Chemical Engineering Hand Written_Notes_Part_36

Chemical Engineering Hand Written_Notes_Part_36 - 74 3....

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74 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES Requirements on S and T ma t r ice sa rea sfo l low s[ 15 ] : matrix A should be decomposed into A = S T such that Matrix S should be easily invertible Sequence © x ( k ) : k =0 , 1 , 2 , .... ª should converge to x where x is the solution of A x = b . At the k’th iteration step, we can write (4.4) x ( k ) = ¡ S 1 T ¢ k x (0) + h ¡ S 1 T ¢ k 1 + ¡ S 1 T ¢ k 2 + ... + S 1 T + I i S 1 b If we select ( S 1 T ) such that (4.5) lim k →∞ ¡ S 1 T ¢ k =[ 0 ] where [ 0 ] represents null matrix, then, using identity £ I ¡ S 1 T ¢¤ 1 = I + ¡ S 1 T ¢ + .... + ¡ S 1 T ¢ k 1 + ¡ S 1 T ¢ k + ... we can write x ( k ) £ I ¡ S 1 T ¢¤ 1 S 1 b =[ S T ] 1 b = A 1 b for large
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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_36 - 74 3....

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