Learning objectives
A sound balancing of theoretical analysis, description of algorithms and discussion of applications is the primary concern.
Course unit content
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<p><strong>Approximation of Functions and Data. <br />
</strong>Interpolation by linear and cubic splines. Convergence results. Cardinal splines and B-Splines. Trigonometric interpolation. Orthogonal polynomials and least-squares approximation. Least-squares fitting. <br />
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<strong>Numerical Integration. <br />
</strong>Gaussian quadrature on bounded and unbounded intervals. Improper integrals. Adaptive quadrature. Error estimates. Multiple integrals. <br />
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<strong>Numerical linear algebra</strong>. <br />
QR-decomposition. Least-squares solution of overdetermined linear systems. Basic iterative methods. Jacobi, Gauss-Seidel and SOR methods. Convergence results. Stop tests. Richardson iterative method. <br />
Eigenvalue problem. Localization of eigenvalues. Stability analysis. The power method. The inverse power method. Eigenvalues and eigenvectors of a tridiagonal matrix. Reduction of a general matrix to Hessemberg form. Householder transformations. The QR algorithm for real Hessemberg matrices. The LR algorithm. <br />
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<strong>Solution of Nonlinear Equations. <br />
</strong>Secant method, False Position method. Convergence results. Fixed-point methods. Rate of convergence. Zeros of polynomials. The Newton-Horner method; the Bairstow method. Newton’s method in several variables. <br />
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<strong>Numerical Solution of Ordinary Differential Equations. <br />
</strong>Linear multistep methods. Adams methods. Predictor-corrector methods. Order, convergence and stability for multistep methods. Boundary valure problems : shooting method, finite-difference method, collocation method, Galerkin method. <br />
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Bibliography
<p>NALDI G., PARESCHI L:, RUSSO G., Introduzione al calcolo scientifico - Metodi e applicazioni con Matlab, Mc Graw-Hill, 2001. <br />
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QUARTERONI A., SACCO R., SALERI F., Matematica numerica, SPRINGER, 2008.</p>
<p>MONEGATO G., Fondamenti di Calcolo Numerico, CLUT Editrice, 1998 <br />
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