Roots of nonlinear equation Chapter 3
Nizar Salim 1 Lecture 2
3.3.2. False Position (Regula Falsi)
The interval-halving (bisection) method brackets a root in the interval (a, b) and
approximates the root as the midpoint of the interval. In the false position (regula falsi)
method, the nonlinear function f(x) is assumed to be a linear function g(x) in the interval
(a, b), and the root of the linear function g(x), x = c, is taken as the next approximation of
the root of the nonlinear function f(x), x = a. The process is illustrated graphically in
Figure 2.5
Figure 2.5 False Position
This method is also called the linear interpolation method. The root of the linear function
g(x), that is, x = c, is not the root of the nonlinear function f(x). It is a false position (in
Latin, regula falsi), which gives the method its name. We now have two
intervals, (a, c) and (c, b). As in the interval-halving (bisection) method, the interval
containing the root of the nonlinear function/(x) is retained, as described in Section 2.3.1,
so the root remains bracketed.
The equation of the linear function g(x) is
( )
( ) ( ) g x
c b
f c f b
(2.14)
where f( ) = 0, and the slope of the linear function g (x) is given by
b a
f b f a
g x
( ) ( )
( ) (2.15)
Solving Eq. 2.12 for the value of which gives f( ) = 0 yields
Roots of nonlinear equation Chapter 3
Nizar Salim 2 Lecture 2
( )
( )
g x
f b
c b ( 2.16)
Note that f(a) and a could have been used in Eqs. (2.14) and (2.16) instead of f(b) and b.
Equation (2.16) is applied repetitively until either one or both of the following two
convergence criteria are satisfied:
b - a < l and/or f(c) < 2
Example 2.2 False position.
Roots of nonlinear equation Chapter 3
Nizar Salim 3 Lecture 2
3.4 OPEN DOMAIN METHODS
The interval halving (bisection) method and the false position (regula falsi) method
presented in Section 2.3 converge slowly. More efficient methods for finding the roots of
a nonlinear equation are desirable. Four such methods are presented in this section:
1. Newton s method
2. The secant method
These methods are called open domain methods since they are not required to keep the
root bracketed in a closed domain during the refinement process.
The secant method is generally preferred. Newton s method and the secant method are
two of the most efficient methods for refining the roots of a nonlinear equation.
2.4.1. Newton s Method
Newton s method (sometimes called the Newton-Rhapson method) for solving nonlinear
equations is one of the most well-known and powerful procedures in all of numerical
analysis. It always converges if the initial approximation is sufficiently close to the root,
and it converges quadratic ally. Its only disadvantage is that the derivative f (x) of the
Roots of nonlinear equation Chapter 3
Nizar Salim 4 Lecture 2
nonlinear function f(x) must be evaluated.
Newton s method is illustrated graphically in Figure 2.6.
Figure 2.6 Newton s method
The function f(x) is nonlinear. Let s locally approximate f(x) by the linear function g(x),
which is tangent to f(x), and find the solution for g(x) = 0. Newton s method is sometimes
called the tangent method. That solution is then taken as the next approximation to the
solution of f(x) = 0.
The procedure is applied iteratively to convergence. Thus,
i i
i i
i x x
f x f x
f x slopeof f x
1
1 ( ) ( )
( ) ( ) (2.54)
Solving Eq. (2.54) for xi+1 with f(xi+1) = 0 yields
( )
( )
1
i
i
i i f x
f x
x x (2.55)
Equation (2.55) is applied repetitively until either one or both of the following
convergence criteria are satisfied:
xi+1 -xi 1 and/or f(xi+1) 2
Roots of nonlinear equation Chapter 3
Nizar Salim 5 Lecture 2
Example 2.3 Newton s method.
Substituting 2 = 32.118463 deg into Eq. (3.62) gives f( 2) = 0.00214376.
These results and the results of subsequent iterations are presented in Table 2.3.
Table 2.3. Newton s method
Roots of nonlinear equation Chapter 3
Nizar Salim 6 Lecture 2
3.4.2. The Secant Method
When the derivative function, f (x), is unavailable or prohibitively costly to evaluate,
an alternative to Newton s method is required. The preferred alternative is the secant
method.
The secant method is illustrated graphically in Figure 2.7
Figure 2.7 The Secant Method
The nonlinear function f(x) is approximated locally by the linear function g(x), which is
the secant to f(x), and the root of g(x) is taken as an improved approximation to the root
of the nonlinear function f(x). A secant to a curve is the straight line which passes
through two points on the curve.
The procedure is applied repetitively to convergence. Two initial approximations, x0 and
x1, which are not required to bracket the root, are required to initiate the secant method.
The slope of the secant passing through two points, xi-1 and xi, is given by
1
( ) ( 1 )
( )
i i
i i
i x x
f x f x
g x 2.78
The equation of the secant line is given by
( )
( ) ( )
1
1
i
i i
i i g x
x x
f x f x
2.79
Where f(xi+1) = 0. Solving Eq. (2.79) for xi+1 yields
Roots of nonlinear equation Chapter 3
Nizar Salim 7 Lecture 2
( )
( )
1
i
i
i i g x
f x
x x 2.80
Equation (2.80) is applied repetitively until either one or both of the following two
convergence criteria are satisfied:
xi+1 xi 1 and/or f(xi+1) 2
Example 2.4 the secant method
Roots of nonlinear equation Chapter 3
Nizar Salim 8 Lecture 2
Substituting 2 = 31.695228 deg into Eq. (3.85) gives f( 2) = -0.00657688.
These results and the results of subsequent iterations are presented in Table 2.8. The
convergence criterion, i+1 i 0.000001 deg, is satisfied on the fifth iteration, which
is one iteration more than Newton s method requires.
Table 2.8 the secant method
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