An FPGA-based analysis of trade-offs in the presence of ill-conditioning and different precision levels in computations

Several areas, such as physical and health sciences, require the use of matrices as fundamental tools for solving various problems. Matrices are used in real-life contexts, such as control, automation, and optimization, wherein results are expected to improve with increase of computational precision. However, special attention should be paid to ill-conditioned matrices, which can produce unstable systems; an inadequate handling of precision might worsen results since the solution found for data with errors might be too far from the one for data without errors besides increasing other costs in hardware resources and critical paths. In this paper, we make a wake-up call, using 2 × 2 matrices to show how ill-conditioning and precision can affect system design (resources, cost, etc.). We first demonstrate some examples of real-life problems where ill-conditioning is present in matrices obtained from the discretization of the operational equations (ill-posed in the sense of Hadamard) that model these problems. If these matrices are not handled appropriately (i.e., if ill-conditioning is not considered), large errors can result in the computed solutions to the systems of equations in the presence of errors. Furthermore, we illustrate the generated effect in the calculation of the inverse of an ill-conditioned matrix when its elements are approximated by truncation. We present two case studies to illustrate the effects on calculation errors caused by increasing or reducing precision to s digits. To illustrate the costs, we implemented the adjoint matrix inversion algorithm on different field-programmable gate arrays (FPGAs), namely, Spartan-7, Artix-7, Kintex-7, and Virtex-7, using the full-unrolling hardware technique. The implemented architecture is useful for analyzing trade-offs when precision is increased; this also helps analyze performance, efficiency, and energy consumption. By means of a detailed description of the trade-offs among these metrics, concerning precision and ill-conditioning, we conclude that the need for resources seems to grow not linearly when precision is increased. We also conclude that, if error is to be reduced below a certain threshold, it is necessary to determine an optimal precision point. Otherwise, the system becomes more sensitive to measurement errors and a better alternative would be to choose precision carefully, and/or to apply regularization or preconditioning methods, which would also reduce the resources required.