Minimization (Exam 1999, Question 6)

Minimization (Exam 1999, Question 6)

Usage notes

We want to find a local minimum of the function

given a starting point

Use $\phi = -\frac{\sqrt{5}-1}{2}$ for your calculations.

a) Brent's method

Perform three steps of Brent's method with direction d=(1,0) and initial step size h=1. The precision is always tested up to 3 digits.
Starting point x=1 y=0. F(1,0) = ___________________

First point:

x = ___________________
y = ___________________
F(x,y) = ___________________

Second Point:

x = ___________________
y = ___________________
F(x,y) = ___________________

Third point:

x = ___________________
y = ___________________
F(x,y) = ___________________

b) Steepest descent method

Perform one step of the steepest descent method (use the analytical method). Write down the direction d of steepest descent:

dx = ___________________
dy = ___________________

Write down the function f describing F along this direction. Use the abscissa x as parameter.

f(x) = ___________________

We consider only (local) minima where the function value is > -infinity. How many candidates for a minimum do exist ?

0

1

2

3

Write down the approximation after the first step (as decimal numbers):

x = ___________________
y = ___________________
F(x,y) = ___________________

c) Newton's Method

Perform one step of the Newton's method. Compute the gradient of F(x,y) in (x,y) = (1,0).

(F')x|(1,0) = ___________________
(F')y|(1,0) = ___________________

Compute the Hessian of F in (1,0) :

(F'')xx|(1,0) = ___________________ (F'')xy|(1,0) = ___________________
(F'')yx|(1,0) = ___________________ (F'')yy|(1,0) = ___________________

Write down the approximation after the first step of Newton's Method:

x = ___________________
y = ___________________
F(x,y) = ___________________

d) Spectral Method

Perform one step of the Spectral method with a step size h=0.1 . Write down the obtained approximation.

x = ___________________
y = ___________________
F(x,y) = ___________________

What happens in y-direction ?

We are on a minimum in y-direction, therefore we do nothing in this direction (Delta_y = 0).

We are on a maximum in y-direction, therefore we cannot choose an orientation (Delta_y = 0).

We are on a minimum in y-direction, but we neverthelesss move one step size (Delta_y = h).

We are on a maximum in y-direction, so we move one step size in any orientation (Delta_y = +/- h).

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