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Numerical examples |
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The determinant of Hilbert's matrixThe determinant of Hilbert's matrix (11x11) without pivoting strategy is computed.After triangularization, the determinant is the product of the diagonal elements. Hilbert's matrix is defined by: a(i,j) = 1/(i+j-1). Without CADNA:Pivot number 1 = 0.1000000000000000D+01Pivot number 2 = 0.8333333333333331D-01 Pivot number 3 = 0.5555555555555522D-02 Pivot number 4 = 0.3571428571428736D-03 Pivot number 5 = 0.2267573696146732D-04 Pivot number 6 = 0.1431549050481817D-05 Pivot number 7 = 0.9009749236431395D-07 Pivot number 8 = 0.5659970607161749D-08 Pivot number 9 = 0.3551362553328898D-09 Pivot number 10 = 0.2226656943069665D-10 Pivot number 11 = 0.1398301799864147D-11 Determinant = 0.3026439382718219D-64 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 ----------------------------------------------------- Pivot number 1 = 0.100000000000000E+001 Pivot number 2 = 0.833333333333333E-001 Pivot number 3 = 0.55555555555555E-002 Pivot number 4 = 0.3571428571428E-003 Pivot number 5 = 0.22675736961E-004 Pivot number 6 = 0.1431549051E-005 Pivot number 7 = 0.90097493E-007 Pivot number 8 = 0.5659970E-008 Pivot number 9 = 0.35513E-009 Pivot number 10 = 0.2226E-010 Pivot number 11 = 0.14E-011 Determinant = 0.30E-064 ----------------------------------------------------- CADNA software --- University P. et M. Curie --- LIP6 No instability detected CommentsThe gradual loss of accuracy is pointed out by CADNA.One can see that the value of the determinant is significant even if it is very "small". This shows how difficult it is to judge the numerical
quality of a computed result by its magnitude.
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