.. _final: =================== Final Exam =================== The in-class Final exam will be on Tuesday December 13, 2:30 - 4:20pm. The exam is closed book, closed notes. No calculators. The exam will cover all the course material, but with an emphasis on the material since the midterm, roughly Lectures 20-27 in Trefethen and Bau. In the video lecture for December 9, I plan to briefly discuss the QR algorithm and computation of the SVD, but these won't be on the exam. **Note:** - Please complete the `course evaluations `_ by Friday December 9. Feedback to help improve this course is very welcome. - There will be no lecture on Friday, December 9 (the last day of classes). Instead a video lecture will be posted. See the `Canvas page `_ - Some extra office hours will be scheduled, but don't put off understanding this material to the last minute. **Office hours** LeVeque will have office hours December 4,5 as usual. During Finals week: - in Lewis 328 - Monday 12/12, 10:00 - 12:00 - On GoToMeeting - Monday 12/12, 5:00 - 6:00pm PST - Tuesday 12/13, 7:00 - 8:00am PST **Some key concepts and algorithms that you should know:** - Review material from the :ref:`midterm`. - Gaussian elimination without pivoting, :math:`A = LU`. - Gaussian elimination with partial pivoting, :math:`PA = LU`. - Permutation matrices. - Cholesky factorization of a hermitian positive definite matrix. - Basic properties of eigenvalues and eigenvectors, characterization of eigenspace as the nullspace of a matrix. - How to compute eigen-decomposition for a :math:`2 \times 2` matrix. - Determining eigenvalues and eigenvectors of a diagonal or triangular matrix. - Diagonalizable matrices: :math:`A = X\Lambda X^{-1}`. For normal matrices, can choose :math:`X` to be unitary. - Schur decomposition: :math:`A = QTQ^*` with :math:`T` upper triangular. Agrees with eigen-decomposition (i.e. :math:`T` is diagonal) if :math:`A` is normal. - Relation between Schur decomposition and SVD when :math:`A` is normal. - Rayleigh quotient and why this approximates an eigenvalue if :math:`x` is close to an eigenvector. - Power method, inverse power method, Rayleigh quotient method. - Decomposition of a vector into eigen-components and use in analyzing power method. - Basic idea of reduction to Hessenberg form via Householder reflector similarity transformations. - Operation counts for algorithms such as Gaussian elimination, back substitution, Gram-Schmidt, Householder reduction. You should know order of magnitude for basic algorithms (e.g. :math:`{\cal O}(m^3)` to factor a general :math:`m\times m` matrix) and how to compute for something involving nested loops.