Related papers: Global Convergence of Hessenberg Shifted QR I: Exa…
We develop a framework for proving rapid convergence of shifted QR algorithms which use Ritz values as shifts, in finite arithmetic. Our key contribution is a dichotomy result which addresses the known forward-instability issues surrounding…
We give a self-contained randomized algorithm based on shifted inverse iteration which provably computes the eigenvalues of an arbitrary matrix $M\in\mathbb{C}^{n\times n}$ up to backward error $\delta\|M\|$ in…
This work presents a novel approach to compute the eigenvalues of non-Hermitian matrices using an enhanced shifted QR algorithm. The existing QR algorithms fail to converge early in the case of non-hermitian matrices, and our approach shows…
Although QR iterations dominate in eigenvalue computations, there are several important cases when alternative LR-type algorithms may be preferable. In particular, in the symmetric tridiagonal case where differential qd algorithm with…
Some fast algorithms for computing the eigenvalues of a block companion matrix $A = U + XY^H$, where $U\in \mathbb C^{n\times n}$ is unitary block circulant and $X, Y \in\mathbb{C}^{n \times k}$, have recently appeared in the literature.…
The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to real…
We develop an exact coordinate descent algorithm for high-dimensional regularized Huber regression. In contrast to composite gradient descent methods, our algorithm fully exploits the advantages of coordinate descent when the underlying…
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…
We develop two fast algorithms for Hessenberg reduction of a structured matrix $A = D + UV^H$ where $D$ is a real or unitary $n \times n$ diagonal matrix and $U, V \in\mathbb{C}^{n \times k}$. The proposed algorithm for the real case…
Solving and visualizing the potential roots of complex functions is essential in both theoretical and applied domains, yet often computationally intensive. We present a hardware-accelerated algorithm for complex function roots density graph…
The Cholesky QR algorithm is an efficient communication-minimizing algorithm for computing the QR factorization of a tall-skinny matrix. Unfortunately it has the inherent numerical instability and breakdown when the matrix is…
The QR-algorithm is one of the most important algorithms in linear algebra. Its several variants make feasible the computation of the eigenvalues and eigenvectors of a numerical real or complex matrix, even when the dimensions of the matrix…
The computation of eigenvalues of real symmetric tridiagonal matrices frequently proceeds by a sequence of QR steps with shifts. We introduce simple shift strategies, functions sigma satisfying natural conditions, taking each n x n matrix T…
One of the most widely used methods for eigenvalue computation is the $QR$ iteration with Wilkinson's shift: here the shift $s$ is the eigenvalue of the bottom $2\times 2$ principal minor closest to the corner entry. It has been a…
We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low-rank matrix $A=G+U V^H$, where $G\in \mathbb C^{n\times n}$ is a unitary matrix represented in some compressed format using $O(nk)$ parameters…
We analyse some QR decomposition algorithms, and show that the I/O complexity of the tile based algorithm is asymptotically the same as that of matrix multiplication. This algorithm, we show, performs the best when the tile size is chosen…
We extend the celebrated QR algorithm for matrices to symmetric tensors. The algorithm, named QR algorithm for symmetric tensors (QRST), exhibits similar properties to its matrix version, and allows the derivation of a shifted…
We revisit Matrix Balancing, a pre-conditioning task used ubiquitously for computing eigenvalues and matrix exponentials. Since 1960, Osborne's algorithm has been the practitioners' algorithm of choice and is now implemented in most…
We analyse an iterative algorithm to minimize quadratic functions whose Hessian matrix $H$ is the expectation of a random symmetric $d\times d$ matrix. The algorithm is a variant of the stochastic variance reduced gradient (SVRG). In…
The Levenberg-Marquardt algorithm is one of the most popular algorithms for finding the solution of nonlinear least squares problems. Across different modified variations of the basic procedure, the algorithm enjoys global convergence, a…