Algorithms For Sparse Linear Systems Research Articles

A dynamic programming approach for quickly estimating large network-based MEV models

We propose a way to estimate a family of static Multivariate Extreme Value (MEV) models with large choice sets in short computational time. The resulting model is also straightforward and fast to use for prediction. Following Daly and Bierlaire (2006), the correlation structure is defined by a rooted, directed graph where each node without successor is an alternative. We formulate a family of MEV models as dynamic discrete choice models on graphs of correlation structures and show that the dynamic models are consistent with MEV theory and generalize the network MEV model (Daly and Bierlaire, 2006). Moreover, we show that these models can be estimated quickly using the concept of network flows and the nested fixed point algorithm (Rust, 1987). The main reason for the short computational time is that the new formulation allows to benefit from existing efficient solution algorithms for sparse linear systems of equations.We present numerical results based on simulated data with varying number of alternatives and nesting structures. We estimate large models, for example, a cross-nested model with 200 nests and 500,000 alternatives and 210 parameters that needs between 100–200 iterations to converge (4.3 h on an Intel(R) 3.2 GHz machine using a non-parallelized code). We also show that our approach allows to estimate a cross-nested logit model of 111 nests with a real data set of more than 100,000 observations in 14 h.

Bounds on the largest singular value of a matrix and the convergence of simultaneous and block‐iterative algorithms for sparse linear systems

AbstractWe obtain the following upper bounds for the eigenvalues of the matrix A†A. For any a in the interval [0, 2] let image image and ca and ra the maxima of the caj and rai, respectively. Then no eigenvalue of the matrix A†A exceeds the maximum of image over all i, nor the maximum of image over all j. Therefore, no eigenvalue of A†A exceeds cara.Using these bounds, it follows that, for the matrix G with entries image no eigenvalue of G†G exceeds one, provided that, for some a in the interval [0, 2], we have image and image Using this result, we obtain convergence theorems for several iterative algorithms for solving the problem Ax=b, including the CAV, BICAV, CARP1, SART, SIRT, and the block‐iterative DROP and SART methods.

Complexity transitions in global algorithms for sparse linear systems over finite fields

We study the computational complexity of a very basic problem, namely that of finding solutions to a very large set of random linear equations in a finite Galois Field modulo q. Using tools from statistical mechanics we are able to identify phase transitions in the structure of the solution space and to connect them to changes in performance of a global algorithm, namely Gaussian elimination. Crossing phase boundaries produces a dramatic increase in memory and CPU requirements necessary to the algorithms. In turn, this causes the saturation of the upper bounds for the running time. We illustrate the results on the specific problem of integer factorization, which is of central interest for deciphering messages encrypted with the RSA cryptosystem.