Re: Unstable Simultaneous Linear Equations!

On Thu, 3 Apr 2008 07:10:08 -0700 (PDT), monir <monirg@xxxxxxxxxxxx> wrote:

<SNIP>
Rodger;

Okay.
Given: (exact 7 significant digits)
..........x.........y........d=(dy/dx)
P1:: -0.0266255 0.9981449 -0.1308565
P2:: -0.0248588 0.9847863 132.5335679

what poly coeff A, B, C, D do you get for:
y = A + B*x +C*x^2 + D*x^3

Upon receipt, I'll provide you with an Excel plot showing the
comparison and how to construct the correct elliptic interpolation
curve.

Regards.
Monir

I calculated the coefficients in Mathematica with 50 digit arithmetic. Other
high precision arithmetic packages could be used, such as Matlab, Maple, Scilab,
etc. Here is a link to an image showing the whole thing, including a plot of
the cubic passing through the endpoints plus the elliptic function you gave in
another post:

http://img2.freeimagehosting.net/image.php?8ada8a8e27.gif

The coefficients I got are:

A = 830.9589672389913
B = 95 850.51921961399
C = 3 687 666.010250186
D = 47 265 155.70059824

I think Excel won't be able to get this result even using double precision
arithmetic, and probably won't even be able to accurately evaluate the cubic and
the derivatives with double precision.

The results and calculations shown in the Mathematica result are quite accurate,
and can be relied on to be correct, for the starting values:

Given: (exact 7 significant digits)
...........x.........y........d=(dy/dx)
P1:: -0.0266255 0.9981449 -0.1308565
P2:: -0.0248588 0.9847863 132.5335679

This all shows that extreme care and high precision arithmetic is required to
get a good result for the cubic because the condition number for the system is
very high. The cubic is not a good function for the interpolation, but it can
be accurately calculated with high precision arithmetic.
.

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