Re: Real and Complex Roots presented in a single plot

*To*: mathgroup at smc.vnet.net*Subject*: [mg92070] Re: Real and Complex Roots presented in a single plot*From*: Narasimham <mathma18 at hotmail.com>*Date*: Fri, 19 Sep 2008 05:15:58 -0400 (EDT)*References*: <ga5koj$r82$1@smc.vnet.net> <gadclf$rkb$1@smc.vnet.net>

On Sep 12, 2:27 pm, magma <mader... at gmail.com> wrote: > > We can also recognize and see the real parts of all = the > > complex roots of z where the curve is nearest to x -axis at {1.4, > > 4.2, 6.6, 9.6, 12.3, 15.3, 17.7}. They are near to x-values where the > > local maxima/minima occur.But we cannot 'see' their complex parts, as > > they need to be computed. > > Please note that your claim that the real part of the complex roots is > found at the local minima of the given function as the indipendent > variable moves on the real axis, is not valid in general and wrong in > this particular instance. > > For example in the interval {1,2} there are 2 local extrema found > using the derivative: > > f[z_] := 1.3 Sin[1.7 z] + 0.6 Sin[4 z] > > The derivative: > > In[43]:= fp[z_] := D[f[z], z] > > In[44]:= fp[z] > > Out[44]= 2.21 Cos[1.7 z] + 2.4 Cos[4 z] > > In[73]:= extrema = z /. FindRoot[fp[z], {z, #}] & /@ {1.4, 1.6} > > Out[73]= {1.33751, 1.69029} > > The first is the local minimum (see the plot). > But the complex roots in interval {1,2} are > > 1.27946 + 0.374308 I and 1.27946 - 0.374308 I > > The real part is 1.27946, while the minimum was at 1.33751 (I do not have Presentations). My point is that 1.33751 is sufficiently near to and corresponding to the root place holder 1.27946 so that succesive tangents drawn in complex Newton-Raphson procedure the roots would not settle anywhere else. Narasimham > This can be visualized on the complex plane using Presentations: > > f[z_] := 1.3 Sin[1.7 z] + 0.6 Sin[4 z] > > Needs["Presentations`Master`"] > > fp[z_] := D[f[z], z] > > extrema = z /. FindRoot[fp[z], {z, #}] & /@ {1.4, 1.6} > > roots = z /. > FindRoot[ > f[z], {z, ToComplex@#}] & /@ {{-0.005546, -0.02102}, {1.28= 5, > 0.3771}, {1.279, -0.3678}, {2.081, > 0.01109}, {3.462, -0.008173}, {4.194, -0.3485}, {4.213, > 0.345}} // Chop; > > In the following plot we see the roots (blu) and the extrema (red) in > the interval {1,2} > > With[{zmin = -.5 - I, zmax = 5 + I, > contourlist = {0, 0.1, 0.2, .5, .505, .5095, .51, .53, 0.55, 0.57, > 0.6, .61, .62, .6205, 1}}, > Draw2D[{ComplexCartesianContour[f[z], {z, zmin, zmax}, Abs, > Contours -> contourlist, ColorFunctionScaling -> False, > ColorFunction -> (ContourColors[contourlist, > ColorData["SolarColors"]][#] &), PlotPoints -> {30, 15= }, > MaxRecursion -> 3, PlotRange -> {0, 11}], > ComplexCirclePoint[#, 3, Black, Blue] & /@ roots, > ComplexCirclePoint[#, 3, Black, Red] & /@ extrema, > ComplexLine[{-5, 5}]}, AspectRatio -> Automatic, > PlotRange -> {{1, 2}, {-.7, .7}}, Frame -> True, > FrameLabel -> {Re, Im}, RotateLabel -> False, > PlotLabel -> Row[{"Modulus of ", f[z]}], ImageSize -> 900, > Background -> Legacy@Linen]]