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...

These lectures cover three loosely related topics in random matrix theory. First we discuss the techniques used to bound the least singular value of (nonHermitian) random matrices, focusing particularly on the matrices with jointly independent entries. We then use these bounds to obtain the circular law for the spectrum of matrices with iid entries of finite variance. Finally, we discuss the Li...

We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates. It appears that one can compute the limits of all noncommutative moments (thus all spectral properties) of the random matrices we consider because, when embedded in a space of larger square matrices, independent rectangular random matrices are asymptotically fre...

We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates. It appears that one can compute the limits of all non commutative moments (thus all spectral properties) of the random matrices we consider because, when embedded in a space of larger square matrices, independent rectangular random matrices are asymptotically fr...

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...

Pattern matrices are a special construction of matrices from a given function φ with the property that the singular values of the matrices correspond nicely to the Fourier coefficients of φ. Also, the eigenvalues “corresponding” to φ̂(S) have an inverse exponential dependence on the size of S. Hence, if φ has mass only on the high-degree Fourier coefficients, then the corresponding pattern matri...

Multiplicative backward stability results are presented for two algorithms which compute the singular value decomposition of dense matrices. These algorithms are the classical onesided Jacobi algorithm, with a stringent stopping criterion, and an algorithm which uses one-sided Jacobi to compute high accurate singular value decompositions of matrices given as rank-revealing factorizations. When ...

Notions of externally and internally positive singular discrete-time linear systems are introduced. It is shown that a singular discrete-time linear system is externally positive if and only if its impulse response matrix is non-negative. Sufficient conditions are established under which a single-output singular discrete-time system with matrices in canonical forms is internally positive. It is...