# Solution for Imaginary part of equation

I have a function Il12i[-0.001, -0.001, w, 0, 1, 0, 0.1002] which is depends on 1 variable w and it is real if w > -3666 (this number I found from graphs). This function is very complicated numerical function as result of numeric integration, some algebraic system solution and NDSolve. I cannot put it here, because it takes something like 10 pages of text.

I need to solve something like this Abs[ Im[ Il12i[-0.001, -0.001, w, 0, 1, 0, 0.1002] ]] > 0

In other words I need to find point, where this function becomes complex.

NSolve does not work at all, as I understand from documentation for complicated function it is better to use FindRoot.

I tried something like this

FindRoot[Abs[Im[Il12i[-0.001, -0.001, w, 0, 1, 0, 0.1002]]] == 0.001, {w, -3700.0}]


but it gives me wrong result {w -> -3639.24} and error message "Encountered a singular Jacobian at the point {w} = {-3639.24}. Try \ perturbing the initial point(s). "

I found a little bit similar problem here HarmonicNumber problem and I understand that I can try to play with WorkingPrecision like this

FindRoot[Abs[Im[Il12i[-0.001, -0.001, w, 0, 1, 0, 0.1002]]] == 0.001, {w, -3700.0}, AccuracyGoal -> Infinity, PrecisionGoal -> 5, WorkingPrecision -> 10000]


but it did not helped.

I made a plot to understand how Im and Re of this function looks like and it it below and I know that this equation has only one solution, I even know where it is approximately from graph (w=-3666), but I cannot find it :(.

My final goal to find many solutions of similar equations to find how this point where function becomes complex depends on parameters a, b, c like this Il12i[a, b, w, 0, 1, 0, c], this is why I cannot use solution from graph or I will need to make hundreds graphs with different parameters a, b, c.

UPDATE:

If I try to use this function in ^2, it is still complex and it is very similar and returns the same error. Here is the graph of normal function and ^2. I prepared some numerical test data from this function and upload it here https://drive.google.com/open?id=0B8TaBGM8Qgp5ZHROUlRscjVKbVk

It can be loaded and returns the same error by this code

Get["C:\\data\\temp\\ccc\\1.zzz"]

Plot[{Im[IIINf[w]]}, {w, -3700, -3650}, PlotLegends -> {"Im"},
PlotStyle -> {Red}]

FindRoot[Abs[Im[IIINf[w]]] == 0.001, {w, -3700},
WorkingPrecision -> 1000]

• Since the imaginary part of an expression is real, instead of Abs[expr] > 0 try Sqrt[expr^2] > 0. The derivatives will then exist and Mathematica may do better. – Bob Hanlon May 7 '16 at 23:05
• Given the character of the curves in the plot, try solving Il12i[w]^2 == 0 with FindRoot.. – bbgodfrey May 8 '16 at 2:58
• Looking at the graph I would surmise Newton's method will shoot from -3700 to a part where Im[...] is zero, hence that singular Jacobian. You might try constraining it, maybe use FindMinimum[{(Im[expr]-.001)^2,x<=-3667},{w,-3700}] and see if that avoids the trouble zone. – Daniel Lichtblau May 8 '16 at 15:04
• Thanks a lot to everyone for answers. But Unfortunately it did not help. This function in power 2 is still complex and graphs looks almost the same like shown in my update. FindMinimum returns almost the same value as I put here x<=-3667. I am going to upload some data as file, where it is possible to reproduce the issue. – Zlelik May 8 '16 at 21:26
• Looks like I found solution. I made Numerical Differentiation with ND[] function and because derivative has some singularity at this point in numerical approach it comes like a maximum and later FindMaximum[] function did the task. I will try to post code later today. – Zlelik May 9 '16 at 14:42

I found a solution by myself. If I continue my uploaded example, where I dump function values to the file and load it later, it will be like this.

1. Calculate Numerical derivative from source function

Needs["NumericalCalculus"]

IIIDNf[x_] = ND[IIINf[w], w, x]

2. Just to see how it looks like I plot the graph of derivative (green) and source function (red). 1. Find maximum for derivative of my function

FindMaximum[Im[IIIDNf[w]], {w, -3700}]

I check it with many parameters and it works very good if I put start point on the left side of the maximum.

my real example with some more tuning for better precision.

FindMaximum[ Im[DNIl12i[-0.001, -0.001, w]], {w, -1200000}, AccuracyGoal -> Infinity, PrecisionGoal -> 5, WorkingPrecision -> 50]
`

I am open for comments. I think it can be used to find solution of some equations with sharp (not smooth) functions. Because calculation of numeric derivative makes function more smooth.