SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A663-A684, January 2021.
A widely used form of test matrix is the randsvd matrix constructed as the product $A = U \Sigma V^*$, where $U$ and $V$ are random orthogonal or unitary matrices from the Haar distribution and $\Sigma$ is a diagonal matrix of singular values. Such matrices are random but have a specified singular value distribution. The cost of forming an $m\times n$ randsvd matrix is $m^3 + n^3$ flops, which is prohibitively expensive at extreme scale; moreover, the randsvd construction requires a significant amount of communication, making it unsuitable for distributed memory environments. By dropping the requirement that $U$ and $V$ be Haar distributed and that both be random, we derive new algorithms for forming $A$ that have cost linear in the number of matrix elements and require a low amount of communication and synchronization. We specialize these algorithms to generating matrices with a specified 2-norm condition number. Numerical experiments show that the algorithms have excellent efficiency and scalability.

中文翻译：

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生成具有指定奇异值或条件数的极端尺度矩阵

SIAM科学计算杂志，第43卷，第1期，第A663-A684页，2021年1月。
广泛使用的测试矩阵形式是构造为乘积$ A = U \ Sigma V ^ * $的randsvd矩阵，其中$ U $和$ V $是来自Haar分布的随机正交或unit矩阵，而$ \ Sigma $是奇异值的对角矩阵。这样的矩阵是随机的，但是具有指定的奇异值分布。形成一个$ m×n $ randsvd矩阵的成本是$ m ^ 3 + n ^ 3 $触发器，在极端规模上是非常昂贵的。此外，randsvd构造需要大量的通信，使其不适用于分布式内存环境。通过放弃将$ U $和$ V $都进行Haar分布并且都必须是随机的要求，我们导出了用于形成$ A $的新算法，这些算法在矩阵元素的数量上具有线性成本，并且需要少量的通信和同步。我们将这些算法专用于生成具有指定2范数条件编号的矩阵。数值实验表明，该算法具有良好的效率和可扩展性。