 # Polynomial function of two variables: Rump equation

This example was proposed by S. M. Rump.
The following polynomial is computed with two couples of values: P(x,y)= 9x4 - y4+ 2y2.

The first computed result is erroneous.
The second value is computed with the best accuracy.
It is impossible, a priori, to distinguish the numerical quality of the two results.

CADNA allows to point out immediatly this difference.
For a computed value without any significant digit, CADNA prints the symbol @.0 .
In the other cases, CADNA only prints the exact significant digits.

P(10864,18817) = 2.0000000000000000 (exact value: 1)
P(1/3,2/3) = 0.8024691358024691

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CADNA software --- University P. et M. Curie --- LIP6
Self-validation detection: ON
Mathematical instabilities detection: ON
Branching instabilities detection: ON
Intrinsic instabilities detection: ON
Cancellation instabilities detection: ON
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P(10864,18817) = @.0 (exact value: 1)
P(1/3,2/3) = 0.802469135802469E+000
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CADNA software --- University P. et M. Curie --- LIP6
There are 2 numerical instabilities
0 UNSTABLE DIVISION(S)
0 UNSTABLE POWER FUNCTION(S)
0 UNSTABLE MULTIPLICATION(S)
0 UNSTABLE BRANCHING(S)
0 UNSTABLE MATHEMATICAL FUNCTION(S)
0 UNSTABLE INTRINSIC FUNCTION(S)
2 UNSTABLE CANCELLATION(S)

The symbol @.0 means that the first result was computed with no significant digit.
The second result was computed with 15 exact significant digits
One can see the difference of accuracy between the two results even if they were provided with the same algorithm.
It points out the importance of the data influence on the numerical quality of results given by an algorithm.

## References

1
S. M. Rump, "how reliable are results of computers", Jahrbuch Uberblicke Mathematik 1983, pp. 163-168.

the classical FORTRAN source code.
the FORTRAN source code with CADNA.