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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


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