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It is well known that improper scaling of linear equations can result in catastrophic loss of accuracy from Gauss elimination. The scaling process is not well understood and the commonly used “scaling rules” can fail. We study the scaling problem for the linear equations that arise from solving elliptic partial differential equations by collocation using Hermite bicubics. We present an a priori scaling rule that is effective but not foolproof. We conclude that one should use scaled partial pivoting for such equations. We also explore the relationship between the ordering used during Gauss elimination and the underlying geometry of the elliptic problem; we conjecture that this ordering must maintain the geometric integrity of the problem in order to avoid severe round-off problems.

The Importance of Scaling for the Hermite Bicubic Collocation Equations