Question

7 significant digits please

(1 point) Starting with a =-1, b = 1, do 4 terations of golden section search to estimate where f(x)-(r-sin()) reaches a minimum. f(c) f(d)

Answer 1

Solution: Note that
5-1 = 0.6180340 R=
is called the golden-ratio. It is the positive root of the quadratic equation $R^2+R-1=0$.

The given funcion is
f()-sin()
. The Golden Section Search Method for finding the minimum of the function
f()-sin()
in the interval [-1,1] is given in the following table. The necessary steps are

Step1:

n=0

If $f(x_1^0)<f(x_2^0)\\$ , then set $a^1=x_2^0,b^1=b^0$ .

Step 2:

rl R(b -a)

Check  
f(r) < f(x))
, then set $a^2=a^1,b^2=x_2^1$ .
Continuing in this way we can find the minimum.

All the computations reported here are rounded to seven decimal of places

	a	b	c	d	f(c)	f(d)
i=0	-1	1	0.236068	-.236068	-0.1781534	0.2896096
i=1	-0.236068	1	0.5278641	0.2360679	-0.2250488	-0.1781534
i=2	-0.236068	0.5278641	0.2360680	0.0557281	-.01781534	-0.05259364
i=3	0.2360680	0.5278641	0.4164079	0.3475242	-0.2310824	-0.2197980
i=4	0.4164079	0.5278641	0.4852916	0.4589804	-0.2309584	-0.2323713