# Scaling

Changing the units of measurement, usually for the numerical stability of an algorithm. The variables are transformed as $LaTeX: \textstyle x' = \mbox{S}x,$ where $LaTeX: \textstyle \mbox{S} = \mbox{diag}(s_j).$ The diagonal elements are the scale values, which are positive: $LaTeX: \textstyle s_1, \dots , s_n > 0.$ Constraint function values can also be scaled. For example, in an LP, the constraints $LaTeX: \textstyle \mbox{A}x = b,$ can be scaled by $LaTeX: \textstyle \mbox{RA} x = \mbox{R} b ,$ where $LaTeX: \textstyle \mbox{R} = \mbox{diag}(r_i)$ such that $LaTeX: \textstyle r > 0.$ (This affects the dual values.) Some LP scaling methods simply scale each column of A by dividing by its greatest magnitude (null columns are identified and removed).

Example A column scaling A row scaling
+ = + 10x 100y 500 –30x .3y 0.2
+ = + .3333x y 500 –x .003y 0.2
+ = + x 10y 50 –300x 3y 2

Another method is logarithmic scaling, which scales by the logarithm of the greatest magnitude. More sophisitcated methods are algorithmic, taking both row and column extremes into account.