Numerical linear algebra software
نویسنده
چکیده
at the current iterate x ∈ Rn. In practice, we find that the bulk of computational resources (memory, time) consumed by these algorithms is spent constructing and solving these linear systems, and not in the higher-level algorithmic details. That is not to say that the higher-level details do not impact performance; for example, algorithmic improvements could result in a reduction in the number of iterations, yielding a proportional reduction in computation time. But the fact remains that the overall performance of many optimization algorithms depends heavily on the performance of the underlying matrix computations. How do we know this? While theoretical complexity analysis can be used to provide some indication, such determinations are generally made through profiling. Profiling is the practice of running a program in a slightly modified manner so that its execution can be carefully monitored; for example, how often each subroutine is called, and how much time is spent in each one. In general, profiling reveals that the performance of a program is dominated by a handful of key subroutines. In the case of numerical optimization software, these critical subroutines are almost always those devoted to numerical linear algebra. Therefore, anyone who wishes to become proficient in the construction of efficient optimization software must develop competence in the area of numerical linear algebra software as well. The most important step towards this competence is accepting this one principle:
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