Computational Methods for Nonlinear Equations: Optimization Techniques and Golden Search -, Study notes of Computer Science

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Computational Methods CMSC/AMSC/MAPL 460

Solving nonlinear equations and zero finding

Ramani Duraiswami,

Dept. of Computer Science

Optimization analog of bisection

•^

Optimization involves finding maximum and minimum offunctions

-^

At these point first derivative vanishes

-^

So optimization typically involves use of differential methods

-^

Here we consider an algorithm like bisection

-^

Suppose we are given an interval

[a,b]

and have to find the

minimum in this interval

-^

We could look at

f(a), f(b)

and

f((a+b)/2)

•^

Even if

f((a+b)/2) < f(a)

and

f((a+b)/2) < f(b)

don’t know if

[a,(a+b)/2]

or

[(a+b)/2,b]

contains the minimum

•^

Could divide domain into three regions

-^

f(a), f(b), f((a+b)/3),

and

f(2(a+b)/3).

•^

Then we know which interval

[a,2(a+b)/3]

or

[(a+b)/3, b]

contains the minimum

Improved Golden Search:

fminbd

•^

As the search proceeds, we will have three points in the intervalwith the minimum

-^

Fit a parabola and find the minimum

-^

If the minimum is within the interval, we can choose it as the nextpoint

-^

To stop: recall near a minimum derivative vanishes

-^

So

f(x)=f(x

)+b(x-x*

2

•^

Let

x-x

δ^

and

f(x

)=a*

f(x)=a+b

(^2) δ

•^

If the interval

δ^

is as small as machine

ε

, then the change in the

value of f will of the order of machine

ε

•^

SO it is not computable

-^

Rather change can at most be about the square root of machine

ε

•^

This is employed in Matlab function

fminbd

and in the book

software function

fmintx

Systems of Nonlinear equations

-^

Analog for 1d bisection: Too hard to find bracketed zero

-^

Analog for Newton is what is used

-^

Derivative is now

f