Solving for the best X , Y , and Z is a multivariable optimization problem. One way to solve them is Gaussian elimination. Note that the formulation involves quantities of the form as well as quantities of the form . For , which is quite reasonable in your measured data, this leads to magnitudes on the order of and .
Computers find it hard to manipulate very large numbers and very small numbers in the same problem.
A way out of this numerical difficulty is to normalize your frequency values so they span a smaller range. Choose a normalizing frequency k , and define . Once X ', Y ', and Z ' are computed, relate them back to X , Y , Z . Then the desired quantities , and Q can be calculated from X , Y , Z .