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We consider the solution of linear systems of equations Ax=b, with A a symmetric positive-definite matrix in ℝn×n, through Richardson-type iterations or, equivalently, the minimization of convex quadratic functions (1/2)(Ax,x)−(b,x) with a gradient algorithm. The use of step-sizes asymptotically distributed with the arcsine distribution on the spectrum of A then yields an asymptotic rate of convergence...
We consider gradient algorithms for minimizing a quadratic function in $${\mathbb{R}^n}$$ with large n. We suggest a particular sequence of step-sizes and demonstrate that the resulting gradient algorithm has a convergence rate comparable with that of Conjugate Gradients and other methods based on the use of Krylov spaces. When the matrix is large and sparse, the proposed algorithm can be more...
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