7.2 Solving a Nonlinear Equation
This section describes some numerical methods for finding real roots
of the one-dimensional nonlinear function
, i.e., solving the nonlinear equation
.
The multidimensional case is treated in Section 7.3.
Our aim is to construct a sequence
of real numbers or real vectors that converges
to the root we search for.
-point iterative methods are used, i.e.,
is computed
as an
-point iteration function depending upon the
previous values:
One-point iterative methods with the same function
for all
are used most often.
In commonly used methods, the sequence is known to converge
if the initial approximation is close enough to the root.
7.2.1 Termination of Iterative Methods
For any iterative method, one has to specify termination criteria.
Natural ones are:
- Function value is close enough to zero, i.e., the procedure stops in
-th step
if
for given
.
- Successive values of
are close to each other (hence, we are approaching the root
probably), i.e., stop the procedure if
for given
.
- The procedure should be stopped if the number of iterations reaches a given maximal
number of iterations; in this case, one should inspect the sequences
and
for eventual loops and possibly start with a different initial approximation.
7.2.2 Newton-Raphson Method for One-dimensional Problems
The Newton-Raphson method (or just Newton method) is one of the most popular methods for root finding.
Let
be
-th approximation of the root. The function
in the vicinity of the root
is approximated by the tangent to the curve
at
; the intersection
of the tangent with the
-axis defines the next approximation,
(see Fig. 7.1).
The formula describing this procedure is
 |
(7.1) |
This formula can be also derived as follows: start with the Taylor series of
centered in
:
and take
; then
Now we neglect the term
and setting
implies the formula (7.1).
REMARK 7.1
The Newton-Raphson method converges quadratically in case of distinct roots.
Hence, it is very efficient for functions that have continuous and nonzero derivative in some
neighbourhood of a root, having a good initial guess.
REMARK 7.2
If

is not known (or its computation is time-consuming too much), it can be approximated
by differences as shown, e.g., in (
Rektorys; 1995); see Section
7.6.1 for details.
However, the convergence rate is lower in this case.