QR decomposition

About: QR decomposition is a research topic. Over the lifetime, 3504 publications have been published within this topic receiving 100599 citations. The topic is also known as: QR factorization.

Chasing Bulges or Rotations? A Metamorphosis of the QR-Algorithm

Abstract: The $QR$-algorithm is a renowned method for computing all eigenvalues of an arbitrary matrix A preliminary unitary similarity transformation to Hessenberg form is indispensable for keeping the computational complexity of the subsequent $QR$-steps under control When restraining computing time is the vital issue, we observe that the prominent role played by the Hessenberg matrix is sufficient but perhaps not necessary to fulfill this goal In this paper, a whole new family of matrices, sharing the major qualities of Hessenberg matrices, will be put forward This gives rise to the development of innovative implicit $QR$-type algorithms, pursuing rotations instead of bulges The key idea is to benefit from the $QR$-factorization of the matrices involved The prescribed order of rotations in the decomposition of the $Q$-factor uniquely characterizes several matrix types such as Hessenberg, inverse Hessenberg, and $CMV$ matrices Loosening the fixed ordering of these rotations provides us the class of matrices under consideration Establishing a new implicit $QR$-type algorithm for these matrices requires a generalization of diverse well-established concepts We consider the preliminary unitary similarity transformation, a proof of uniqueness of this reduction, an extension of the $CMV$-decomposition to a double Hessenberg factorization, and an explicit and implicit $QR$-type algorithm A detailed complexity analysis illustrates the competitiveness of the novel method with the traditional Hessenberg approach The numerical experiments show comparable accuracy for a wide variety of matrix types, but disclose an intriguing difference between the average number of iterations before deflation can be applied

Joint eigenvalue decomposition using polar matrix factorization

Abstract: In this paper we propose a new algorithm for the joint eigenvalue decomposition of a set of real non-defective matrices. Our approach resorts to a Jacobi-like procedure based on polar matrix decomposition. We introduce a new criterion in this context for the optimization of the hyperbolic matrices, giving birth to an original algorithm called JDTM. This algorithm is described in detail and a comparison study with reference algorithms is performed. Comparison results show that our approach provides quicker and more accurate results in all the considered situations.

A data structure for sparse QR and LU factorizations

Abstract: For a general m by n sparse matrix A, a new scheme is proposed for the structural representation of the factors of its sparse orthogonal decomposition by Householder transformations. The storage scheme is row-oriented and is based on the structure of the upper triangular factor obtained in the decomposition. The storage of the orthogonal matrix factor is particularly efficient in that the overhead required is only $m + n$ items, independent of the actual number of nonzeros in the factor. The same scheme is applicable to sparse orthogonal factorization by Givens rotations, and also to the recent implementation of sparse Gaussian elimination with partial pivoting developed by George and Ng (this Journal, 1987, to appear). Experimental results are provided to compare the sparse Gaussian elimination using the new storage scheme with that proposed by George and Ng.

Numerical stability of orthogonalization methods with a non-standard inner product

Abstract: In this paper we study the numerical properties of several orthogonalization schemes where the inner product is induced by a nontrivial symmetric and positive definite matrix. We analyze the effect of its conditioning on the factorization and the loss of orthogonality between vectors computed in finite precision arithmetic. We consider the implementation based on the backward stable eigendecomposition, modified and classical Gram–Schmidt algorithms, Gram–Schmidt process with reorthogonalization as well as the implementation motivated by the AINV approximate inverse preconditioner.

Double image encryption using 3D Lorenz chaotic system, 2D non-separable linear canonical transform and QR decomposition

Abstract: In this paper, a double image encryption scheme using 3D Lorenz chaotic system and QR decomposition in 2D non-separable linear canonical transform domain is proposed. Here the independent parameters of the 2D non-separable canonical transform enlarge the key-space of the proposed scheme and enhance robustness against brute-force attack. 3D Lorenz chaotic system is employed for creating a permutation keystream for pixel transaction process. The proposed scheme is non-linear and asymmetric in nature. To validate and verify the proposed cryptosystem, the numerical simulations have been performed on grayscale images. Results display that the proposed scheme has greater robustness to occlusion and special attacks.