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Numerical examples




Jacobi's method

This example shows how the non ended loops of iterative methods (because of non verified stopping criteria) can be avoided by using CADNA. The number of iterations is optimized for a same numerical quality of results.

This program solves a linear system of order 20 by using Jacobi's method.

The stopping criterion is

|| X(n+1) - X(n) || <= eps

where ||X|| is the maximum norm and eps=0.0001 . (for the matrix and the right hand-side, see the source code).

Without CADNA:

niter = 1000
x_sol( 1) = 0.1700196E+01 (right value : 0.1700000E+01), residu( 1) = 0.3841369E-03
x_sol( 2) = -0.4746890E+04 (right value : -0.4746890E+04), residu( 2) = -0.2979056E-02
x_sol( 3) = 0.5023017E+02 (right value : 0.5023000E+02), residu( 3) = 0.8360886E-03
x_sol( 4) = -0.2453201E+03 (right value : -0.2453200E+03), residu( 4) = -0.1759439E-02
x_sol( 5) = 0.4778290E+04 (right value : 0.4778290E+04), residu( 5) = 0.9692840E-03
x_sol( 6) = -0.7572979E+02 (right value : -0.7573000E+02), residu( 6) = 0.1725890E-02
x_sol( 7) = 0.3495430E+04 (right value : 0.3495430E+04), residu( 7) = 0.4875915E-02
x_sol( 8) = 0.4350791E+01 (right value : 0.4350000E+01), residu( 8) = 0.4516218E-03
x_sol( 9) = 0.4529802E+03 (right value : 0.4529800E+03), residu( 9) = 0.1504409E-02
x_sol(10) = -0.2759527E+01 (right value : -0.2760000E+01), residu(10) = 0.2770665E-02
x_sol(11) = 0.8239240E+04 (right value : 0.8239240E+04), residu(11) = -0.2254617E-02
x_sol(12) = 0.3460363E+01 (right value : 0.3460000E+01), residu(12) = 0.1678284E-02
x_sol(13) = 0.1000000E+04 (right value : 0.1000000E+04), residu(13) = -0.2312908E-02
x_sol(14) = -0.5000558E+01 (right value : -0.5000000E+01), residu(14) = -0.2520537E-02
x_sol(15) = 0.3642399E+04 (right value : 0.3642400E+04), residu(15) = -0.1822454E-02
x_sol(16) = 0.7353602E+03 (right value : 0.7353600E+03), residu(16) = 0.2192364E-02
x_sol(17) = 0.1699791E+01 (right value : 0.1700000E+01), residu(17) = 0.5826736E-03
x_sol(18) = -0.2349170E+04 (right value : -0.2349170E+04), residu(18) = 0.1020119E-02
x_sol(19) = -0.8247519E+04 (right value : -0.8247520E+04), residu(19) = 0.2886578E-02
x_sol(20) = 0.9843569E+04 (right value : 0.9843570E+04), residu(20) = 0.3246601E-02

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
-----------------------------------------------------
niter = 30
x_sol( 1) = 0.169E+01 (right value : 0.1700000E+01), residu( 1) = @.0
x_sol( 2) = -0.474689E+04 (right value : -0.4746890E+04), residu( 2) = @.0
x_sol( 3) = 0.50229E+02 (right value : 0.5023000E+02), residu( 3) = @.0
x_sol( 4) = -0.24532E+03 (right value : -0.2453200E+03), residu( 4) = @.0
x_sol( 5) = 0.477829E+04 (right value : 0.4778290E+04), residu( 5) = @.0
x_sol( 6) = -0.7573E+02 (right value : -0.7573000E+02), residu( 6) = @.0
x_sol( 7) = 0.349543E+04 (right value : 0.3495430E+04), residu( 7) = @.0
x_sol( 8) = 0.435E+01 (right value : 0.4350000E+01), residu( 8) = @.0
x_sol( 9) = 0.45298E+03 (right value : 0.4529801E+03), residu( 9) = @.0
x_sol(10) = -0.276E+01 (right value : -0.2760000E+01), residu(10) = @.0
x_sol(11) = 0.823924E+04 (right value : 0.8239240E+04), residu(11) = @.0
x_sol(12) = 0.346E+01 (right value : 0.3460000E+01), residu(12) = @.0
x_sol(13) = 0.999999E+04 (right value : 0.9999999E+03), residu(13) = @.0
x_sol(14) = -0.500E+01 (right value : -0.5000000E+01), residu(14) = @.0
x_sol(15) = 0.3642399E+04 (right value : 0.3642400E+04), residu(15) = @.0
x_sol(16) = 0.73536E+03 (right value : 0.7353601E+03), residu(16) = @.0
x_sol(17) = 0.170E+01 (right value : 0.1700000E+01), residu(17) = @.0
x_sol(18) = -0.234917E+04 (right value : -0.2349170E+04), residu(18) = @.0
x_sol(19) = -0.824752E+04 (right value : -0.8247519E+04), residu(19) = @.0
x_sol(20) = 0.984357E+04 (right value : 0.9843570E+04), residu(20) = @.0
-----------------------------------------------------
CADNA software --- University P. et M. Curie --- LIP6
There are 38 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)
38 UNSTABLE CANCELLATION(S)

Comments

With the classical arithmetic, the stopping criterion is satisfied at the 1000-th iteration. But, from the 30th iteration there is no numerical improvement of the results. A kind of "trampling" appears. the value of eps is too small.

With CADNA, the program is stopped when, for all components, difference between two iterations becomes no significant. It happens at the 30-th iteration. For a same numerical quality, CADNA has divided the number of iteration by a factor 30.


the classical FORTRAN source code..

the FORTRAN source code with CADNA..

More information can be requested to the CADNA team
Thanks to Baptiste Mary for the CADNA logo