- (Mo, Sep 4)
- Errors in computations
- Absolute and relative error and their change by basic
operations

- Finite precision, floating numbers, the error of number
representations

- (Tu, Sep 5 )
- Linear algebra review
- Vector norm definition

- inner product definition
- 1-norm, 2-norm, p-norm, infinity norm and their relations
- Matrix norm definition

- (Mo, Sep 11)

- Matrix norm induced by a vector norm
- Application to 1-norm, infinity norm

- Adjoint of a matrix, self-adjoint matrices and their basic properties
- Quadratic forms, positive (semi)definite, negative
(semi)definite and indefinite matrices

- (Tu, Sep 12)
- Rayleigh quotient

- Connection between theinduced 2-norm of matrix A and eigenvalues of A*A
- Its application for self-adjoint matrices

- Spectral radius
- Gershgorim circles
- (Mo, Sep 18)
- Proof of Gershgorin's Theorem
- Singular values and singular value decomposition
- Definition of Moore-Penrose inverse

- (Tu, Sep 19)

- Application: computation of Moore-Penrose inverse by SVD
- Linear systems Ax=b
- The best approximation if there is no solution
- Relation between the relative error of b and of x (when A is exact)
- Relation between the relative error of A and of x (when b is exact)
- Condition number of a regular matrix
- Direct method to solve the system: by rank-one-decomposition
- A space efficient way to do it
- The special case of tridiagonal matices, formulas for the solution
- (Mo, Sep 25)
- Iterative improvement of the solution
- Estimate of the norm of error, condition to ensure fast convergence
- General iteration for linear systems, condition for
convergence

- Gauss-Seidel iteration and how to compute it
- (Tu, Sep 26)
- Successive over relaxation
- a special case
- (Mo, Oct 2)
- spec. case cont'd
- Tensor product of matrices

- Poisson equation (partial differential equation) and its discrete version
- (Tu, Oct 3)
- Trade-off between the density of the grid and the sped of convergence
- Gradient method

- (Mo, Oct 9)
- Conjugate gradient method
- Eigenvalue computation: power iteration

- (Tu, Oct 10) -- the class is cancelled, see you on Monday
- (Mo, Oct 16)
- Inverse iteration
- Householder transformation
- Sturm sequence
- (Tu, Oct 17)
- Theorem of Sturm
- Eigenvalue localization of tridiagonal matrices by Sturm's theorem
- (Tu, Oct 24)
- Numerical SVD (via bidiagonal form)
- QR decomposition
- QR algorithm
- Hessenberg matrices and their QR decompositions
- (Mo, Oct 30)
- Computation of Hessenberg form
- QR with a shift

- (Tu, Oct 31)
- Eigenvalues of Hessenberg matrices
- Lanczos method

- (Mo, Nov 6)
- Lanczos method cont'd (proof of correctness, speed of convergence - w/o proof)
- Courant-Fisher theorem
- (Tu, Nov 7)
- Proof of Courant-Fisher theorem
- Theorem of Weyl
- Eigenvalue estimates when the matrix is modified by a matrix
of rank one

- Interlacing property

- (Mo, Nov 13)
- Proof of interlacing property
- Wielandt-Hoffman theorem

- (Tu, Nov 14)
- Theorem of Birkhoff on doubly stochastic matrices
- Wielandt-Hoffman for singular values
- Least squares problem -- some examples

- (Mo, Nov 20)
- Normal equation of Least squares problem
- Application: fitting line to data points
- (In)stability -- example
- Full rank case: solution by QR transformation

- (Tu, Nov 21)
- Least squares for matriix of not full rank - by SVD
- Generalized eigenvalues

- properties, examples
- the triangle case

- (Mo, Nov 27)
- Generalized eigenvalues cont'd

- algorithm for the symmetrical case
- by Cholesky decomposition when B is positive definite

- Intro to Linear Matrix Inequalities from the application side (by Bálint)
- (Tu, Nov 28)
- Linear Matrix inequalities
- basic properties
- sketch of algorithms
- (Mo, Dec 4) -- the class is canceled
- (Tu. Dec 5) -- the class is canceled