lec26_11132006 - 10.34 Numerical Methods Applied to...

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Unformatted text preview: 10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #26: TA led Review Singular Value Decomposition A mxn = U mxm S mxn V nxn T U-1 = U T V-1 = V T T n V S U ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 2 1 σ σ σ % A = U ·S ·V T σ 1 ~ 0 U T ·A = (U T U )S ·V T S-1 = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 1 1 1 3 2 1 % σ σ σ 1/ σ 1 Æ ∞ (treat it as zero) U T ·A = I ·S ·V T V ·S-1 U T ·A = I ·S ·V T V ·S-1 A ·x = b x = A-1 b A-1 = V ·S-1 U T x = V ·S-1 U T ·b Ordinary Differential Equations dx nm /dt = F n (x ) x = x (t o ) + ( ) ∫ t t o dt t x F ) ( Trapezoidal Rule: t o t 1 t 2 f 3 f 2 f 1 Error = O( ∆ t 3 )xN Î O( ∆ t 2 ) Figure 1. Integration by the Trapezoidal Rule. Simpson’s Error Error = O( ∆ t 5 )xN Î O( ∆ t 4 ) Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD 2006....
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lec26_11132006 - 10.34 Numerical Methods Applied to...

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