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Chemical Engineering Hand Written_Notes_Part_44

# Chemical Engineering Hand Written_Notes_Part_44 - 90 3...

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90 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES and the Guassian elimination yields (5.9) " 1 1 0 0 . 9999 # " x 1 x 2 # = " 2 0 . 9998 # and again due to three digit round o ff in our computer, the solution becomes h x 1 x 2 i T = h 1 1 i Thus, A is a well conditioned numerically only thing that can cause calculation blunders is wrong pivoting strategy. If maximum pivoting is used then natural resistance to round o ff the errors is no longer compromised. Now, in order to understand di culties with ill conditioned systems, con- sider another system ( System-2 ) (5.10) " 1 1 1 1 . 0001 # " x 1 x 2 # = " 2 2 # By Guassian elimination (5.11) " 1 1 0 0 . 0001 # " x 1 x 2 # = " 2 0 # = " x 1 x 2 # = " 2 0 # If we change R.H.S. of the system 2 by a small amount (5.12) " 1 1 1 1 . 0001 # " x 1 x 2 # = " 2 2 . 0001 # (5.13) " 1 1 0 0 . 0001 # " x 1 x 2 # = " 2 0 . 0001 # = " x 1 x 2 # = " 1 1 # Note that change in the fi fth digit of second element of vector b was ampli fi ed to change in the fi rst digit of the solution. Thus, matrix A in the system 2 is ill conditioned as it is near singular.

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