Numerical techniques and applications

Code 536AA

Credits 6

Learning outcomes

Objectives

In the course numerical methods are proposed for solving various applicative problems.

Major emphasis is given to the techniques of numerical linear algebra mostly used in applications.

Syllabus

1. Basic notions in linear algebra: similar reduction to diagonal and other canonical forms, positive definite matrices, singular value decomposition, norms, condition number

2. Direct methods for linear systems: elementary matrices, LU, LLh, QR, factorizations, Givens rotations, Cholesky and Householder methods

3. Iterative methods for linear systems: classic methods, overrelaxation, conjugate gradient method

4. Iterative methods for nonlinear systems: Newton and quasi-Newton methods

5. Iterative methods for eigenvalues: conditioning of the problem, power method, LR and QR methods, reduction to tridiagonal form of a symmetric matrix

6. Linear least squares problem: normal equations and SVD

7. Methods for tridiagonal matrices: cyclic reduction, Sturm sequences, divide-and-conquer techniques

8. Non-negative matrices: Perron-Frobenius results, stochastic matrices

9. Discrete Fourier Transform: some applications

Oral examination.

In the course numerical methods are proposed for solving various applicative problems.

Major emphasis is given to the techniques of numerical linear algebra mostly used in applications.

Syllabus

1. Basic notions in linear algebra: similar reduction to diagonal and other canonical forms, positive definite matrices, singular value decomposition, norms, condition number

2. Direct methods for linear systems: elementary matrices, LU, LLh, QR, factorizations, Givens rotations, Cholesky and Householder methods

3. Iterative methods for linear systems: classic methods, overrelaxation, conjugate gradient method

4. Iterative methods for nonlinear systems: Newton and quasi-Newton methods

5. Iterative methods for eigenvalues: conditioning of the problem, power method, LR and QR methods, reduction to tridiagonal form of a symmetric matrix

6. Linear least squares problem: normal equations and SVD

7. Methods for tridiagonal matrices: cyclic reduction, Sturm sequences, divide-and-conquer techniques

8. Non-negative matrices: Perron-Frobenius results, stochastic matrices

9. Discrete Fourier Transform: some applications

Oral examination.