Civil Engineering Reference
In-Depth Information
Since the sign of
f
(
x
) changed between
x
= 1 and
x
= 1.25, it is assumed
that a root was passed between those values. The actual root occurs at
x
= 1.2. In this example, five digits of precision were used, but in most
cases it is a good rule to carry one more digit in the calculations than in the
desired accuracy of the answer.
Once the roots have been bounded by the incremental search method,
other methods can be utilized in finding more accurate roots: The follow-
ing sections will cover the refined incremental search, bisection, false
position, secant, Newton-Raphson, and Newton's second order meth-
ods to determine more accurate roots of algebraic and transcendental
equations.
1.6
REfinED incREMEntAL SEARcH MEtHOD
Closer approximations of the root may be obtained by the refined incre-
mental search method. This method is a variation of the incremental
search method. Once a root has been bounded by a search, the last value
of
x
preceding the sign change is used to perform another search using
a smaller increment such as Δ
x
/10 as shown in Figure 1.2 until the sign
changes again.
This process can be repeated with smaller increments of
x
until the
desired accuracy of the root is obtained. Usually the accuracy on the
f(x)
∆
x/10
f(x)
f(x
i
)
f(x
i+1
)
(x)
x
i
x
x
i+1
∆x
∆x
Figure 1.2.
Refined incremental search method.
