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MA60053 - Computational Linear Algebra

Topics covered

Week Contents
Week 1 Course overview, discretization of ODE(motivation for linear system of equations), Reiview of basic properties about matrices
Week 2 Reivew of diagonalizability of matrices, spectral theorem symmetric matrices, positive square root of positive semidefinte matrices, Generalized eigenvalue problem, Sensitivity and condtioning, absolute and relative error, floating point arithemetic
Week 3 Norms, matrix norms, induced norms, Matrix p-norms, condtion number
Week 4 condition number, perturbing the coefficient matrix A and/or the vector b in the linear system Ax=b, Scaling and condition number
Week 5 Back substitution, Gaussinan elimination, LU decomposition, pivoting
Week 6 Cholesky decomposition, Plane rotator(Givens rotator), Reflector(Householder transformation), QR decomposition(Proofs using rotator and reflectors)
Week 7 Least squares problems, revision for mid-semester
Mid semester 18.02.2019 to 26.02.2019
Week 8 Singular value decomposition(SVD)
Week 9 SVD and least squares problem, Low rank approximation
Week 10 Pseudo inverse, sensitivity analysis for SVD
Week 11 Eigenvalue problems, Gershgorin's theorem, Improved Gershgorin's theorem
Week 12 Improved Gershgorin's theorem(cont.), Rayleigh principle, Courant Fischer min-max principle
Week 13 Sylvester's law of inertia, Bauer-Fike theorem, Sensitivity analysis for eigenvalues
Week 14 & 15 The power method, inverse iteration by von Wielandt, Jacobi method, Householder reduction to Hessenberg form, QR algorithm.
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  • Problem sheets
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