Learning objectives
A sound balancing of theoretical analysis, description of algorithms and discussion of applications is the primary concern.
Prerequisites
Numerical Analysis 1, Computational Numerical Lab.
Course unit content
Approximation of Functions and Data: Trigonometric interpolation. Rational interpolation. Least-squares approximationof functions and data.
Numerical Integration:
Orthogonal polynomials. Gaussian quadrature on bounded and unbounded intervals. Error estimates. Multiple integrals.
Numerical linear algebra:
QR-decomposition. Basic iterative methods. Jacobi, Gauss-Seidel and SOR methods. Convergence results. Coniugate gradient method. Stop tests.
Eigenvalue and eigenvector problem. Localization of eigenvalues. Stability analysis. The power method. The inverse power method. Eigenvalues and eigenvectors of a tridiagonal matrix. Householder transformations. Reduction of a general matrix to Hessemberg form. The LR algorithm.
The QR algorithm.
Solution of Nonlinear Equations:
Secant method, False Position method. Convergence results. Fixed-point methods. Rate of convergence. Zeros of polynomials. The Newton-Horner method; the Bairstow method. Stop tests. Newton’s method in several variables.
Numerical Solution of Ordinary Differential Equations:
Linear multistep methods. Order, convergence and stability analysis. Adams methods. Predictor-corrector methods.
Boundary valure problems: shooting method, finite-difference method, Galerkin method.
Full programme
- Approximation of Functions and Data. Trigonometric interpolation. Rational interpolation. Least-squares approximation: the continuous and the discrete cases.
- Numerical Integration. Orthogonal polynomials. Gaussian quadrature on bounded and unbounded intervals. Error estimates. Multiple integrals. Adaptive algorithms.
- Numerical linear algebra. Basic iterative methods. Jacobi, Gauss-Seidel and SOR methods. Richardson methods. Coniugate gradient method. GMRES and Bi_CGStab. Convergence results. Stop tests.
- Eigenvalue and eigenvector problem. Localization of eigenvalues. Stability analysis. The power method. The inverse power method. Eigenvalues and eigenvectors of a symmetric matrix: Sturm technique. Householder transformations. Reduction of a general matrix to Hessemberg form. The LR algorithm. The QR algorithm.
- Solution of Nonlinear Equations: Fixed-point methods.Convergence results. Stop tests. Newton’s method in several variables.
- Numerical Solution of Ordinary Differential Equations: Linear multistep methods for Cauchy problems. Order, convergence and stability analysis. Adams methods. Predictor-corrector methods.
- Boundary valure problems: shooting method, finite difference method, Galerkin method.
Bibliography
A.Quarteroni, R.Sacco, F.Saleri, Matematica Numerica, SPRINGER, (2008).
G.Naldi, L.Pareschi, G.Russo, Introduzione al Calcolo Scientifico, McGraw-Hill, (2001)
Teaching methods
Oral lessons and Lab.
Assessment methods and criteria
Oral exam and Lab test.
Other information
During the teaching course, students are asked to resolve some theorical and practical exercises, with the help of computing machines and using Matlab programming language, already introduced in Numerical Analysis course in the previous years.
