<<
I don't get it.  Wouldn't a 2d function (real+imaginary) have a
tangent-plane, not a tangent line?  And the intersection of that with
another plane is a line.  You'd want a point...  explain this to me.
>>

Okay; take the function that generates this image:

we want the roots of z^4 + 4.  so we need z^4 = -4

it's to the 4th power, so it has 4 roots.  we can directly solve for the
roots by doing (-4)^(1/4) and using the formula:

r^(1/n) * [cos( (theta + 2*pi*k) / n ) + i * sin( (theta + 2*pi*k) / n ) ]

where r is the radius of the complex number (here we're using -4 as our
complex number; it has no imaginary part so the radius is just 4) and theta
is the angle from the +x axis around to that point in the complex plane( a
point at 1 + i would have a theta of pi/4, for our example, our point -4 +
0i we have a theta of pi).

n is the number of roots, 4, and k ranges from 0 to n-1.  this gives us all


You're right in that newton's method is a slope method, but don't get too
caught up on the visualizations.  what we do with NM is guess a root, and
then subtract from that guess the function evaluated at that guess divided
by the derivative of the function evaluated at that guess, or

z - f(z) / f'(z) gives us our new guess.

this will work just fine to find complex roots.  lets say we guess 2 + i as
a root, for instance.

f(z) = z^4 + 4 = (2+i)^4 + 4 = -3 + 24i
f'(z) = 4*z^3 = 8 + 4i

(2+i) - (-3+24i)/(8+4i) = 1.1 - 1.55i

Making this our new guess, our next iteration gives us approximately 0.965 -
1.20i, our third iteration gives 0.968 - 1.02i, etc etc.  You see that we
are converging to one of the roots, namely 1 - i.

However, if you were to start with something like 2 as your guess, your new
approximation would diverge; and bounce around all over the place like crazy
with this function; since there are no real roots.  So NM works just fine to
find complex roots, but you have to start with a complex number in the first
place, otherwise you will not converge to a complex number.

I think you're just getting a bit hung up on the graphical description of
newton's method (pick a point, draw a tangent line from that point to the
axis, use this as your new x value, and repeat).

--


Mike Metheny

"He that breaks a thing to find out what it is has left the path of wisdom."

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