Square matrix

About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.

Solving a Quadratic Matrix Equation by Newton's Method with Exact Line Searches

Abstract: We show how to incorporate exact line searches into Newton's method for solving the quadratic matrix equation AX2 + BX + C = 0, where A, B and C are square matrices. The line searches are relatively inexpensive and improve the global convergence properties of Newton's method in theory and in practice. We also derive a condition number for the problem and show how to compute the backward error of an approximate solution.

Communication-Optimal Parallel Recursive Rectangular Matrix Multiplication

Abstract: Communication-optimal algorithms are known for square matrix multiplication. Here, we obtain the first communication-optimal algorithm for all dimensions of rectangular matrices. Combining the dimension-splitting technique of Frigo, Leiserson, Prokop and Ramachandran (1999) with the recursive BFS/DFS approach of Ballard, Demmel, Holtz, Lipshitz and Schwartz (2012) allows for a communication-optimal as well as cache and network-oblivious algorithm. Moreover, the implementation is simple: approximately 50 lines of code for the shared-memory version. Since the new algorithm minimizes communication across the network, between NUMA domains, and between levels of cache, it performs well in practice on both shared and distributed-memory machines. We show significant speedups over existing parallel linear algebra libraries both on a 32-core shared-memory machine and on a distributed-memory supercomputer.

Sensitivity Analysis of Periodic Matrix Models

Abstract: Periodic matrix models are used to describe the effects of cyclic environmental variation, both seasonal and interannual, on population dynamics. If the environmental cycle is of length m, with matrices B(1), B(2),...., B(m) describing population growth during the m phases of the cycle, then population growth over the whole cycle is given by the product matrix A = B(m)B(m—1)...B(1). The sensitivity analysis of such models is complicated because the entries in A are complicated combinations of the entries in the matrices B(i), and thus do not correspond to easily interpreted life history parameters. In this paper we show how to calculate the sensitivity and elasticity of population growth rate to changes in the entries in the individual matrices B(i) making up a periodic matrix product. These calculations reveal seasonal patterns in sensitivity that are impossible to detect with sensitivity analysis based on the matrix A. We also show that the vital rates interact in important ways: the sensitivity to changes in a rate at one point in the cycle may depend strongly on changes in other rates at other points in the cycle.

On sparse and symmetric matrix updating subject to a linear equation

Abstract: A procedure for symmetric matrix updating subject to a linear equation and retaining any sparsity present in the original matrix is derived. The main feature of this procedure is the reduction of the problem to the solution of an n dimensional sparse system of linear equations. The matrix of this system is shown to be symmetric and positive definite. The method depends on the Frobenius matrix norm. Comments are made on the difficulties of extending the technique so that it uses more general norms, the main points being shown by a numerical example.

The permanent of a square matrix

Abstract: We investigate the permanent of a square matrix over a field and calculate it using ways different from Ryser's formula or the standard definition. One formula is related to symmetric tensors and has the same efficiency O(2^mm) as Ryser's method. Another algebraic method in the prime characteristic case uses partial differentiation.