Spectral Method

Usage notes

In this exercise, the spectral method will be applied to three different functions.

For the results please use the Maple 6 syntax. E.g.: Matrix(2, 2, [[1,2], [3,4]]); and Vector([1,2,3]).

a)

Given the function

and the starting point
(x₀,y₀) = (1,-1)
Verify that the function value at the starting point is 1 and compute the gradient at the starting point.
f₁'(x₀,y₀) = ___________________

Compute the Hessian at the starting point.

f₁''(x₀,y₀)= ___________________

Compute the eigenvalue / eigenvector decomposition of this matrix. Sort the eigenvalues in decreasing order.

___________________

U = ___________________

Compute the vector d defined in the lecture.

d = ___________________

Assume that we set the stepsize of the spectral method to h = 1. Compute the vector z.

z = ___________________

Now compute the increment vector, the new position and the function value at the new position.

___________________
(x₁, y₁) = ___________________
f₁ (x₁, y₁) = ___________________

Verify that the function value decreased.

The following graph illustrates what happens in the direction of the first eigenvector. (In the direction of the second eigenvector, nothing happens.) It shows a section of the function graph of f₁ along the plane going through the starting point and having the direction of the first eigenvector, i.e. the plane y = -x. Additionally it shows the second order tangent (a parabola) in this plane. The section is projected onto the plane y = 0.

The minimum of the parabola is 0.75 and thus greater than the function value of f₁.

b)

Now consider a second function

and the starting point

(x₀, y₀) = (1, 1)

Verify that the function value at the starting point is 2/3 and compute the gradient.

f₂'(x₀,y₀) = ___________________

Compute the Hessian at the starting point.

f₂''(x₀,y₀) = ___________________

Compute the eigenvalue / eigenvector decomposition of this matrix. Sort the eigenvalues in decreasing order.

___________________
U = ___________________

Compute the vector d defined in the lecture.

d = ___________________

Assume that the stepsize of the spectral method is h. Compute the vector z.

z = ___________________

Now compute the increment vector. Group the terms that depend on the step size and those that do not.

___________________ + h * ___________________

Set

and compute the new position and the function value at the new postion.

(x₁, y₁) = ___________________
f₂ (x₁, y₁) = ___________________

Verify that the function value decreased.

c)

Finally consider a third function

and the starting point

(x₀,y₀)=(-2,-2)

Verify that the function value at the starting point is 3/4 and compute the gradient at the starting point.

f₃'(x₀,y₀) = ___________________

Compute the Hessian at the starting point.

f₃''(x₀,y₀) = ___________________

Compute the eigenvalue / eigenvector decomposition of this matrix. Sort the eigenvalues in decreasing order.

___________________
U = ___________________

Compute the vector d defined in the lecture.

d = ___________________

Assume that we set the stepsize of the spectral method to h = 1. Compute the vector z.

z = ___________________

Now compute the increment vector. Group the terms that depend on the step size and those that do not.

___________________ + h * ___________________

Set

and compute the new position and the function value at the new position.

(x₁, y₁) = ___________________
f₃(x₁, y₁) = ___________________

Verify that the function value decreased.

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