Related papers: An Enhanced Shifted QR Algorithm for Efficient Eig…
In this paper, we present the QR Algorithm with Permutations that shows an improved convergence rate compared to the classical QR algorithm. We determine a bound for performance based on best instantaneous convergence, and develop low…
In this paper we present several additions to the quaternion QR algorithm, including algorithms for eigenvector computation and eigenvalue reordering. A key outcome of the eigenvalue reordering algorithm is that the aggressive early…
We propose a quantum algorithm for finding eigenvalues of non-unitary matrices. We show how to construct, through interactions in a quantum system and projective measurements, a non-Hermitian or non-unitary matrix and obtain its eigenvalues…
Rapid convergence of the shifted QR algorithm on symmetric matrices was shown more than fifty years ago. Since then, despite significant interest and its practical relevance, an understanding of the dynamics and convergence properties of…
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…
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…
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…
Large-scale eigenvalue problems pose a significant challenge to classical computers. While there are efficient quantum algorithms for unitary or Hermitian matrices, eigenvalue problems for non-normal matrices remain open in quantum…
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…
Estimating the eigenvalues of non-normal matrices is a foundational problem with far-reaching implications, from modeling non-Hermitian quantum systems to analyzing complex fluid dynamics. Yet, this task remains beyond the reach of standard…
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…
Non-Hermitian physics has emerged as a rich field of study, with applications ranging from $PT$-symmetry breaking and skin effects to non-Hermitian topological phase transitions. Yet most studies remain restricted to small-scale or…
A majority of numerical scientific computation relies heavily on handling and manipulating matrices, such as solving linear equations, finding eigenvalues and eigenvectors, and so on. Many quantum algorithms have been developed to advance…
We present a new algorithm for solving an eigenvalue problem for a real symmetric arrowhead matrix. The algorithm computes all eigenvalues and all components of the corresponding eigenvectors with high relative accuracy in $O(n^{2})$…
Quantum algorithms for estimating the eigenvalues of matrices, including the phase estimation algorithm, serve as core subroutines in a wide range of quantum algorithms, including those in quantum chemistry and quantum machine learning. The…
A fast implicit QR algorithm for eigenvalue computation of low rank corrections of unitary matrices is adjusted to work with matrix pencils arising from polynomial zerofinding problems . The modified QZ algorithm computes the generalized…
In this paper, we consider the problem of efficiently computing the eigenvalues of limited-memory quasi-Newton matrices that exhibit a compact formulation. In addition, we produce a compact formula for quasi-Newton matrices generated by any…
The problem of estimating the smallest singular value of random square matrices is important in connection with matrix computations and analysis of the spectral distribution. In this survey, we consider recent developments in the study of…
We describe a quantum algorithm for finding the smallest eigenvalue of a Hermitian matrix. This algorithm combines Quantum Phase Estimation and Quantum Amplitude Estimation to achieve a quadratic speedup with respect to the best classical…
The eigenvalue density of a matrix plays an important role in various types of scientific computing such as electronic-structure calculations. In this paper, we propose a quantum algorithm for computing the eigenvalue density in a given…