The SPQR RANK package contains routines that calculate the numerical rank of large, sparse, numerically rank-deficient matrices. The routines can also calculate orthonormal bases for numerical null spaces, approximate pseudoinverse solutions to least squares problems involving rankdeficient matrices, and basic solutions to these problems. The algorithms are based on SPQR from SuiteSparseQR (ACM...

We consider the problem of commutative rank computation of a given matrix space, B ⊆ Fn×n. The problem is fundamental, as it generalizes several computational problems from algebra and combinatorics. For instance, checking if the commutative rank of the space is n, subsumes problems such as testing perfect matching in graphs and identity testing of algebraic branching programs. An efficient det...

We propose a simple variant of kd-trees, called rankbased kd-trees, for sets of points in R. We show that a rank-based kd-tree, like an ordinary kd-tree, supports range search queries in O(n1−1/d + k) time, where k is the output size. The main advantage of rank-based kd-trees is that they can be efficiently kinetized: the KDS processes O(n) events in the worst case, assuming that the points fol...

The matrix separation problem aims to separate a low-rank matrix and a sparse matrix from their sum. This problem has recently attracted considerable research attention due to its wide range of potential applications. Nuclear-norm minimization models have been proposed for matrix separation and proved to yield exact separations under suitable conditions. These models, however, typically require...

This paper is concerned with the fast computation of Fourier integral operators of the general form ∫ Rd e f(k)dk, where k is a frequency variable, Φ(x, k) is a phase function obeying a standard homogeneity condition, and f is a given input. This is of interest for such fundamental computations are connected with the problem of finding numerical solutions to wave equations, and also frequently ...