Goal Seek with Complex Numbers

[monirg]

Hello;

I'm trying to understand how complex numbers are handled/processed in Excel.
As related to my application, an interesting exercise would be to use Goal Seek w/s command to find the roots of the equation:
X^2 + 4 = 0
setting the (rounded) value in cell A2 to 0 by changing A1

A1:: 1+i
A2:: =COMPLEX(ROUND(IMREAL(IMSUM(IMPOWER(A1,2),4)),6),
              ROUND(IMAGINARY(IMSUM(IMPOWER(A1,2),4)),6))

Obviously a conventional or direct use of Goal Seek wouldn't work since Excel treats complex numbers as text.

Perhaps, one should use Goal Seek twice in this case:
first: find the coefficient "a" for IMREAL(A2) = 0
second: find the coefficient "b" for IMAGINARY(A2) = 0
and the root would be "a+bi".

There might be an easier way to do it. Any thought ??

Thank you kindly.

 

[pgc01]

Hi Monir

I believe that the difference from zero in the target may be just related to the fact that excel reached its limit in terms of precision.

I did a small test with your example of the 3rd degree equation.

After I ran the solver I got the root

327.992927018071-3.32215428669284E-11i

For this root I get the target (real part of the polynomial evaluation using the root):

-2.08616256713867E-07

Now, the question is: Can I get the target closer to zero?

To do that I'd have to get a better aproximation to the root. I did the test manually. I varied the real part of the root using the smallest possible increment/decrement, 1E-12 (remember excel works with a maximum of 15 significant digits).

327.992927018071 ->-2.08616256713867E-07 - target found by solver
327.992927018071-1E12 ->-1.10268592834473E-06 - target is farther from 0.
327.992927018071+1E12 ->1.04308128356934E-07 - target changed sign. Means that the solution is between value and value + 1E-12

The values for the target that I got for the three cases were clear. By varying the real part of the root I see that I'd get a better result if I could have a value between 327.992927018071 and 327.992927018071 + 1E-12. But that's not possible, in this case 1E-12 is the smallest increment possible.

You can also do this test for the imaginary part of the root, but I think that the problem may excel's 15 significant digit limit.

If I didn't mess up the calculations, the answer to your question "Do you consider the above resulting values of target and constraint to be close enough to zero ??" is: you can get no better using excel.

I had no time to do this test for your other example, the 4th degree equation, but I suspect I'd get similar results.

Remark: this is just to understand the mathematics of this. Outside the mathematical realm, I believe that the 15 significant digits is usually more than enough to for ex. physics or financial applications. I agree, however, that we must understand the tools we use, in this case the solver.

I'd like to hear your comment on all this, maybe you can try other examples?
I used excel 2007.

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PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Polynomial</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; 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BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">327.992927018072</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-0.0000000000332215428669284</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.04308128356934E-07-0.0000225063349822479i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">5</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; 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BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="PADDING-LEFT: 1em; BACKGROUND: #9cf" colSpan=16>[ComplexRoots.xlsb]3rd</TD></TR></TBODY></TABLE>

 

[monirg]

Hi PGC;

1) It is very encouraging that you're getting "very reasonable results" on your system.
For the same cubic equation (and I assume with the same Solver model) you're getting:
..Root: 327.992927018071-3.32215428669284E-11i
..Target : -2.08616256713867E-07

I'm getting after running Solver and "Solver found a solution"
..Root: 327.992927010735+7.68419772256834E-011i
..Target: -0.00437801

As you could see, there's a considerable difference in the resulting Target value between the two sets of results.
Could it be that Solver in your XL 2007 is different (more reliable!) than that in my XL 2003 ??
Did you use different Solver model or Options ?? I've tried almost all available options in different combinations!

2) Same cubic equation and same Solver model as in my previous reply:
x^3 + p x^2 + q x + r = 0
where:
p = 660.96866353
q = - 78862.11028966
r = - 80525640.7629787
and the analytically-derived roots:
...x1 = 327.992927017581
...x2 =-494.480795275332 + 31.6075878056574i
...x3 =-494.480795275332 - 31.6075878056574i

3) Entered initial values:
K31:: 300
L31:: 0

4) Ran Solver
"Solver found a solution. All constraints and optimality conditions are satisfied."
Root M31:: 327.99292619951
Target K34:: -0.55421242
Constraint L34:: 0
Notice the specified Target K34 = 0 was not satisfied and its resulting value is far off !!

