Various applications in numerical linear algebra and computer science are related to selecting the \(r\times r\) submatrix of maximum volume contained in a given matrix \(A\in \mathbb R^{n\times n}\). 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 \(\mathcal O(n^3)\).

在数值线性代数和计算机科学中的各种应用与选择给定矩阵\（A \ in \ mathbb R ^ {n \ times n} \）中包含的最大体积的\（r \ times r \）子矩阵有关。我们提出了成本的新的贪心算法\（\ mathcal为O（n）\） ，为案件一个对称半正定（SPSD）和我们讨论其推广到相关的优化问题，如卷的最大比例。在本文的第二部分中，我们证明了任何SPSD矩阵都允许建立在主子矩阵上的交叉逼近，该主子矩阵的逼近误差由\（（r + 1）\）乘以最佳秩r的误差核规范中的近似值。根据Cortinovis和Kressner最近的工作精神，我们得出了一些确定性算法，该算法能够以成本\（\ mathcal O（n ^ 3）\）检索拟最佳交叉近似。