Square matrix

About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.

Characterization of rank-constrained feasibility problems via a finite number of convex programs

Abstract: In this paper, the rank-constrained matrix feasibility problem is considered, where an unknown positive semidefinite (PSD) matrix is to be found based on a set of linear specifications. First, we consider a scenario for which the number of given linear specifications is at least equal to the dimension of the corresponding space of rank-constrained matrices. Given a nominal symmetric and PSD matrix, we design a convex program with the property that every arbitrary matrix could be recovered by this convex program based on its specifications if: i) the unknown matrix has the same size and rank as the nominal matrix, and ii) the distance between the nominal and unknown matrices is less than a positive constant number. It is also shown that if the number of specifications is nearly doubled, then it is possible to recover all rank-constrained PSD matrices through a finite number of convex programs. The results of this paper are demonstrated on many randomly generated matrices.

Exact Maximum Singular Value Calculation of an Interval Matrix

Abstract: In this note, we present a method for calculating the maximum singular value of an interval matrix. First, we provide an algorithm for calculating the maximum singular value of a square interval matrix. Then, based on this algorithm, we extend the result to non-square interval matrix case and to the case of computing the minimum singular value. Through numerical examples, the validity of the suggested methods is illustrated. Particularly, we compare the newly-proposed method with an existing method to show that the new method finds the correct bound of the maximum singular value with no exception

Increasing the Efficiency of Sparse Matrix-Matrix Multiplication with a 2.5D Algorithm and One-Sided MPI

Abstract: Matrix-matrix multiplication is a basic operation in linear algebra and an essential building block for a wide range of algorithms in various scientific fields. Theory and implementation for the dense, square matrix case are well-developed. If matrices are sparse, with application-specific sparsity patterns, the optimal implementation remains an open question. Here, we explore the performance of communication reducing 2.5D algorithms and one-sided MPI communication in the context of linear scaling electronic structure theory. In particular, we extend the DBCSR sparse matrix library, which is the basic building block for linear scaling electronic structure theory and low scaling correlated methods in CP2K. The library is specifically designed to efficiently perform block-sparse matrix-matrix multiplication of matrices with a relatively large occupation. Here, we compare the performance of the original implementation based on Cannon's algorithm and MPI point-to-point communication, with an implementation based on MPI one-sided communications (RMA), in both a 2D and a 2.5D approach. The 2.5D approach trades memory and auxiliary operations for reduced communication, which can lead to a speedup if communication is dominant. The 2.5D algorithm is somewhat easier to implement with one-sided communications. A detailed description of the implementation is provided, also for non ideal processor topologies, since this is important for actual applications. Given the importance of the precise sparsity pattern, and even the actual matrix data, which decides the effective fill-in upon multiplication, the tests are performed within the CP2K package with application benchmarks. Results show a substantial boost in performance for the RMA based 2.5D algorithm, up to 1.80x, which is observed to increase with the number of processes involved in the parallelization.