Page 85 - Standard Handbook Of Petroleum & Natural Gas Engineering

74 Mathematics

then, by replacement of the next three equations,

and finally

Gauss-Jordan elimination is a variation of the preceding method, which by
continuation of the same procedures yields

Therefore, x, = r,", etc., Le., the r vector is the solution vector. If the element
in the current pivot position is zero or very small, switch the position of the
entire pivot row with any row below it, including the x vector element, but not
the r vector element.
If det C # 0, C-' exists and can be found by matrix inversion (a modification
of the Gauss-Jordan method), by writing C and I (the identity matrix) and then
performing the same operations on each to transform C into I and, therefore,
I into C-I.
If a matrix is ill-conditioned, its inverse may be inaccurate or the solution
vector for its set of equations may be inaccurate. Two of the many ways to
recognize possible ill-conditioning are

1. If there are elements of the inverse of the matrix that are larger than
elements of the original matrix.
2. If the magnitude of the determinant is small, i.e., if

det C
,- 5 1

Gauss-Siedel method is an iterative technique for the solution of sets of equa-
tions. Given, for example, a set of three linear equations