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Numerical examples




J.M. Muller sequence

This example was proposed by J.-M. Muller [1]. The 25 first iterations of the following recurrent sequence are computed:

U(n+1) = 111 - 1130/U(n) + 3000/(U(n)*U(n-1))

with U(0) = 5.5 and U(1) = 61/11

The exact value for the limit is 6.

Without CADNA:

U( 3) = 0.5590163934426229D+01
U( 4) = 0.5633431085043980D+01
U( 5) = 0.5674648620510026D+01
U( 6) = 0.5713329052378341D+01
U( 7) = 0.5749120919664605D+01
U( 8) = 0.5781810919824309D+01
U( 9) = 0.5811314226859892D+01
U(10) = 0.5837656352257866D+01
U(11) = 0.5860948153832367D+01
U(12) = 0.5881319751541141D+01
U(13) = 0.5898177025615012D+01
U(14) = 0.5897965247556456D+01
U(15) = 0.5647011084038567D+01
U(16) = 0.9683399445297453D+00
U(17) =-0.5073216051624674D+03
U(18) = 0.1071206352328062D+03
U(19) = 0.1003959421894409D+03
U(20) = 0.1000235186060601D+03
U(21) = 0.1000014035745554D+03
U(22) = 0.1000000838527958D+03
U(23) = 0.1000000050131387D+03
U(24) = 0.1000000002998870D+03
U(25) = 0.1000000000179481D+03

With CADNA:

-----------------------------------------------------
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
-----------------------------------------------------
U( 3) = 0.55901639344262E+001
U( 4) = 0.5633431085044E+001
U( 5) = 0.56746486205E+001
U( 6) = 0.5713329052E+001
U( 7) = 0.574912092E+001
U( 8) = 0.57818109E+001
U( 9) = 0.581131E+001
U(10) = 0.58377E+001
U(11) = 0.5861E+001
U(12) = 0.588E+001
U(13) = 0.6E+001
U(14) =@.0
U(15) =@.0
U(16) =@.0
U(17) = 0.9E+002
U(18) = 0.999E+002
U(19) = 0.9999E+002
U(20) = 0.99999E+002
U(21) = 0.999999E+002
U(22) = 0.9999999E+002
U(23) = 0.999999999E+002
U(24) = 0.9999999999E+002
U(25) = 0.99999999999E+002
The true limit is 6.
-----------------------------------------------------
CADNA software --- University P. et M. Curie --- LIP6
BE CAREFUL : the self-validation detects major problem(s).
The results are NOT guaranteed
There are 9 numerical instabilities
7 UNSTABLE DIVISION(S)
0 UNSTABLE POWER FUNCTION(S)
2 UNSTABLE MULTIPLICATION(S)
0 UNSTABLE BRANCHING(S)
0 UNSTABLE MATHEMATICAL FUNCTION(S)
0 UNSTABLE INTRINSIC FUNCTION(S)
0 UNSTABLE CANCELLATION(S)

Comments

The traces UNSTABLE DIVISION(S) are generated by divisions where the denominator is a computed zero in the meaning of J. Vignes [2] (no significant value or mathematical zero).

Such operations make the computed trajectory turn off the exact trajectory and then, the estimation of accuracy is not possible any more.

Even using the double precision, the computer cannot give any significant result after the iteration number 15.


References

1
J.-M. Muller, "Arithmetique des ordinateurs", Ed. Masson, 1987.

2
J. Vignes, Zero mathematique et zero informatique, CRAS, Paris, t. 303, serie I, pp. 1-13, 1986.



the classical FORTRAN source code..

the FORTRAN source code with CADNA..

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