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42 Solution Procedures
The following subsections describe the solution methods used for the various iterations, as shown
in Figure 5-2.
5-2.1 Successive-Substitution Method
The successive-substitution method (with relaxation) is an example of a fixed point solution. A
direct iteration is simply xn+1 = G(xn), but|G ′ | > 1 for many practical problems. A relaxed iteration
uses F =(1− λ)x + λG:
xn+1 =(1− λ)xn + λG(xn) =xn − λf(xn)
with relaxation factor λ. The convergence criterion is then
|F ′ (α)| = |1 − λ + λG ′ | < 1
so a value of λ can be found to ensure convergence for any finite G ′ . Specifically, the iteration converges
if the magnitude of λ is less than the magnitude of 2/(1 − G ′ )=2/f ′ (and λ has the same sign as
1 − G ′ = f ′ ). Quadratic convergence (F ′ =0) is obtained with λ =1/(1 − G ′ )=1/f ′ . Over-relaxation
(λ >1) can be used if |G ′ | < 1. Since the correct solution x = α is not known, convergence must be
tested by comparing the values of two successive iterations:
error = xn+1 − xn ≤tolerance
where the error is some norm of the difference between iterations (typically absolute value for scalar x).
Note that the effect of the relaxation factor is to reduce the difference between iterations:
xn+1 − xn = λ
G(xn) − xn
Hence the convergence test is applied to (xn+1 − xn)/λ in order to maintain the definition of tolerance
independent of relaxation. The process for the successive-substitution method is shown in Figure 5-3.
5-2.2 Newton–Raphson Method
The Newton–Raphson method (with relaxation and identification) is an example of a zero-point
solution. The Taylor series expansion of f(x) =0leads to the iteration operator F = x − f/f ′ :
xn+1 = xn − [f ′ (xn)] −1 f(xn)
which gives quadratic convergence. The behavior of this iteration depends on the accuracy of the
derivative f ′ . Here it is assumed that the analysis can evaluate directly f, but not f ′ . It is necessary to
evaluate f ′ by numerical perturbation of f, and for efficiency the derivatives may not be evaluated for
each xn. These approximations compromise the convergence of the method, so a relaxation factor λ is
introduced to compensate. Hence a modified Newton–Raphson iteration is used, F = x − Cf:
xn+1 = xn − Cf(xn) =xn − λD −1 f(xn)
where the derivative matrix D is an estimate of f ′ . The convergence criterion is then
|F ′ (α)| = |1 − Cf ′ | = |1 − λD −1 f ′ | < 1
since f(α) =0. The iteration converges if the magnitude of λ is less than the magnitude of 2D/f ′ (and λ
has the same sign as D/f ′ ). Quadratic convergence is obtained with λ = D/f ′ (which would require λ