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2 Aug 2019 7.5 Systems of Linear Equations and Determinants . 49 General Results of Optimization Theory. 1591 49.2 Existence of Solutions of an Optimization Problem . If d is a square-free positive integer and if d ≥ 2, the ring Z[ This allows us to transfer properties Wilson [178] (Chapters 11 and 12). direct methods for solving linear systems and least-squares problems, and to ods for sparse matrix problems, from theory to algorithms and data struc- tures to working When the right-hand side is sparse, the solution x is also sparse, and not all of generating loop-free code requires significant memory for the compiled.