Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distributions P(s)(x), such that their moments are equal to the Fuss-Catalan numbers of order s. We find a representation of the Fuss-Catalan distributions P(s)(x) in terms of a combination of s hypergeometric functions of the type (s)F(s-1). The explicit formula derived here...

For matrices with a well defined numerical rank in the sense that there is a large gap in the singular value spectrum we compare three rank revealing QR algorithms and four rank revealing LU algorithms with the singular value decomposition. The fastest algorithms are those that construct LU factorizations using rook pivoting. For matrices with a sufficiently large gap in the singular values all...

We investigate the properties of uniform doubly stochastic random matrices, that is non-negative matrices conditioned to have their rows and columns sum to 1. The rescaled marginal distributions are shown to converge to exponential distributions and indeed even large sub-matrices of side-length o(n) behave like independent exponentials. We determine the limiting empirical distribution of the si...

In this paper, we connect rectangular free probability theory and spherical integrals. In this way, we prove the analogue, for rectangular or square non symmetric real matrices, of a result that Guionnet and Mäıda proved for symmetric matrices in [GM05]. More specifically, we study the limit, as n,m tend to infinity, of 1 n logE{exp[nmθXn]}, where Xn is an entry of UnMnVm, θ ∈ R, Mn is a certai...

Lyapunov exponents can be estimated by accurately computing the singular values of long products of matrices, with perhaps 1000 or more factor matrices. These products have extremely large ratios between the largest and smallest eigenvalues. A variant of Rutishauser’s Cholesky LR algorithm for computing eigenvalues of symmetric matrices is used to obtain a new algorithm for computing the singul...

We present new O(n) algorithms to compute very accurate SVDs of Cauchy matrices, Vandermonde matrices, and related \unit-displacement-rank" matrices. These algorithms compute all the singular values with guaranteed relative accuracy, independent of their dynamic range. In contrast, previous O(n) algorithms can potentially lose all relative accuracy in the tiniest singular values. LAPACK Working...

The orthogonal qd algorithm with shifts (oqds algorithm), proposed by von Matt, is an algorithm for computing the singular values of bidiagonal matrices. This algorithm is accurate in terms of relative error, and it is also applicable to general triangular matrices. In particular, for lower tridiagonal matrices, BLAS Level 2.5 routines are available in preprocessing stage for this algorithm. BL...

t ) t 0 be a (memoryless) source, i.e. a sequence of identically distributed, independent random variables. Informally, a source is an object that emits symbols (in general from a finite alphabet) according to some random mechanism. This could be, for example, a physical random generator or in first-order approximation a natural alphabet-based (plain) text. The objective of this paper is to inv...