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Numerical computation in thorny regions
It is desirable to have uniform best possible accuracy of the numerical result throughout the domain of computation under a fixed precision computation. In general, in some regions of the domain, the accuracy of numerical computation is good while in some other regions it is not so good or even considered bad. This is the situation in fixed precision computation. Special effort is called for to tackle the low accuracy in the later regions so that the accuracy is improved and it is comparable with the accuracy in other good regions. Alternatively, the low accuracy is improved to the best possible accuracy in the given context, that may not be comparable with good accuracy in other regions, but this best possible accuracy (which is the lowest accuracy) will be considered the order of accuracy for the whole domain. We will discuss this uniform accuracy issue or the best attainable order of accuracy (the lowest one) and attempt to prescribe remedies for several numerical problems such as those involving singular/near-singular linear systems, optimization, and differential equations involving singularity.