Course Details for A.Y. 2013/2014

Course Objectives

Basic notions of functions of complex variables, Laplace and Fourier transforms and their applications. Basic notions of numerical analysis, numerical methods for linear systems and for Cauchy problems for ordinary differential equations. Basic MatLab programming.

Course Content

• Elements of complex analysis. The field of complex numbers. Functions of complex variable. Holomorphic functions. Line integrals. Primitives of functions of complex variable. Power series.  Analyticity of holomorphic functions. Laurent's series. Zeroes of holomorphic functions. Isolated singularity for  holomorphic functions. Residues. Residues' Theorem and its applications. 
• Fourier Transform. Definition. Properties. Transform of convolutions.  Applications of the Fourier Transform.
• Laplace Transform. Functions which are Laplace-transformable. Regularity and properties of Laplace transform. Convolutions theorem and its consequences. The problem of the Laplace antitransform. Applications of the Laplace Transform. 
• Computer representation of numbers: Floating point representation. Accuracy of floating point representation. Error analysis: Chopping and rounding. Loss of significance errors. Error and relative error. Condition number of a problem and stability of a numerical algorithm. 
• Linear systems. Complements of linear Algebra. Vector and matrices' norms.  Condition number of a linear system. Direct methods to solve a linear system: Gauss-naive, Gauss with pivoting. Factorization of a matrix. Iterative methods to solve a linear system: generalities. Convergence condition for an iterative method. Speed of convergence,  stopping criteria.  Jacobi, Gauss-Seidel, JOR, and SOR methods.
• Cauchy problems for ordinary differential equations. Generalities. Transformation of a scalar Cauchy problem of order n in a first order vectorial problem. Explicit and implicit one-step methods;  fixed stepsize algorithms. Local truncation and global errors. Analysis of the local unitary truncation error. Explicit Runge-Kutta methods with r stages. Convergence of the Euler method and of the one-step methods.

Prerequisites and Learning Activities

Functions of two real variables, ordinary differential equations; linear algebra, linear systems.

Assessment Methods and Criteria

Written and oral exam and laboratory practical exam

Textbooks

• A. Quarteroni, Elementi di Calcolo Numerico , Progetto Leonardo, Bologna.    
• A. Quarteroni, R. Sacco, F. Saleri, Esercizi di Calcolo Numerico risolti con MATLAB , Progetto Leonardo, Bologna.    
• W.J. Palm III, MATLAB 6 per l'Ingegneria e le Scienze , Mc. Graw-Hill.    
• G. Di Fazio, M. Frasca, Metodi Matematici per l'Ingegneria , Monduzzi.    
• M. Codegone, Metodi Matematici per l’Ingegneria , Zanichelli.    
• F. Tomarelli, Metodi Matematici per l’Ingegneria (esercizi) , Città Studi Edizioni.    

Notes

• Further available teaching material: E. Santi. Appunti delle lezioni di Analisi Numerica  http://www.mathmods.eu/resources/downloads/viewcategory/17-lecture-notes-appunti [for Complex variables - see Analisi 3]  http://ing.univaq.it/calcolnu/areastud.html [to download: slides for Matlab and exercises about numerical analysis; functions for numerical methods;  exams exercises]

Course page updates

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