Newton’s Method - Numerical Methods - Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Newton’s Method for One-Dimensional

Optimization

Newton’s Method-Overview

• Open search method

• A good initial estimate of the solution is

required

• The objective function must be twice

differentiable

• Unlike Golden Section Search method

• Lower and upper search boundaries are not

required (open vs. bracketing)

• May not converge to the optimal solution

Newton’s Method-Algorithm

I nitialization: Determine a reasonably good estimate for

the maxima or the minima of the function.

Step 1. Determine and.

Step 2. Substitute (initial estimate for the first iteration)

and into

to determine and the function value in iteration i.

Step 3.If the value of the first derivative of the function is

zero then you have reached the optimum (maxima or minima). Otherwise, repeat Step 2 with the new value of

f ( ) x

f '^ ( ) x f ''( ) x

x i x 0

f '^ ( ) x f ''( ) x

''

' 1 i

i i i f x

f x x (^) + = x −

xi + 1

x i

Example

The cross-sectional area A of a gutter with equal base and edge length of 2 is given by

A = 4 sinθ ( 1 +cos θ )

.

Find the angle θ which maximizes the cross-sectional area of the gutter.

2

2

2

θ (^) θ

Solution Cont.

Iteration 2:

1. 0472 4 sin 1. 0466 ( 1 4 cos 1. 0466 )

4 (cos 1. 0466 cos 1. 0466 sin 1. 0466 )

1. 0466

2 2 (^2) − + =

• − θ = −

Iteration θ 1 0.7854 2.8284 -10.8284 1.0466 5. 2 1.0466 0.0062 -10.3959 1.0472 5. 3 1.0472 1.06E-06 -10.3923 1.0472 5. 4 1.0472 3.06E-14 -10.3923 1.0472 5. 5 1.0472 1.3322E-15 -10.3923 1.0472 5.

Summary of iterations

Remember that the actual solution to the problem is at 60

degrees or 1.0472 radians.

f '( θ) f ''( θ) θ estimate f ( θ )