and the approximate value, i.e.,The Newton-Raphson method is suitable for implementation on a computer (pseudo-code). It is a process for the determination of a real root of an equation f (x) = 0, given just one point close to the desired root. It can be viewed as a limiting case of the secant method (Step 8) or as a special case of simple iteration ( Step 9).
Let x0 denote the known
approximate value of the root of f(x) = 0 and h the
difference between the true value and the approximate value, i.e.,
The second degree, terminated Taylor expansion ( STEP 5) about x0 is
where 
lies between 
Ignoring the remainder term and writing 
whence
and, consequently,
should be a better estimate of the root than x0. Even better approximations may be obtained by repetition (iteration) of the process, which then becomes
Note that if f is a polynomial, you can
use the recursive procedure of STEP 5 to compute 
The geometrical interpretation is
that each iteration provides the point at which the tangent at
the original point cuts the x-axis (Figure 9). Thus the
equation of the tangent at (xn, f (xn))
is
y - f(x0) = f '(x0)(x - x0)
so that (x1, 0) corresponds to
-f(x0) = f '(x0)(x1 - x0),
whence x1 = x0 - f(x0)/f '(x0).
We will use the Newton-Raphson method to find the positive root of the equation sin x = x2, correct to 3D.
It will be convenient to use the method of false position to obtain an initial approximation. Tabulation yields
With numbers displayed to 4D, we see that there is a root in the interval 0.75 < x < 1 at approximately
Next, we will use the Newton-Raphson method:
and
yielding
Consequently, a better approximation is
Repeating this step, we obtain
and
so that
Since f(x2) = 0.0000, we conclude that the root is 0.877 to 3D.
If we write
,
the Newton-Raphson iteration expression
may be rewritten
We have observed (STEP 9) that, in general, the iteration method converges when 
near
the root. In the case of Newton-Raphson, we have
,
so that the criterion for convergence is
,
i.e., convergence is not as assured as, say, for the bisection method.
The second degree terminated Taylor expansion about xn is
where 
is the error at the n-th iteration
and 
.
Since 
, we find
But, by the Newton-Raphson formula,
whence the error at the (n + 1)-th iteration is
provided en is sufficiently small.
This result states that the error at the (n
+ 1)-th iteration is proportional to the square of the error
at the nth iteration; hence, if 
, an answer correct to one decimal place
at one iteration should be accurate to two places at the next
iteration, four at the next, eight at the next, etc. This
quadratic - second-order convergence - outstrips the rate
of convergence of the methods of bisection and false position!
In relatively little used computer programs, it may be wise to prefer the methods of bisection or false position, since convergence is virtually assured. However, for hand calculations or for computer routines in constant use, the Newton-Raphson method is usually preferred.
One application of the Newton-Raphson method is in the computation of square roots. Since a½ is equivalent to finding the positive root of x2 = a. i.e.,
f(x) = x2 - a = 0.
Since f '(x) = 2x, we have the Newton-Raphson iteration formula:
xn+1 = xn - (x²n - a)/2xn = ½(xn + a/xn),
a formula known to the ancient Greeks. Thus, if a = 16 and x0 = 5, we find to 3D
x1 = (5 + 3.2)/2 = 4.1, x2 = (4.1 + 3.9022)/2 = 4.0012, and x3 = (4.0012 + 3.9988)/2 = 4.0000.
1. Use the Newton-Raphson method to find to 4S the (positive) root of 3xex=1?
Note that Tables of natural logarithms (Naperian) are more more readily available than tables of the exponential, so that one might prefer to solve the equivalent equation f(x) = loge 3x + x = log 3 + loge x + x = 0.
2.Derive the Newton-Raphson iteration formula
xn + 1 = xn - (xkn - a)/k xk-1n
for finding the k-th root
of a.
3. Compute the square root of 10 to 5 significant digits from an initial guess.
4. Use the Newton-Raphson method to find to 4D the root of the equation
x cos x = 0.
Use the Newton-Raphson method to find to 4D the root of each equation in the exercises 1 - 3.