Chemical Engineering Hand Written_Notes_Part_35

Chemical Engineering Hand Written_Notes_Part_35 - 72 3....

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72 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES 3.2.2. Triangular and Block Triangular Matrices. A triangular matrix is a sparse matrix with zero-valued elements above the diagonal,i.e., L = l 11 0 .. 0 l 12 l 22 0 . . . . . . l n 1 . l nn To solve a system Lx = b , the following algorithm is used (3.79) x 1 = b 1 /l 11 (3.80) x i = [ b i i 1 P j =1 l ij x j ] l ii ; i =2 , 3 , ..... n The operational count ϕ i.e., the number of multiplications and divisions, for this elimination process is (3.81) ϕ = n ( n +1) / 2 which is considerably smaller than the Gaussian elimination. . In some applications we encounter equations with a block triangular matri- ces. For example, A 11 [0] .... [0] A 12 A 22 .... [0] ..... . .... ..... A n 1 A n 2 .... A nn η (1) η (2) .... η ( n ) = b (1) b (2) .... b ( n ) Where A ij are m × m sub-matrices while η ( i ) R m and b ( i ) R m are sub- vectors for i =1 , 2 ,..n . The solution of this type of systems is completely
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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_35 - 72 3....

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