How Can a Condition Number Be Defined for Optimizing an Equation's Variables?

[hermano]

Condition of an equation

Hi,

I have an equation (transfer function) which I must calculate for several harmonics 'k' (from 1 till 100) namely:

H(k) = 1 + a * exp(i*k*alpha) + b * exp(i*k*beta)

Depending on the variables (a, b, alpha, beta), H(k) becomes zero or almost zero depending on the value of k. Because this undesired for my problem (I divide by this H(k), thus the result can go to infinity), I want to do an optimisation to search for the best variables (a, b, alpha, beta) in such a way that the values of H(k), for k 1 till 100, are not in the vicinity of zero. I want to give a score on each value of H(k) depending of the value reaches zero or not, a sort of condition number, which I sum in the end for each value of H(k). Depending on the value of the condition number I can make then a choice which variables are most appropriate.

Can anybody help me how I can define such a condition number for this problem?

Thanks

 

[anathema]

for your post and for sharing your problem with the community. I understand your concern about the condition of your equation and the need for an optimization process to find the best variables.

To define a condition number for your problem, you can use the concept of sensitivity analysis. This involves calculating the sensitivity of your equation to changes in the variables (a, b, alpha, beta). In other words, how much does the value of H(k) change when you vary one of the variables by a small amount?

To do this, you can use partial derivatives to calculate the sensitivity of H(k) with respect to each variable. This will give you a measure of how sensitive H(k) is to changes in each variable. You can then sum these sensitivities for each value of k to get a total sensitivity for H(k).

A higher sensitivity means that a small change in the variables can greatly affect the value of H(k), which is undesirable in your case. Therefore, you can use this total sensitivity as your condition number. A lower value of the condition number indicates that your equation is more robust and less sensitive to changes in the variables.

I hope this helps and good luck with your optimization process! Don't hesitate to reach out if you need further assistance.