University:
Massachusetts Institute of Technology
Course:
10.34 | Numerical Methods Applied to Chemical Engineering
Academic year:

2019
Views:

32

Pages:

3
Author:

Zimml83

1.2.2 Gauss-Jordan Elimination In the method of Gaussian elimination, starting from a system A x = b of the general form a11 a21  :  : an1 a12 a13 ... a22 a23 ... : : : : an2 an3 ... a1n  a2n  :   :  ann   x1   b1   x2   b2      :  = :      :  :   xN  bn  is converted to an equivalent system A’ x = b ’ after (1.2.2-1) 2 3 N FLOP’s that is of upper 3 triangular form a      ' 11 a ' 12 a '22 ' 13 ... a '23 ... ' 33 ... a a   a '2N   ' a 3N  ' a NN  a ' 1N '  x1   b 1   x2] b '     2 :  = b 3'      :  :   xN  b '   N  (1.2.2-2) At this point, it is possible, through backward substitution, to solve for the unknowns in N2 the order xN, xN-1, xN-2, … in steps. 2 In the method of Gauss-Jordan elimination, one continues the work of elimination, placing zeros above the diagonal. To “zero” the element at (N-1, N), we write the last two equations of (1.2.2-2) a 'N −1, N −1 x N −1 + a 'N −1, N x N = b 'N −1   a 'N, N x N = b 'N  (1.2.2-3) We then define λ N −1, N = a 'N −1, N (1.2.2-4) a 'NN And replace the N-1st row with the equation obtained after performing the row operation (a 'N −1, N −1 x N −1 + a 'N −1, N x N = b 'N −1 ) − λ N −1, N (a 'NN x N = b 'N ) (1.2.2-5) a 'N −1, N −1 x N −1 + (a 'N −1, N − λ N −1, N a 'NN )x N = b 'N −1 − b 'N λ N −1, N Defining b 'N' −1 = b 'N −1 − b 'N λ N −1, N (1.2.2-6) and noting a '' N −1 =a ' N −1, N − λ N −1, N a ' NN =a ' N -1, N −( a 'N −1, N a ' NN )a 'NN = 0 (1.2.2-7) After this row operation the set of equations becomes ' a 11          ' a 12 ' a 13 ... a 1,' N-1 a '22 a '23 ... a '2, N-1 ' a 33 ... a 3,' N-1 : a 'N' -1, N-1 ' '  x1 a 1.N    b1     a '2, N   x2  b '2    a 3,' N   x 3  b 3'   = :  :  :      ''  0   x N -1  b N -1   x  '  a 'NN   N  b N  (1.2.2-2) We can continue this process until the set of equations is in diagonal form  a   '''   a 22   ''' a 33   '''   a NN  ''' 11 '''  x1   b 1   x2   b ' ' '     2 :  = b 3' ' '      :  :   xN  b ' ' '   N  (1.2.2-9) Dividing each equation by the value of its single coefficient yields  b1' ' '  '''  a 11 1  x 1   b '2' ' 1  x   a ' ' '    2   22 1  x 3  =  b 3' ' '     ''' :  :   a 33 1  x N  :   b'''  ' 'N'  a NN               (1.2.2-10) The matrix on the left that has a one everywhere along the principal diagonal and zeros everywhere else is called the identity matrix, and has the property that for any vector v, Iv = v (1.2.2-11) The form (1.2.2-10) therefore immediately gives the solution to the problem. In practice, we use Gaussian Elimination, stopping at (1.2.2-2) to begin backward substitution rather than continue the elimination process because backward substitution is so fast, N2 << 2N3/3 for all but small problems. We therefore do not consider the method of Gauss-Jordan Elimination further.

Gauss-Jordan Elimination

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