Page 83 - Numerical methods for chemical engineering

The secant method 69

so that the errors at successive iterations are related by

[k]
f x
ε k+1 = ε k + (2.32)
f (1) x [k]
[k]
If we divide the Taylor series approximation (2.30) by f (1) (x ),
[k] (2) [k]
f x f x
0 ≈ + ε k + ε 2 k (2.33)
f (1) x [k] 2 f (1) x [k]
and use (2.32), we obtain

f (2) x [k]
ε k+1 =−ε 2 (2.34)
k (1) [k]
2 f x
This means that if we are very close to the solution, Newton’s method converges quadrat-
ically. For example, assume that we are sufficiently close to a solution for this quadratic
−1
convergence to hold and that |ε k |= 10 . Then, the sequence of errors in the next few
iterations is approximately

|ε k+1 | = 10 −2 |ε k+2 | = 10 −4 |ε k+3 | = 10 −8 |ε k+4 | = 10 −16 (2.35)

Once Newton’s method is close enough to the real solution for the second-order Taylor series
approximation to be accurate, the sequence of estimates converges very rapidly (quadrati-
cally) to the solution.

The secant method

Each iteration of Newton’s method requires not only an evaluation of the function, but
also an evaluation of the first derivative. In some cases, the algebraic function may be of
such complexity that it is inconvenient to derive the analytical form of the derivative. One
alternative would be to use a finite difference approximation,

[k] [k]
f x + ε − f x
df √
≈ ε ≈ eps (2.36)
dx ε
x [k]
where eps is the machine precision. This approach, however, requires two function evalua-
tions per iteration, so that in practice it is more common to use the secant method, in which
the value of the derivative is approximated by the two most recent function evaluations,

[k] [k−1]
f x − f x
df
≈ (2.37)
dx x [k] − x [k−1]
x [k]