Advantages of Newton

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