, a fully fledged implementation of polynomial Jacobi–Davidson has to consider several issues, including deflation to compute more than one eigenpair, use of non-monomial bases for the case of large degree polynomials, and handling of complex eigenvalues when computing in real arithmetic. We discuss these aspects and present computational results of a parallel implementation in the SLEPc library.

中文翻译：

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支持非单项式基和通货紧缩的多项式 Jacobi-Davidson 求解器

大规模多项式特征值问题可以通过对大小为 $$d\cdot n$$ d · n 的等效线性特征问题（线性化）进行运算的 Krylov 方法来解决，其中 d 是多项式次数，n 是问题大小，或者通过将计算保持在 n 维空间中的投影方法。Jacobi-Davidson 属于后一类方法，并且由于它是一个预处理的特征求解器，因此在显式计算矩阵分解非常昂贵的情况下，它可能具有竞争力。然而，多项式 Jacobi-Davidson 的成熟实现必须考虑几个问题，包括计算多个特征对的紧缩，在大次多项式的情况下使用非单项式基，以及在实际算术中计算时处理复特征值.