“Ill-Conditioned” Vertices

The round-off errors tend to increase particularly rapidly after pivoting at “ill-conditioned” vertices. Those vertices where two or more hyperplanes, each representing one constraint, intersect at a very slight angle are considered as “ill-conditioned”. An “ill-conditioned” vertex is for instance given by the intersection of the two constraints: $$\eqalign{ & 3\,{{\rm{x}}_{\rm{1}}}\, + \,{{\rm{x}}_{\rm{2}}}\, \le \,6 \cr & {{\rm{x}}_{\rm{1}}}\, + \,.354\,{{\rm{x}}_{\rm{2}}}\, \le \,2.001 \cr} $$

Some General Remarks

The Zero-Check for Eliminating Non-Significant Elements

During continued matrix operations like the simplex method a lot of small non-significant elements, the actual value of which is zero,usually augment the working coefficient matrix. These elements are caused by round-off errors. They arise in the following manner in a computation of the type: $${\rm{d}}\, = \,{\rm{a}}\,{\rm{ - }}\,{\rm{b}}{\rm{.c}}$$ with e.g. the data (in FORTRAN notation) $${\rm{a}}\, = \,{\rm{2}}\,{\rm{ = }}\,{\rm{.20000000}}\,{\rm{E}}\,{\rm{01}}$$ $${\rm{b}}\, = \,{\rm{6}}\,{\rm{ = }}\,{\rm{.60000000}}\,{\rm{E}}\,{\rm{01}}$$ $${\rm{c}}\, = \,{\rm{1/3}}\,{\rm{ = }}\,{\rm{.33333333}}\,{\rm{E}}\,{\rm{00}}$$

The Use of Easily Computed Checks as a Trigger for Error Elimination

Several checks have been inserted in the three main programs in order to learn about their correlation to the average relative error.

The Increase and Cumulation of Round-Off Errors

To give an impression of how fast round-off errors may increase even in a not really ill-conditioned case, a short numerical example shall be discussed before reporting the results of the computer runs. The problem is to compute $${\rm{e}}\,{\rm{ = }}\,{\rm{a}}\,{\rm{ - }}\,{\rm{b}}{\rm{.c}}$$ and $${\rm{g}}\,{\rm{ = }}\,{\rm{d}}\,{\rm{ - }}\,{\rm{e}}{\rm{.f}}$$