Thursday, May 31, 2007

TK Solver Laws

The late Dr. Milos Konopasek, inventor of TK Solver, wrote a paper several years ago entitled TK Solver Laws. That paper and a related TK model file are available via the TK Model Share section of the UTS web site, in the Miscellaneous group. The paper has been modified a few times over the years to reflect some of the new features that were added to the program. Here is the introduction and a quick summary of the laws from that paper.

The purpose of the following material is to contribute to the collective understanding of the essence of TK Solver. It does not answer the question, “What is TK?”, and it does not attempt to catalog or explain TK techniques and tricks to any extent. It is a set of problem solving laws which are good to know, respect and live by.

The “laws” are supplemented with a sample model FRUSTUM in the Appendix. It deals with the frustum of a cone and it has 18 variables and 15+ rules. It is an extension of the cone model frequently used in TK Solver demos and training materials.

1. As many independent equations are needed as there are unknown variables to be solved for.

2. A set of TK rules equivalent to m independent equations linking n variables provides a framework for solving all feasible combinations of input and output variables.

3. A TK model can be solved for a certain set of unknowns by the Direct Solver if the reduced variable-equation-coincidence (VEC) matrix can be triangulized by reshuffling of rows and columns. Every diagonal element points to a variable and an equation resolvable in that variable.

4. Simultaneous equations break down the propagation of solution by the Direct Solver.

5. The simultaneous equations for TK’s Iterative Solver to be concerned with are those left after resolving the triangulized head and removing the reverse-triangulized tail of the model.

6. A set of simultaneous equations can always be solved by the Direct Solver, after assigning input values to some of the unknowns and dealing with ensuing inconsistencies, by editing error terms into affected rules.

7. There is always a minimum number of unknown variables to be assigned input (guess) values in order to make the set of simultaneous equations resolvable by the Direct Solver.

8. The number of guess variables may be reduced, or the use of the Iterative Solver avoided altogether for some output variables, by presolving the subsystems of simultaneous equations symbolically and adding redundant equations to the set of rules, or by using the techniques of local root finding or iteration.

9. The invertibility of the operations and functions is the name of the game with TK.

10. The lexical analysis (with VEC matrices and other tools) helps us to use TK efficiently. It may assist with, but it never substitutesfor, the necessary mathematical and subject-related analysis.

11. List solving or block solving is a way of performing a series of simple solutions automatically for a series of values of input variable(s); it should not be launched before simple solving works satisfactorily.

12. Respect and take advantage of the difference between values of variables (that are initialized at the beginning and cannot be overwritten during the solution process) and values of list elements (no automatic initialization, free to overwrite any time).

13. Beware of multiple solutions.

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