Kim Day Jessie Twigger Christian Zelenka How is this technique conducted The NewtonRaphson formula consists geometrically of extending the tangent line at a current point until it crosses zero then ID: 202990
Download Presentation The PPT/PDF document "Newton-Raphson Method" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Newton-Raphson Method
Kim Day
Jessie Twigger
Christian ZelenkaSlide2
How is this technique conducted
The Newton-Raphson formula consists geometrically of
extending the tangent line at a current point
until it crosses zero, then setting the next guess to the abscissa of that zero crossing.This technique derives from the Taylor series expansion of a function near a pointEx)
Slide3
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
approximatelySo by the Newton-Raphson formulaSlide4
Example for Clarity
Courtesy of WikipediaSlide5
Advantages of Newton-Raphson
One of the fastest convergences to the root
Converges on the root
quadraticlyNear 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 techniqueEasy to convert to multiple dimensionsCan be used to “polish” a root found by other methodsSlide6
Disadvantages of Newton-Raphson
Must find the derivative
Poor global convergence properties
Dependent on initial guessMay be too far from local rootMay encounter a zero derivativeMay loop indefinitelySlide7
Examples of Disadvantages
On the left, we have Newton’s Method finding
a
local maxima, in such cases the method will
s
hoot off into negative infinity
Figure 9.4.2
Figure 9.4.3
Newton's Method has entered an
i
nfinite cycle. Better initial guesses
m
ay be able to alleviate this problemSlide8
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. Slide9
Example: Square Root of a NumberSlide10
Example: Solving EquationsSlide11
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 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
for all subsequent steps