This requires some elaborated work. The basic idea is to make all function "algebraically-ready" before they are inserted into the plot function. Here is an example:

f[x] = Det[Inverse[{{1,2},{3,x}}]]
g[x] := Det[Inverse[{{1,2},{3,x}}]]
Plot[f[x],{x,100,200}]
Plot[g[x],{x,100,200}]

The first plot should be significantly faster than the second because f has the rhs evaluated already during definition while the g looks for the inversion first and compute the determination. Check the downvalues of f and g respectively:

Use this idea to precompute functions. For example, Yiidealtotal is thus converted to a rational function before it is inserted into the plot funciton:

Some notes about the optimization: Numeric computation may yield imaginary residues in coefficients, which are difficult for simplification. You want to take a look Simplify[Chop[#]]& function to knock down the complicated expressions.

To simplify a list of function, you can choose the parallel function to accelerate. For Yi[x]:

Yi[s_] = ParallelMap[Simplify[Chop[#]] &, nciM.Inverse[ZLRC[s]].nci];

Use the attached notebook to take a look at the details.

PS I use high precision numbers throughout the notebook to handle the large coefficients.

Attachments:

Hi Shenghui

I have been playing around and it looks very promising. The only drawback I have found is that the curves are different from the slow ones I got. It's probably a precision problem and/or the Chop function. I will continue researching on the item to see if I can improve the curves.

I will try also to force ListPlot to first substitute the f value and then compute the Inverse matrix. This could be faster, I hope.

Stefan

Dear Shenghui

I see that the code can be improved, but how?

Could you please explain how to do it? What should I take care off?

Thank you very much.

Stefan

A count of the number of constants, operators and variables in just one of the items you want to plot

In[40]:= LeafCount[20 Log10[Abs[ Yiideal[2. Pi I f]]]]

Out[40]= 547063

I tried an assortment of simple things, like replacing 0.+0.I by 0, etc. and, while that removed a few tens of thousands of redundant terms, nothing made a significant dent in the half million items needed to be evaluated hundreds or likely thousands of times to make each plot.

When the function to plot is expensive I have sometimes been able to significantly speed things up by constructing a Table of equally spaced evaluations (Log spaced in this case) and then do a ListPlot with Joined->True to reduce the number of evaluations by sometimes an order of magnitude with modest reduction in the publication quality of the graph. Unfortunately with the appearance of your plots I don't think this will likely be acceptable.