Chapter Three.
Numerical Analysis
Roots of nonlinear functions.
Let f (x) be a real valued function defined for a ≤ x ≤ b. A number ξ is called a root of
the function f (x) in this interval if f (ξ ) = 0 and, correspondingly, a number x = ξ that makes
f (x) vanish is called a zero of f (x). The need to find roots of functions is fundamental to the
development and application of mathematics, and only in simple cases can the roots be
determined necessary to find them numerically. Many different methods exist for the
numerical analytically, so in all other cases it is determination of roots of functions, but of
these only the iteration method and Newton’s method will be described in any detail, as they
are in everyday use and are easily implemented on a computer.
Nature of roots
To estimate approximately an initial value of x approach to the exact value, separate
the function f(x) = 0 to two functions f1(x) and f2(x) and sketch each function, the
intersection point of them is the initial vale of x denoted as x˳.
Chapter Three.
Numerical Analysis
Roots of nonlinear functions.
1. Iteration method.
This method is well suited to machine computation provided numerical values of the
function involved are easily calculated, and a good approximation to the root is used to start
the iteration process. The idea is straightforward, and its success depends on rewriting the
given function f (x) whose root is required in the form
𝑓(𝑥) = 𝑥 − 𝑔(𝑥), 𝑓(𝑥) = 0
𝑥 = 𝑔(𝑥)
Starting with x = x°
𝑥1 = 𝑓(𝑥° )
or, in general:
𝑥𝑛+1 = 𝑓(𝑥𝑛 )
Iteration method
Ex: Find the roots of the function f(x) = x – e-x?
Sol.
f1(x) = x, f2(x) = e-x
Chapter Three.
Numerical Analysis
Roots of nonlinear functions.
From the graph, X˳ ≈ 0.5
f(x) = 0
x – e-x = 0
x = e-x
x1 = e(-0.5) = 0.607
x2 = e(-0.607) = 0.545
x3 = e(-0.545) = 0.5797
x4 = e(-0.5797) = 0.560
x5 = e(-0.560) = 0.571
x6 = e(-0.571) = 0.565
x7 = e(-0.565) = 0.568
x8 = e(-0.568) = 0.566
x9 = e(-0.566) = 0.568
x10 = e(-0.568) = 0.567
x11 = e(-0.567) = 0.567
stop
Check:
F(0.567) = 0.567 - e(-0.567) = - 0.0002 ≈ 0
2. Newton’s Method.
Our starting point for the derivation of Newton’s method for the determination of a
zero of a differentiable function f (x), also known as the Newton–Raphson method, is the
mean value theorem representation of f (x) about a point x = x˳ that can be written
Where ξ is a point between x and x˳
Chapter Three.
Numerical Analysis
Roots of nonlinear functions.