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MATHEMATICALINSTITUTE

PROF. DR. CHRISTIANEHELZEL

DAVIDKERKMANN

JUNE4TH2020

12 13 Σ

NAME: MAT-NR.:

Numerical Methods for Data Science – Exercise Sheet 5

Exercise 12:

LetAbe a matrix,xbe a vector withkxk2 = 1 andλbe a scalar. Definer=Ax−λx. Show that there exists a matrixEsuch thatA+E has eigenvalueλwith corresponding eigenvectorxandkEkF =krk2, where k · kF denotes the Frobenius matrix norm.

Exercise 13:

Implement the Power Method, the Inverse Iteration as well as the Rayleigh Quotient Iteration.

Test your implementations on the matrix Tn+I ∈R10×10, where Tn is the matrix from chapter 1,

|| 1 and I is the identity matrix, by always searching for the largest eigenvalue in magnitude.

Plot the error after each iteration of all methods in a combined plot. How does the convergence speed depend on for the different methods? Always start with the same initial vector v(0) for all methods which is a good guess for the eigenvector that belongs to the largest eigenvalues in magnitude. For the Inverse Iteration, chooseµ to be a good guess on the largest eigenvalue in magnitude as well.

Submit until June 18th 2020, 2:00 pm in the ILIAS.

Review in the exercise course on June 19th 2020.

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