Finding the $r\times r$ submatrix of maximum volume of a matrix $A\in\mathbb R^{n\times n}$ is an NP hard problem that arises in a variety of applications. We propose a new greedy algorithm of cost $\mathcal O(n)$, for the case $A$ symmetric positive semidefinite (SPSD) and we discuss its extension to related optimization problems such as the maximum ratio of volumes. In the second part of the paper we prove that any SPSD matrix admits a cross approximation built on a principal submatrix whose approximation error is bounded by $(r+1)$ times the error of the best rank $r$ approximation in the nuclear norm. In the spirit of recent work by Cortinovis and Kressner we derive some deterministic algorithms which are capable to retrieve a quasi optimal cross approximation with cost $\mathcal O(n^3)$.
Positive semidefinite and positive semidefinite splitting iteration methods for solving nonsingular positive semidefinite systems
Minab Higher Education Center, University of Hormozgan（霍尔木甘大学米纳布高等教育学院）；Department of Mathematics, University of Hormozgan（霍尔木甘大学数学系）；Faculty of Mathematical Sciences, University of Guilan（吉兰大学数理学院）
LAAS-CNRS, University of Toulouse, France（图卢兹大学 LAAS-CNRS）；Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic（布拉格捷克理工大学电气工程学院）
