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Register a new userA New Real Structure-preserving Quaternion QR Algorithm
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New real structure-preserving decompositions are introduced to develop fast and robust algorithms for the (right) eigenproblem of general quaternion matrices. Under the orthogonally JRS-symplectic transformations, the Francis JRS-QR step and the JRS-QR algorithm are firstly proposed for JRS-symmetric matrices and then applied to calculate the Schur forms of quaternion matrices. A novel quaternion Givens matrix is defined and utilized to compute the QR factorization of quaternion Hessenberg matrices. An implicit double shift quaternion QR algorithm is presented with a technique for automatically choosing shifts and within real operations. Numerical experiments are provided to demonstrate the efficiency and accuracy of newly proposed algorithms.
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Spectral computations of infinite-dimensional operators are notoriously difficult, yet ubiquitous in the sciences. Indeed, despite more than half a century of research, it is still unknown which classes of operators allow for computation of spectra and eigenvectors with convergence rates and error control. Recent progress in classifying the difficulty of spectral problems into complexity hierarchies has revealed that the most difficult spectral problems are so hard that one needs three limits in the computation, and no convergence rates nor error control is possible. This begs the question: which classes of operators allow for computations with convergence rates and error control? In this paper we address this basic question, and the algorithm used is an infinite-dimensional version of the QR algorithm. Indeed, we generalise the QR algorithm to infinite-dimensional operators. We prove that not only is the algorithm executable on a finite machine, but one can also recover the extremal parts of the spectrum and corresponding eigenvectors, with convergence rates and error control. This allows for new classification results in the hierarchy of computational problems that existing algorithms have not been able to capture. The algorithm and convergence theorems are demonstrated on a wealth of examples with comparisons to standard approaches (that are notorious for providing false solutions).We also find that in some cases the IQR algorithm performs better than predicted by theory and make conjectures for future study.
We propose a new Lagrange multiplier approach to construct positivity preserving schemes for parabolic type equations. The new approach introduces a space-time Lagrange multiplier to enforce the positivity with the Karush-Kuhn-Tucker (KKT) conditions. We then use a predictor-corrector approach to construct a class of positivity schemes: with a generic semi-implicit or implicit scheme as the prediction step, and the correction step, which enforces the positivity, can be implemented with negligible cost. We also present a modification which allows us to construct schemes which, in addition to positivity preserving, is also mass conserving. This new approach is not restricted to any particular spatial discretization and can be combined with various time discretization schemes. We establish stability results for our first- and second-order schemes under a general setting, and present ample numerical results to validate the new approach.
The structure-preserving doubling algorithm (SDA) is a fairly efficient method for solving problems closely related to Hamiltonian (or Hamiltonian-like) matrices, such as computing the required solutions to algebraic Riccati equations. However, for large-scale problems in $mathbb{C}^n$ (also $mathbb{R}^n$), the SDA with an $O(n^3)$ computational complexity does not work well. In this paper, we propose a new decoupled form of the SDA (we name it as dSDA), building on the associated Krylov subspaces thus leading to the inherent low-rank structures. Importantly, the approach decouples the original two to four iteration formulae. The resulting dSDA is much more efficient since only one quantity (instead of the original two to four) is computed iteratively. For large-scale problems, further efficiency is gained from the low-rank structures. This paper presents the theoretical aspects of the dSDA. A practical algorithm dSDA t with truncation and many illustrative numerical results will appear in a second paper.
In emph{Guo et al, arXiv:2005.08288}, we propose a decoupled form of the structure-preserving doubling algorithm (dSDA). The method decouples the original two to four coupled recursions, enabling it to solve large-scale algebraic Riccati equations and other related problems. In this paper, we consider the numerical computations of the novel dSDA for solving large-scale continuous-time algebraic Riccati equations with low-rank structures (thus possessing numerically low-rank solutions). With the help of a new truncation strategy, the rank of the approximate solution is controlled. Consequently, large-scale problems can be treated efficiently. Illustrative numerical examples are presented to demonstrate and confirm our claims.
We design a fast implicit real QZ algorithm for eigenvalue computation of structured companion pencils arising from linearizations of polynomial rootfinding problems. The modified QZ algorithm computes the generalized eigenvalues of an $Ntimes N$ structured matrix pencil using $O(N)$ flops per iteration and $O(N)$ memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.
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