lec05_09152006

# lec05_09152006 - 10.34 Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #5: Introduction to Systems of Nonlinear Equations. Ill-Conditioning and Condition Numbers A ·x true = b true If measurements perfect Know exactly If one can solve exactly, x true gives the actual flows in a reactor system, for example. In reality, there are errors in the measurements b = (b true + δ b ), so: A (x true + δ x ) = (b true + δ b ) Assume A is non-singular (A has an inverse) A x true + A δ x = b true + δ b Î A δ x = δ b δ x = A -1 δ b || δ x || ||A -1 ||·|| δ b || cond(A ) = ||A ||·||A -1 || || || || || ) cond( || || || || || || || || || || || || true 1 b b A b A b A x x δ δ δ = true || x || || A || || b || true true || || || || ) cond( || || || || b b A x x δ δ true 10 -2 ,10 -3 : # of sig figs in b if cond(A ) = 1: is what you expect if cond(A ) > 10 4 : no accuracy Need cond(A ) to be small to bound error cond(A ) 1 Means error is always amplified. cond(cA ) = cond(A ) cond(A ) = | λ | max /| λ | min Æ cond(A ) Æ , if λ i = 0 λ : eigenvalue + + mach ε δ δ δ A A b b A x x ) cond( || || || || ε mach : machine stores 10 -15 ; still have machine error even when you measure everything perfectly.

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