Chemical Engineering Hand Written_Notes_Part_38

Chemical Engineering Hand Written_Notes_Part_38 - 78 3....

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78 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES factor ω> 1 and get (4.29) e y y = ω ( b y y ) (4.30) or e y = ω b y +(1 ω ) y This ampli f cation process is an extrapolation and is an example of over- relaxation . If the intermediate value b y tends to overshoot target y ,th en we may have to use ω< 1 ;th isisca l led under-relaxation . Application of over-relaxation to Gauss-Seidel method leads to the fol- lowing set of equations ˜ x ( k +1) i = x ( k ) i + ω [ x ( k +1) i x ( k ) i ] (4.31) i =1 , 2 , ..... n where x ( k +1) i are generated by Gauss-Seidel method, i.e., x ( k +1) i = μ 1 a ii " b i i 1 X j =1 a ij x ( k +1) j n X j = i +1 a ij x ( k ) j # (4.32) i =1 , 2 , ..... n With some algebraic manipulations, the above equations can be rearranged in vector matrix form as follows (4.33) ( D + ωL ) x ( k +1) =[(1 ω ) D ωU ] x ( k ) + ω b Relaxation Algorithm INITIALIZE : b ,A, x ,k max
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical Engineering Hand Written_Notes_Part_38 - 78 3....

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