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Topic: same error in your applet?
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Subject:   roots for TI app (Explanation)
Author: sione
Date: Dec 11 2003

The precision I used for that applet is 1.490116119384766E-8, that means that
complex number solution that comes up where
the absolute of imaginary part is less than or equal the
precision (which is 1.490116119384766E-8), then
it is counted as a REAL solution.

Now I ran im my machine locally the roots of your
equation :


and the answers came up (including complex number solutions)

1)  -1.41421356   (ie, -sqr(2))
2)  -1 + 3.9484299147897906E-8 * i
3)  -1 - 3.9484299147897906E-8 * i
4)  1.41421356     (ie, sqr(2))

Notice that roots 2 & 3 are conjugate pairs of complex
number roots (meaning plotting them on the complex plane , they reflect each
other along the real-axis). You can SEE clearly WHY my applet missed the  -1
as a real solution because , the ABSOLUTE(3.9484299147897906E-8) or
ABSOLUTE(-3.9484299147897906E-8) are definitely greater than my precision
which is 1.490116119384766E-8 : This leads to the ommision of -1 as a real

When I changed my precision to 5E-8 , I got all the REAL  roots correct, which

1)  -1.41421356   (ie, -sqr(2))
2)  -1
3)  1.41421356     (ie, sqr(2))

This is because ABSOLUTE(3.9484299147897906E-8) or
ABSOLUTE(-3.9484299147897906E-8) is LESS THAN my precision which is 5E-8

Algorithm used:
The alogrithm used in solving ROOTS for polynomial is called  EIGENVALUE
DECOMPOSITION and there is no doubt
that TI-89 used this algorithm too. You can find more about the EIGENVALUE
DECOMPOSITION in many advanced matrix algebra text books or any numerical
analysis books.

Now , I am glad that you brought this issue up, because in the next version of
my polynomial roots applet , I will add a text box so that the user will enter
the PRECISION number themselves, so that the applet can allow any precision
value, instead of being fixed at 1.490116119384766E-8 (default precision).

I have not used  TI-89 before to see whether the functionality allows the user
to specify the precision or not. If this flexibility is not available in TI-89
perhaps, you and others who have bought it bring this issue up with the
manufacturer (I assume it is Texas Instrument). Ask the manufacturer to include
such functionalities (user specified precision) in the next version of TI-89


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