The speed of convergence of the R-linear GMRES is bounded in terms of a polynomial approximation problem on a finite subset of the spectrum. This result resembles the classical GMRES convergence estimate except that the matrix involved is assumed to be condiagonalizable. The bounds obtained are applicable to the CSYM method, in which case they are sharp. Then a three term recurrence for generating a family of orthogonal polynomials is shown to exist, yielding a natural link with complex symmetric Jacobi matrices. This shows that a mathematical framework analogous to the one appearing with the Hermitian Lanczos method exists in the complex symmetric case. The probability of being condiagonalizable is estimated with random matrices.
Randomized linear algebra for model reduction. Part II: minimal residual methods and dictionary-based approximation
[Balabanov and Nouy 2019，Part I]中介绍了在高维参数相关方程组的模型降阶背景下使用随机草图的方法。在这个框架下，我们在这里构建一个降阶模型从一个小的，有效的可计算的随机对象称为草图的降阶模型，使用最小残差方法。我们介绍了一个草图版本的最小残差投影以及一种新的非线性逼近方法，其中对于每个参数值，该解决方案是近似的最小残差投影到一个子空间上的几个向量挑选（在线）从候选基向量的字典。结果表明，随机绘制技术不仅可以提高计算效率，而且可以提高数值稳定性。严格分析了获得给定精度所需的随机草图上的条件。通过考虑草绘矩阵的足够大尺寸的不经意嵌入，可以以高概率先验地确保这些条件。此外，提供了一个简单可靠的程序，用于后验验证的质量的草图。该方法可用于近似性的证明以及随机草图矩阵大小的自适应选择。
Probabilistic Analysis of Least Squares, Orthogonal Projection, and QR Factorization Algorithms Subject to Gaussian Noise
Department of Computer Science, University of Saskatchewan（萨斯喀彻温大学计算机科学系）；Department of Applied Mathematics, University of Colorado Denver（科罗拉多大学丹佛分校应用数学系）；Department of Mathematics, The University of Tulsa（塔尔萨大学数学系）
