| Abstract: | Algorithms for formulating colorant combinations of more than three dyes are based on a process of conditioned-least-squares minimization. Using a technique of matrix partitioning, rather than the classical procedure which uses undetermined Lagrangian multipliers, it is possible to reduce the problem to a lower-order one of unconstrained minimization followed by matrix substitution. The system also yields a correction matrix which may be employed to improve the metameric match of a trial dyeing by adjusting the first three dyes of the set. The algorithms have been incorporated into a colour-matching program using a PDP11-VO3 minicomputer. Included in a number of subroutines for matrix manipulation is one called GINV, which produces a generalized inverse with features designed to achieve best possible accuracy. It is demonstrated that the generalized inverse provides versatility without requiring elaborate modifications of programming, since the generalized inverse may be directly applied to solving least-squares problems. This feature may be useful when setting up the initial recipe approximation for more than three dyes, and it can yield directly the best approximation to a tristimulus match where the colour to be matched calls for two dyes or a single dye. |