5) Entered initial values:
K31:: -400
L31:: 10

6) Ran Solver
"Solver found a solution. All constraints and optimality conditions are satisfied."
Root M31:: -494.48079613214+31.6075850754991i
Target K34:: -0.14023702
Constraint L34:: 0.05000284
Again, the specified Target K34 = 0 was not satisfied !!

7) So it seems to me that there is something fundamentally inconsistent with the optimizer!

8) I agree that 15 significant digits is usually sufficient for most applications including this one. A while back, however, I had to consider the possibility of using 40 significant digits in a Fortran application in order to meet specific mathematical challenges in a discretization forward-marching ODE scheme.

Regards.

 

[pgc01]

Hi Monir

1) These are the options I changed in the solver:
Precision: 1e-10
Convergence: 1e-10
Use automatic scaling
Estimates: quadratic
Derivatives: Central
Search: Conjugate

Guess: 321+1i
Root: 327.992927018071-3.32215428669284E-11i
Target:-2.08616256713867000E-07

2)
I evaluated the polynomial using your analytical roots and mine (from CubicEq)

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BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">CubicEq</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">11</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Root</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Polynomial</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">12</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">327.992927018072-8.93896477040421E-15i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.04308128356934E-07-6.05580951863333E-09i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">13</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-494.480795274036+31.6075878201226i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.04308128356934E-07+8.00937414169312E-08i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">14</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-494.480795274036-31.6075878201225i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-8.94069671630859E-07+1.02445483207703E-08i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">15</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">16</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">17</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Yours</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">18</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Root</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Polynomial</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">19</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">327.992927017581+0i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-0.000332489609718323</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">20</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-494.480795275332 + 31.6075878056574i </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-0.000749707221984863+0.0000963332131505013i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">21</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-494.480795275332 - 31.6075878056574i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-0.000749707221984863-0.0000963332131505013i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">22</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="PADDING-LEFT: 1em; BACKGROUND: #9cf" colSpan=5>[ComplexRoots.xlsb]3rd</TD></TR></TBODY></TABLE>

3-4)
Guess: 300+0i
Root: 327.992927018579+0i
Target: 3.43799591064453000E-04

5-6)
Gess: -400+10i
Root: -494.480795274775+31.607587816931900i
Target: -1.64598226547241000E-04

7) OK

----

With my settings I'm getting better results than you.
We see that a small change in the guess has a clear impact in the final value of the target.


My conclusion (in this we may disagree):

- the differences we got from these experiences reinforce my previous impression that the solver is a more than capable tool to solve most of the problems that an excel user is going to face. I find it a great tool for what it is intended to do. Doesn't mean it can't be improved, the results in this thread show it clearly.

- excel is not a pure mathematical tool. This, however, we already knew. For applications like the one you described or matematics/physics/other applications that really require high precision there are specific tools like Wolfram's Mathematica.

 

[monirg]

Hi PGC;

Thank you once again for your tremendous help, time and patience.

1) At last, I've just been able to run the Solver procedure successfully and almost in full agreement with your latest results!
Here's what I did. I copied the Excel file from my main computer A (XL 2003 & Win XP Home) to a 2nd computer B (XL 2000 & Win XP Home) and to a 3rd computer C (XL 2003 & Win XP Prof). The Solver procedure worked fine and produced reasonable results on computers B and C. I saved the file and copied it back to computer A from computer C. Now it works fine and produces very reasonable results.
For the cubic equation real root:
..Root: 327.992927017581
..Target: -3.12924385070801E-007
I don't have a clue on what was the problem and I've no intention (at present) to investigate it!

2) At this point I believe that the Solver procedure has run its course and there would be no added value in pursuing it any further. You said it correctly: "you can get no better using solver"
I suppose we should move on to a general and more reliable procedure.

