Kim Day
Jessie Twigger
Christian Zelenka
How is this technique conducted
 The Newton-Raphson formula consists geometrically
of extending the tangent line at a current point 𝑥𝑖 until
it crosses zero, then setting the next guess 𝑥𝑖+1 to the
abscissa of that zero crossing.
 This technique derives from the Taylor series
expansion of a function near a point
 Ex)
The iterative technique
 Why is Newton-Raphson so powerful? The answer is
its rate of convergence:
 Within a small distance of x, the function and its
derivative are approximately
 So by the Newton-Raphson formula
Example for Clarity
Courtesy of Wikipedia
Advantages of Newton-Raphson
 One of the fastest convergences to the root
 Converges on the root quadraticly
 Near a root, the number of significant digits
approximately doubles with each step.
 This leads to the ability of the Newton-Raphson Method
to “polish” a root from another convergence technique
 Easy to convert to multiple dimensions
 Can be used to “polish” a root found by other methods
Disadvantages of Newton-Raphson
 Must find the derivative
 Poor global convergence properties
 Dependent on initial guess
 May be too far from local root
 May encounter a zero derivative
 May loop indefinitely
Examples of Disadvantages
Figure 9.4.2
On the left, we have Newton’s Method finding
a local maxima, in such cases the method will
shoot off into negative infinity
Figure 9.4.3
Newton's Method has entered an
infinite cycle. Better initial guesses
may be able to alleviate this problem
Unfortunate Scenarios
• Newton’s method will obviously not converge in those scenarios where no
root is present.
• Thus functions with discontinuity at zero are impossible to analyze.
Example: Square Root of a Number
x  612
2
f ( x )  x  612
2
f '( x)  2 x
Example: Solving Equations
cos( x )  x
3
f ( x )  cos( x )  x
3
f ' ( x )   sin( x )  3 x
2
Notes on Efficiency
 For efficiency the user provides the routine that evaluates
both f(x) and its first derivative at the point x.
 The Newton-Raphson formula can also be applied using a
numerical difference to approximate the true local
derivative but this is not recommended.
 You are doing two function evaluations per step, so at best the
super-linear order of convergence will be only 2
 If you take dx too small, you will be wiped out by round off,
while if you take it too large, your order of convergence will be
only linear, no better than using the initial evaluation 𝑓 ` (𝑥0 )
for all subsequent steps