Multiplierless iteration for the resolution of semidefinite linear systems

Algorithms of numerical analysis assume by default that all numbers manipulated by the computer are real numbers. We introduce for the first time in this talk a numerical method that accommodates the internal coarse binary operations of a computer to increase the efficiency of the algorithm. We show that a linear system of equations with a matrix that is symmetric and positive semidefinite can be iteratively solved with an algorithm where every multiplication is reduced to a scaling by a power of 2, which simply amounts to bit shifts in binary.

We will then see how this multiplierless algorithm can be used in various problems, such as least squares, l1-regularized least squares and the minimal-norm resolution of any consistent linear system. A particular application of focus will be the minimal-norm reconstruction of a bandlimited signal from generalized nonuniform samples.