lizfcm/homeworks/hw-7.org

Homework 7

Question One

See UTEST(eigen, dominant_eigenvalue) in test/eigen.t.c and the entry Eigen-Adjacent -> dominant_eigenvalue in the LIZFCM API documentation.

Question Two

See UTEST(eigen, leslie_matrix_dominant_eigenvalue) in test/eigen.t.c and the entry Eigen-Adjacent -> leslie_matrix in the LIZFCM API documentation.

Question Three

See UTEST(eigen, least_dominant_eigenvalue) in test/eigen.t.c which finds the least dominant eigenvalue on the matrix:

\begin{bmatrix} 2 & 2 & 4 \\ 1 & 4 & 7 \\ 0 & 2 & 6 \end{bmatrix}

which has eigenvalues: $5 + \sqrt{17}, 2, 5 - \sqrt{17}$ and should thus produce $5 - \sqrt{17}$.

See also the entry Eigen-Adjacent -> least_dominant_eigenvalue in the LIZFCM API documentation.

Question Four

See UTEST(eigen, shifted_eigenvalue) in test/eigen.t.c which finds the least dominant eigenvalue on the matrix:

\begin{bmatrix} 2 & 2 & 4 \\ 1 & 4 & 7 \\ 0 & 2 & 6 \end{bmatrix}

which has eigenvalues: $5 + \sqrt{17}, 2, 5 - \sqrt{17}$ and should thus produce $2.0$.

With the initial guess: $[0.5, 1.0, 0.75]$.

See also the entry Eigen-Adjacent -> shift_inverse_power_eigenvalue in the LIZFCM API documentation.

Question Five

See UTEST(eigen, partition_find_eigenvalues) in test/eigen.t.c which finds the eigenvalues in a partition of 10 on the matrix:

\begin{bmatrix} 2 & 2 & 4 \\ 1 & 4 & 7 \\ 0 & 2 & 6 \end{bmatrix}

which has eigenvalues: $5 + \sqrt{17}, 2, 5 - \sqrt{17}$, and should produce all three from the partitions when given the guesses $[0.5, 1.0, 0.75]$ from the questions above.

See also the entry Eigen-Adjacent -> partition_find_eigenvalues in the LIZFCM API documentation.

Question Six

Consider we have the results of two methods developed in this homework: least_dominant_eigenvalue, and dominant_eigenvalue into lambda_0, lambda_n, respectively. Also assume that we have the method implemented as we've introduced, shift_inverse_power_eigenvalue.

Then, we begin at the midpoint of lambda_0 and lambda_n, and compute the new_lambda = shift_inverse_power_eigenvalue with a shift at the midpoint, and some given initial guess.

1. If the result is equal (or within some tolerance) to lambda_n then the closest eigenvalue to the midpoint is still the dominant eigenvalue, and thus the next most dominant will be on the left. Set lambda_n to the midpoint and reiterate.
2. If the result is greater or equal to lambda_0 we know an eigenvalue of greater or equal magnitude exists on the right. So, we set lambda_0 to this eigenvalue associated with the midpoint, and re-iterate.
3. Continue re-iterating until we hit some given maximum number of iterations. Finally we will return new_lambda.