Re: Real and Complex Roots presented in a single plot

*To*: mathgroup at smc.vnet.net*Subject*: [mg92601] Re: Real and Complex Roots presented in a single plot*From*: Narasimham <mathma18 at hotmail.com>*Date*: Tue, 7 Oct 2008 07:07:13 -0400 (EDT)*References*: <ga5koj$r82$1@smc.vnet.net> <gadclf$rkb$1@smc.vnet.net>

On Sep 19, 2:15 pm, Narasimham <mathm... at hotmail.com> wrote: > On Sep 12, 2:27 pm, magma <mader... at gmail.com> wrote: > > > We can also recognize and see the real parts of al= l = > 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 The point/ claim in a simple case is that e.g. in y = (x - x1)^2 + a^2 there is a min at x = x1 and a complex root there with real part x1, the full root being x1 (+/-) a *I . Narasimham