A Geometric Algorithm for Solving Linear Systems
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Based on the geometric Triangle Algorithm for testing membership of a point in a convex set, we present a novel iterative algorithm for testing the solvability of a real linear system Ax=b, where A is an m × n matrix of arbitrary rank. Let C_A,r be the ellipsoid determined as the image of the Euclidean ball of radius r under the linear map A. The basic procedure in our algorithm computes a point in C_A,r that is either within ε distance to b, or acts as a certificate proving b ∉C_A,r. Each iteration takes O(mn) operations and when b is well-situated in C_A,r, the number of iterations is proportional to log(1/ε). If Ax=b is solvable the algorithm computes an approximate solution or the minimum-norm solution. Otherwise, it computes a certificate to unsolvability, or the minimum-norm least-squares solution. It is also applicable to complex input. In a computational comparison with the state-of-the-art algorithm BiCGSTAB ( Bi-conjugate gradient method stabilized), the Triangle Algorithm is very competitive. In fact, when the iterates of BiCGSTAB do not converge, our algorithm can verify Ax=b is unsolvable and approximate the minimum-norm least-squares solution. The Triangle Algorithm is robust, simple to implement, and requires no preconditioner, making it attractive to practitioners, as well as researchers and educators.
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