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## 6FMAI13 Computer Exercises

The computer exercises are to be completed either in a group of two students, or by one student alone. A written report that answers all questions should be written (possibly by hand). Include graphs in the report. The final Matlab programs should be sent by email to `fredrik.berntsson@liu.se`.

#### Exercise 1: Least Squares Problems

In this exercise we will compute the QR decomposition using Householder reflections. Also we will use the QR decomposition to fit an ellipse to given points (xk,yk).

Instructions: E1-TANA15-QR-and-Applications.pdf
Matlab: ApplyReflection.m , HouseholderQR.m, QRUpdate.m, CometTracking.m.

#### Exercise 2: Eigenvalues

The QR is the default choice for computing eigenvalues of non-symmetric matrices. In this exercise the alrogithm will be implemented and tested.

Instructions: E2-TANA15-Eigenvalues.pdf
Matlab: Hessenberg.m, HessEigQR.m.

#### Exercise 2.5: Sylvesters Law of Inertia

In this exercise you will use Sylverster's Law of Inertia to prove simple bounds for the eigenvalues of a symmetric matrix.

Instructions: E2.5-MAI0119-Sylvester.pdf

#### Exercise 4: The Singular Value Decomposition

In this exercise the SVD will be used for a small model fitting problem and also for solving a classification problem.

Instructions: E4-TANA15-Singular-values.pdf
Material: Goose.mat, DataSet.mat, CircleData.m, ExtractDigits.m, DisplayDigit.m, DistanceFromSubspace.m, CreateSubspaces.m and ClassifyDigit.m.

#### Exercise 4.5: The Conjugate Gradient Method

The Conjugate gradient method is the default choice for solving sparse linear systems with positive definite and symmetric matrices. In this exercise we will implement the method and test its properties.

Instructions: E4.5-Sparse.pdf
Material: Bundle1.mat.

Sidansvarig: Fredrik Berntsson