
ARCHITECT: Arbitraryprecision Hardware with Digit Elision for Efficient Iterative Compute
Many algorithms feature an iterative loop that converges to the result o...
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Numerical computation of formal solutions to interval linear systems of equations
The work is devoted to the development of numerical methods for computin...
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Regularized Momentum Iterative Hessian Sketch for Large Scale Linear System of Equations
In this article, Momentum Iterative Hessian Sketch (MIHS) techniques, a...
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Multistage Mixed Precision Iterative Refinement
Low precision arithmetic, in particular half precision (16bit) floating...
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Accelerating Distributed SGD for Linear Regression using Iterative PreConditioning
This paper considers the multiagent distributed linear leastsquares pr...
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Algebraic error analysis for mixedprecision multigrid solvers
This paper establishes the first theoretical framework for analyzing the...
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Random Osborne: a simple, practical algorithm for Matrix Balancing in nearlinear time
We revisit Matrix Balancing, a preconditioning task used ubiquitously f...
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Digit Stability Inference for Iterative Methods Using Redundant Number Representation
In our recent work on iterative computation in hardware, we showed that arbitraryprecision solvers can perform more favorably than their traditional arithmetic equivalents when the latter's precisions are either under or overbudgeted for the solution of the problem at hand. Significant proportions of these performance improvements stem from the ability to infer the existence of identical mostsignificant digits between iterations. This technique uses properties of algorithms operating on redundantly represented numbers to allow the generation of those digits to be skipped, increasing efficiency. It is unable, however, to guarantee that digits will stabilize, i.e., never change in any future iteration. In this article, we address this shortcoming, using interval and forward error analyses to prove that digits of high significance will become stable when computing the approximants of systems of linear equations using stationary iterative methods. We formalize the relationship between matrix conditioning and the rate of growth in mostsignificant digit stability, using this information to converge to our desired results more quickly. Versus our previous work, an exemplary hardware realization of this new technique achieves an upto 2.2x speedup in the solution of a set of variously conditioned systems using the Jacobi method.
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