# Wagon's FindAllCrossings2D[] function to find complex roots in a given complex domain

I am using an modified version of Wagon's FindAllCrossings2D[] function available here Updating Wagon's FindAllCrossings2D[] function. I am defining f[alpha,w] and g[alpha,w] as the real and imaginary part of function da[alpha,w] whose complex roots are to be evaluated.

(*Code for Roots Finding*)
(*Options[FindAllCrossings2D] =
Sort[Join[
Options[FindRoot], {MaxRecursion -> Automatic,
PerformanceGoal :> $PerformanceGoal, PlotPoints -> Automatic}]]; FindAllCrossings2D[funcs_, {x_, xmin_, xmax_}, {y_, ymin_, ymax_}, opts___] := Module[{contourData, seeds, tt, fy = Compile[{x, y}, Evaluate[funcs[[2]]]]}, contourData = Map[First, Cases[Normal[ ContourPlot[funcs[[1]], {x, xmin, xmax}, {y, ymin, ymax}, Contours -> {0}, ContourShading -> False, PlotRange -> {Full, Full, Automatic}, Evaluate[ Sequence @@ FilterRules[Join[{opts}, Options[FindAllCrossings2D]], DeleteCases[Options[ContourPlot], Method -> _]]]]], _Line, Infinity]]; seeds = Flatten[Map[#[[ 1 + Flatten[ Position[Rest[tt = Sign[Apply[fy, #, 2]]] Most[tt], -1]]]] &, contourData], 1]; If[seeds == {}, seeds, Select[Union[ Map[{x, y} /. FindRoot[{funcs[[1]] == 0, funcs[[2]] == 0}, {x, #[[1]]}, {y, #[[2]]}, Evaluate[ Sequence @@ FilterRules[Join[{opts}, Options[FindAllCrossings2D]], Options[FindRoot]]]] &, seeds]], (xmin < #[[1]] < xmax && ymin < #[[2]] < ymax) &]]]*) (*Program For Sliding Control*) Clear[sigma, sigmab, xib, cn, ct, a, l, ei, tau, pb, fb, fc, ks, aa,bb, k, lamda, xi, b, lp, w, alpha, da, an] cn = 0.0034; ct = 0.0017; a = (0.185*10^(-6))/2; l = 58.3*10^(-6); ei = 1700*10^(-24) // N; tau = 0.004; pb = 0.03; fb = 3.8*10^(-12) // N; fc = 2*10^(-12) // N; fp = 1.8*10^(-6) // N; ks = 94.8*10^(-3) // N; gama = 0.273*10^(-3) // N; rho = 150*10^(6) // N; k3 = 800*10^(12) // N; aa = 2*rho*pb*(1 - pb)*fb*fp/fc; bb = 2*rho*pb*fp; k = -1620; lamda = -7.60; sigma[alpha_, w_] := alpha + I*w; xi[alpha_, w_] := (-aa*(alpha + I*w)/(1 + (alpha + I*w)*tau)) + (bb*(alpha + I*w)); sigmab[alpha_, w_] := (alpha + I*w)*cn*l^4/ei ; xib[alpha_, w_] := ((-aa*(alpha + I*w)/(1 + (alpha + I*w)*tau)) + (bb*(alpha + I*w)))*a^2*l^2/ei; lp[alpha_, w_] := ((-aa*(alpha + I*w)/(1 + (alpha + I*w)*tau)) + (bb*(alpha + I*w)))*a^2*l^2/ei; nomodes = 1; (*Calculation of Elements of Matrix*) b = {1.875, 4.694, 7.8548, 10.9955, 14.1372, 17.2788, 20.4204, 23.5619, 26.7035, 29.8451, 32.9867, 36.1283, 39.2699, 42.4115, 45.5531, 48.6947}; an = (-Sin[b] + Sinh[b])/(Cos[b] + Cosh[b]); q = an*(Cos[b*x] - Cosh[b*x]) + (Sin[b*x] + Sinh[b*x]); qx = D[q, x]; qxx = D[qx, x]; qxxx = D[qxx, x]; phi = an*(-Cos[b*x] - Cosh[b*x]) + (-Sin[b*x] + Sinh[b*x]); phix = D[phi, x]; phixx = D[phix, x]; phixxx = D[phixx, x]; cm = Table[ NIntegrate[phi[[i]]*q[[j]], {x, 0, 1}], {i, nomodes}, {j, nomodes}]; coeffMat[alpha_, w_] := Table[ k1m = NIntegrate[phixx[[i]]*qxx[[j]], {x, 0, 1}]; k2m = NIntegrate[phix[[i]]*qx[[j]], {x, 0, 1}]; k3ma = phi[[i]]*qxxx[[j]] + phi[[i]]*lp[alpha, w]*qx[[j]] - phix[[i]]*qxx[[j]] /. x -> 1; k3mb = phi[[i]]*qxxx[[j]] + phi[[i]]*lp[alpha, w]*qx[[j]] - phix[[i]]*qxx[[j]] /. x -> 0; k3m = k3ma - k3mb; km = k1m + lp[alpha, w]*k2m + k3m, {i, nomodes}, {j, nomodes}] da[alpha_, w_] = Det[coeffMat[alpha, w] + sigmab[alpha, w]*cm] f[alpha_, w_] = Re[da[alpha, w]] g[alpha_, w_] = Im[da[alpha, w]] pts = FindAllCrossings2D[{f[alpha, w], g[alpha, w]}, {alpha, -100, 100}, {w, -400, 400}, Method -> {"Newton", "StepControl" -> "LineSearch"}, PlotPoints -> 85, WorkingPrecision -> 20] // Chop ContourPlot[{f[alpha, w], g[alpha, w]}, {alpha, -100, 100}, {w, -400, 400}, Contours -> {0}, ContourShading -> False, Epilog -> {AbsolutePointSize[6], Red, Point /@ pts}]  There are certain problems which I am facing • While calculating matrix form of coeffMat[alpha,w] using Table function its showing error $Failed
• When using the FindAllCrossings2D[] it is unable to find the roots

(I think the first error is leading to the failure in finding the roots because of improper definition of the functionscoeffMat[alpha,w])

• I don't know if that's where the problem comes from, but f[alpha, w] cannot evaluate Im[w] or Re[alpha] (and others) because it did not specify that w and alpha are Reals. Compare your f[alpha, w] with FullSimplify[ComplexExpand@Re[f[alpha, w]], alpha > 0 && w > 0] for example. – anderstood Feb 6 '18 at 15:29