相关论文: Implementation of QR factorization of tall and ver…

In this paper we present a novel algorithm developed for computing the QR factorisation of extremely ill-conditioned tall-and-skinny matrices on distributed memory systems. The algorithm is based on the communication-avoiding CholeskyQR2…

The QR factorization and the SVD are two fundamental matrix decompositions with applications throughout scientific computing and data analysis. For matrices with many more rows than columns, so-called "tall-and-skinny matrices," there is a…

Previous studies have reported that common dense linear algebra operations do not achieve speed up by using multiple geographical sites of a computational grid. Because such operations are the building blocks of most scientific…

We present parallel and sequential dense QR factorization algorithms that are both optimal (up to polylogarithmic factors) in the amount of communication they perform, and just as stable as Householder QR. Our first algorithm, Tall Skinny…

We introduce an algorithmic framework for performing QR factorization with column pivoting (QRCP) on general matrices. The framework enables the design of practical QRCP algorithms through user-controlled choices for the core subroutines.…

A QR factorization of a tall and skinny matrix with n columns can be represented as a reduction. The operation used along the reduction tree has in input two n-by-n upper triangular matrices and in output an n-by-n upper triangular matrix…

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…

CholeskyQR2 and shifted CholeskyQR3 are two state-of-the-art algorithms for computing tall-and-skinny QR factorizations since they attain high performance on current computer architectures. However, to guarantee stability, for some…

Efficient task scheduling is paramount in parallel programming on multi-core architectures, where tasks are fundamental computational units. QR factorization is a critical sub-routine in Sequential Least Squares Quadratic Programming…

Linear algebra operations have been widely used in big data analytics and scientific computations. Many works have been done on optimizing linear algebra operations on GPUs with regular-shaped input. However, few works focus on fully…

This paper describes efficient algorithms for computing rank-revealing factorizations of matrices that are too large to fit in RAM, and must instead be stored on slow external memory devices such as solid-state or spinning disk hard drives…

Scalable QR factorization algorithms for solving least squares and eigenvalue problems are critical given the increasing parallelism within modern machines. We introduce a more general parallelization of the CholeskyQR2 algorithm and show…

Standard rank-revealing factorizations such as the singular value decomposition and column pivoted QR factorization are challenging to implement efficiently on a GPU. A major difficulty in this regard is the inability of standard algorithms…

We present parallel and sequential dense QR factorization algorithms for tall and skinny matrices and general rectangular matrices that both minimize communication, and are as stable as Householder QR. The sequential and parallel algorithms…

This paper develops and analyzes a new algorithm for QR decomposition with column pivoting (QRCP) of rectangular matrices with many more rows than columns. The algorithm carefully combines methods from randomized numerical linear algebra to…

This article proposes and analyzes several variants of the randomized Cholesky QR factorization of a matrix $X$. Instead of computing the R factor from $X^T X$, as is done by standard methods, we obtain it from a small, efficiently…

Numerous algorithms are used for nonnegative matrix factorization under the assumption that the matrix is nearly separable. In this paper, we show how to make these algorithms efficient for data matrices that have many more rows than…

This article presents matrix backpropagation algorithms for the QR decomposition of matrices $A_{m, n}$, that are either square (m = n), wide (m < n), or deep (m > n), with rank $k = min(m, n)$. Furthermore, we derive novel matrix…

This paper introduces fast R updating algorithms specifically designed for statistical applications, including regression, filtering, and model selection, where data structures change frequently. Although traditional QR decomposition is…

Driven by the insatiable needs to process ever larger amount of data with more complex models, modern computer processors and accelerators are beginning to offer half precision floating point arithmetic support, and extremely optimized…