3) One of the apparent difficulties in using the Solver procedure is that its success, if at all, crucially depends on having a good first guess of the root, which by no means is readily available for a general equation. Equally problematic is the fact that the user of the Solver procedure has no control on hunting the root down even if it is accurately bracketed.

4) This brings me to your recent suggestion of writing a UDF to determine all the roots of a polynomial, possibly using the Newton-Raphson method. The method, if I recall correctly, requires the 1st derivative of the function to be provided, which is not a problem, but more importantly one must first bracket the roots either by a different procedure or by providing an array of guesses (as you suggested) and let the procedure zooms in each root to within a specified convergence criterion.

5) Such methods deal with real polynomial coefficients and determine real roots. Instead, I would suggest the Languerre's method. It is guaranteed to converge to all types of roots in the - oo to + oo range, handles polynomials with complex coefficients, and does not require an initial guess or starting point or trial solutions. (Keep in mind, with complex coefficients, complex roots may or may not occur in conjugate pairs.)
Sounds too good to be true ?? Well, it's actually true! I successfully used the procedure years ago and even recommended it to grad. students.

6) NR, p. 264, 265 has the Languerre's routine (~ 30 lines of code) and its deriver ZROOTS (~ 30 lines). Both codes are in F77 and are relatively easy to follow and convert to VBA. You would use your FUNCTION CubicEq() as a template to write the procedure. It is really that simple, specially for someone with your expertise in XL VBA.

7) The developed root-finding procedure would be applicable to any one-dimensional equation of the form:
f(x) = sum [k=1 to k=N+1] A(k) x^(k-1)
where f(x) has only one independent variable "x", N is the degree, and A(N+1) are the complex coefficients of the polynomial.

8) It cannot be overemphasized that one should always try to get some idea of how the function behaves before trying to find its roots. There's really no excuse for not to, since it is a one dimensional and the task can't be more simpler in XL.
The display would identify where the function changes sign and whether the change in sign is associated with a real root, a finite jump discontinuity, or a singular point. Also, potential problems such as double real roots (value of function is zero at its max or min), multiple real roots in close proximity (oscillating function about the x-axis), and other problems, could easily be identified from a simple display automatically generated on the w/s by the UDF (just a thought).

9) One could validate the developed UDF using the analytical data for 3rd and 4th degree equations. There're no general analytical solutions for polynomials of degree higher than 4, though there're (very complicated) solutions for particular families of polynomials of any degree. One can't, however, justify the effort required in pursuing such solutions. Rather, one may manufacture such solutions by multiplying a lower degree equation (with known roots) by a known real or imaginary root.

I'd like to know what you think about the whole thing.
(It would be a good idea to continue the discussion under a new thread "Roots of 1D Polynomials of any Degree with Real or Complex Coefficients", just in case someone might have already done that, and it is a matter of polishing the procedure. Would you like me to do that and copy the above as OP??)

Regards.

 

[pgc01]

Hi Monir

I agree. This thread was about using goal seek or the solver to get the roots and it did its job. It's a good idea to start a new thread focusing on solving the problem NOT using the solver.

I also agree on using your last post in this thread as the OP in the new one. Maybe just removing the points 1 and 2 and 3 and replacing them with a new point introducing the subject.

In my opinion, you should avoid any detailed discussion of this thread in the new first point, as you want that the people that read it focus on the UDF solution. Emphasize that you don't want to use the solver. Maybe at the end of the OP something like: "This subject was already discussed using the solver here:" and give the address of this thread.

One last thing. Until now you have only worked with functions of complex variables but with real coefficients. If you need to have also complex coefficients (do you?) make sure you specify that clearly from the start.

Remark:

... possibly using the Newton-Raphson ... Such methods deal with real polynomial coefficients and determine real roots. ...

I'm not sure I agree with you here. I will check. I studied numerical analysis a long time ago but I'm sure that the Newton-Raphson works with functions with complex variables.

 

[monirg]

Hi PGC;

Will follow your suggestions, and will try to post the new thread: "Roots of 1D Polynomials of any Degree with Real or Complex Coefficients" either today or early tomorrow.

Regards.