Condition Numbers of Gaussian Random Matrices

Let $G_{m times n}$ be an $m times n$ real random matrix whose elements are independent and identically distributed standard normal random variables, and let $kappa_2(G_{m times n})$ be the 2-norm condition number of $G_{m times n}$. We prove that, for any $m geq 2$, $n geq 2$ and $x geq |n-m|+1$, $kappa_2(G_{m times n})$ satisfies $ frac{1}{sqrt{2pi}} ({c}/{x})^{|n-m|+1} < P(frac{kappa_2(G_{m times n})} {{n}/{(|n-m|+1)}}> x) < frac{1}{sqrt{2pi}} ({C}/{x})^{|n-m|+1}, $ where $0.245 leq c leq 2.000$ and $ 5.013 leq C leq 6.414$ are universal positive constants independent of $m$, $n$ and $x$. Moreover, for any $m geq 2$ and $n geq 2$, $ E(logkappa_2(G_{m times n})) < log frac{n}{|n-m|+1} + 2.258. $ A similar pair of results for complex Gaussian random matrices is also established.

Accelerating Scientific Computations with Mixed Precision Algorithms

On modern architectures, the performance of 32-bit operations is often at least twice as fast as the performance of 64-bit operations. By using a combination of 32-bit and 64-bit floating point arithmetic, the performance of many dense and sparse linear algebra algorithms can be significantly enhanced while maintaining the 64-bit accuracy of the resulting solution. The approach presented here can apply not only to conventional processors but also to other technologies such as Field Programmable Gate Arrays (FPGA), Graphical Processing Units (GPU), and the STI Cell BE processor. Results on modern processor architectures and the STI Cell BE are presented.

Computing the Conditioning of the Components of a Linear Least Squares Solution

In this paper, we address the accuracy of the results for the overdetermined full rank linear least squares problem. We recall theoretical results obtained in Arioli, Baboulin and Gratton, SIMAX 29(2):413--433, 2007, on conditioning of the least squares solution and the components of the solution when the matrix perturbations are measured in Frobenius or spectral norms. Then we define computable estimates for these condition numbers and we interpret them in terms of statistical quantities. In particular, we show that, in the classical linear statistical model, the ratio of the variance of one component of the solution by the variance of the right-hand side is exactly the condition number of this solution component when perturbations on the right-hand side are considered. We also provide fragment codes using LAPACK routines to compute the variance-covariance matrix and the least squares conditioning and we give the corresponding computational cost. Finally we present a small historical numerical example that was used by Laplace in Theorie Analytique des Probabilites, 1820, for computing the mass of Jupiter and experiments from the space industry with real physical data.