Computer Aided Engineering Design

342 COMPUTER AIDED ENGINEERING DESIGN

The bisection method can be improved by taking into account the magnitudes of f(xl) and f (xu) as
well. This helps in making judicious decisions with regard to xr which is otherwise computed more
in a brute force fashion as the average of lower and upper limits of the interval.

(b) Method of Regula falsi or False Positioning or Linear Interpolation
As the name suggests, this method uses linear function interpolation between the lower and upper
bounds of the bracket to predict the false position of the root as shown in Figure 12.4. Using
similarity of triangles, we have

fx

xx

fx

xx

l

r l

u

ur

()

• = –

()

or xx

fx x x

rufx fx

u l u

l u

= –

()( – )

() – ( ) (12.2)

The value of xr so computed is used as in the bisection method described above. The algorithm is
continued until the bracket width xu – xl is less than the desired value ε or the function value at the root
is close to zero. Though the root location is more intuitive in case of the linear interpolation method,
it may not guarantee faster convergence when compared with the bisection method. This may be
because one of the bracket limits may stay fixed and thus slow down the bracket shrinkage. If one of
the limits gets stuck for two or more iterations, it is recommended to reduce the corresponding function
value by half, that is, if f(xl) does not change in two iterations, then^12 f(xl) is used in place of f(xl) and
the same holds true for f (xu) as well.

Example 12.2. We solve Example 12.1 with the initial bracket [–0.95, 1.87] using the false positioning
(not modified) method. The results are shown in the following table. We ensure that f(xl) and f(xu)
are of opposite signs when selecting the initial bracket. Also note that the algorithm converges since
f(xr) is close to zero.

xl xu xr f(xl) f (xu) f (xr)|xu–xl|

• 0.9500 1.8700 0.3749 0.2876 – 0.3246 1.3968 2.8200
  0.3749 1.8700 1.5881 1.3968 – 0.3246 –0.6269 1.4951
  0.3749 1.5881 1.2122 1.3968 –0.6269 –0.3698 1.2132
  0.3749 1.2122 1.0369 1.3968 –0.3698 –0.0724 0.8374
  0.3749 1.0369 1.0043 1.3968 –0.0724 –0.0085 0.6621
  0.3749 1.0043 1.0005 1.3968 –0.0085 –0.0009 0.6294

f(x)

f(xl)

xl

xr x

xu

f(xu)

1

2 (xl + xu)

Figure 12.4 Method of Regula falsi