Page 86 - Standard Handbook Of Petroleum & Natural Gas Engineering

Numerical Methods 75

cllxl + cI2xq + cI9xg = rl
C21XI + C?ZX2 + c21x.9 = r2
CllXl + C32X2 + c1:<x9 = r3
solving for the unknowns yields

rz - cPlxl - c2%xg
X? =
c 2'2

By making an initial guess for xl, x2, and xgr denoted as xy, x:, and x:, the value
of x, on the first iteration is

Using the most recently obtained values for each unknown (as opposed to the
fixed point or Jacobi method), then

If the equations have the proper characteristics, the iterative process will
eventually converge. Commonly used convergence criteria are of two types:

1. Absolute convergence criteria of the form

are most useful when approximate magnitudes of xi are known beforehand
so that E may be chosen to be proportional to x,.
2. Relative convergence criteria of the form

is the choice if the magnitudes of xi are uncertain.