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### Numerical Analysis

Professors:
Boffi Daniele, Gardini Francesca
Year:
2015/2016
Course code:
502215
ECTS:
9
SSD:
MAT/08
DM:
270/04
Lessons:
84
Period:
II semester
Language:
Italian

#### Objectives

The aim of the course is to introduce the basics of Numerical Analysis and Scientific Computing so that the students can classify problems and the numerical algorithms suitable for their approximation. The course has a theoretical part as well as some Lab classes which take place at the Computer Lab of the Math Department.

#### Teaching methods

Lectures. Exercises. Labs.

#### Examination

Written and oral exam. Lab report.

#### Prerequisites

First year Calculus courses and the contents of the course "Algebra lineare".

#### Syllabus

1) Error analysis.
Classification of computational problems. Floating point numbers and arithmetics. Error propagation. Conditioning of a problem.

2) Direct methods for the solution of linear systems.
Triangular systems. Gauss elimination. LU factorization. Pivoting. Other factorizations, Choleski factorization. Banded, block and sparse matrices. Condition number. Forward and backward a priori analysis. Stability of LU factorization. Over-constrained systems; QR factorization; modified Gram-Schmidt algorithm and Householder matrices.

3) Iterative methods for the solution of linear systems.
Splitting methods: Jacobi and Gauss-Seidel methods. Iteration matrix and spectral radius. JOR and SOR methods. Convergence study and stopping criteria. Richardson-like methods; analysis of stationary Richardson method. Gradient method (steepest descent). Conjugate gradient method; preconditioned conjugate gradient method. Preconditioners.

4) Eigenvalues and eigenvectors approximation.
Conditioning of eigenproblems and eigenvalue localization. Power method. Inverse power method. Shifting. Deflation. Similarity methods; QR method.

5) Approximation of functions and data.
Lagrange interpolation. Error analysis in polynomial interpolation; Lebesgue constant and error estimate. Runge phenomenon and Chebychev nodes. Newton method and divided differences. Stability analysis of the interpolation. Abstract interpolation: unisolvence. Splines: linear and third order splines. Piecewise polynomial interpolation in three dimensions. General problem of linear interpolation. Linear least squares. Orthogonal polynomials (Legendre, Chebyshev). Best approximation.

6) Non linear equations and optimization.
Bisection method. Regula Falsi and Illinois methods. Newton method. Analysis of Newton method. Chord and secant methods. Fixed point iterations. Convergence of fixed point and error propagation. Newton method as fixed point iteration: multiple zeros. Deflation method in polynomial root finding.

7) Numerical integration.
Midpoint rule. Newton-Cotes integration (trapezoidal rule and Cavalieri-Simpson rule). Error estimate for Newton-Cotes formulae. Composite rules. Gauss integration, Jacobi theorem. Gauss-Legendre, Gauss-Chebyshev, Gauss-Lobatto rules. Adaptive Cavalieri-Simpson rule.

#### Bibliography

A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio. Matematica numerica, ed. Springer (collana UNITEXT)

#### Modules

Professor:
Boffi Daniele
Lessons:
48
ECTS:
6
SSD:
MAT/08

Professor:
Gardini Francesca
Lessons:
36
ECTS:
3
SSD:
MAT/08

Dipartimento di Matematica ''F. Casorati''

Università degli Studi di Pavia - Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602