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44 Solution Procedures
initialize
evaluate h
test convergence: error = |hj − htargetj| ≤tolerance × weight j
initialize derivative matrix D to input matrix
calculate gain matrix: C = λD −1
iteration
identify derivative matrix
optional perturbation identification
perturb each element of x: δxi =Δ× weight i
evaluate h
calculate D
calculate gain matrix: C = λD −1
increment solution: δx = −C(h − htarget)
evaluate h
test convergence: error = |hj − htargetj| ≤tolerance × weight j
Figure 5-4. Outline of Newton-Raphson method.
5-2.3 Secant Method
The secant method (with relaxation) is developed from the Newton–Raphson method. The modified
Newton–Raphson iteration is:
xn+1 = xn − Cf(xn) =xn − λD −1 f(xn)
where the derivative matrix D is an estimate of f ′ . In the secant method, the derivative of f is evaluated
numerically at each step:
f ′ (xn) ∼ = f(xn) − f(xn−1)
xn − xn−1
It can be shown that then the error reduces during the iteration according to:
|ɛn+1| ∼ = |f ′′ /2f ′ ||ɛn||ɛn−1| ∼ = |f ′′ /2f ′ | .62 |ɛn| 1.62
which is slower than the quadratic convergence of the Newton–Raphson method (ɛ 2 n), but still better than
linear convergence. In practical problems, whether the iteration converges at all is often more important
than the rate of convergence. Limiting the maximum amplitude of the derivative estimate may also be
appropriate. Note that with f = x − G(x), the derivative f ′ is dimensionless, so a universal limit (say
maximum |f ′ | =0.3) can be specified. A limit on the maximum increment of x (as a fraction of the x
value) can also be imposed. The process for the secant method is shown in Figure 5-5.
5-2.4 Method of False Position
The method of false position is a derivative of the secant method, based on calculating the derivative
with values that bracket the solution. The iteration starts with values of x0 and x1 such that f(x0) and
f(x1) have opposite signs. Then the derivative f ′ and new estimate xn+1 are
f ′ (xn) ∼ = f(xn) − f(xk)
xn − xk
xn+1 = xn − λD −1 f(xn)