37261 - Numerical Analysis

Learning outcomes

The student learns the numerical methods for differential problems. At the end of the course the student knows the numerical-mathematical aspects and the main algorithmic methodologies that deal with the numerical solution of differential problems of interest in Engineering.

Course contents

Prerequisites:

A prior knowledge and understanding of Geometry and Algebra, Mathematical Analysis and MATLAB programming is required. Moreover, a prior knowledge of the basic topics of Numerical Analysis is also needed.

Program:

1. Iterative methods for solving single nonlinear equations and systems of nonlinear equations

A review of root-finding methods for one equation in one variable: the bisection method, Newton's method, the secant method. Iterative methods for the numerical solution of systems of non-linear equations: Newton's method and Quasi-Newton methods.

2. Numerical Differentiation: derivative approximation by Finite Differences

Approximating the derivatives of a function by finite differences. The method of undetermined coefficients. Determining the step size that minimizes the total error associated with finite-difference derivative approximations. Increasing the accuracy of finite-difference derivative approximations by Richardson's extrapolation.

3. Numerical Integration by Gaussian Quadrature formulas

A short review of Newton-Cotes quadrature formulas and their drawbacks. Open and closed Gaussian quadrature formulas. Numerical algorithms for computing nodes and weights of Gauss-Legendre, Gauss-Lobatto and Gauss-Radau formulas.

4. Numerical methods for solving ODE-IVPs

Ordinary Differential Equations (ODEs) and Initial Value Problems (IVPs). Existence, uniqueness and stability of solutions. Numerical methods for ODE-IVPs: one-step methods (Runge-Kutta), adaptive Runge-Kutta methods, Adams multistep methods,  Predictor-Corrector methods and BDF methods. Accuracy and stability of numerical methods for ODE-IVPs. Stiff problems.

5. Numerical methods for solving ODE-BVPs

Two-point Boundary Value Problems (BVPs) involving second-order linear/non-linear ODEs. Shooting methods and finite difference methods.

Readings/Bibliography

It will be fundamental to use the notes taken during the lectures and the computer material made available at https://virtuale.unibo.it/.

For further study we recommend:

[1] J. Kiusalaas: Numerical Methods in Engineering with MATLAB, 3rd ed., Cambridge University Press, 2015.

[2] K. Atkinson, W. Han, D. Stewart: Numerical Solution of Ordinary Differential Equations, John Wiley and Sons, 2009.

[3] A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics, 2nd ed., Springer, 2007.

[4] R.J. LeVeque: Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.

[5] U.M. Ascher, L.R. Petzold: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, 1998.

 

To fill any knowledge gaps related to the prerequisites of Numerical Analysis and to the fundamentals of MATLAB programming it is advisable to consult:

[1] A. Quarteroni, F. Saleri, P. Gervasio: Scientific Computing with MATLAB and Octave (4th edition), Springer, 2014.

[2] C. Moler: Numerical Computing with MATLAB, SIAM, 2004.

[3] J. Stoer, R. Bulirsch: Introduction to Numerical Analysis (3rd ed.), Springer, 2002.

[4] C.T. Kelley: Iterative Methods for Linear and Nonlinear Equations, SIAM, 1995.

Teaching methods

The course is structured in lectures and exercises in the computer laboratory. More precisely, the lectures on the numerical methods for differential problems described by ordinary differential equations, are followed by laboratory exercises aimed at implementing these methods in MATLAB and developing an adequate sensitivity and awareness of their use.

In consideration of the type of activity and the teaching methods adopted, the attendance of this training activity requires the prior participation of all students in Modules 1 and 2 of Health and Safety training courses [https://elearning-sicurezza.unibo.it/?lang=en] in e-learning mode.

Assessment methods

The end-of-course exam aims to verify the achievement of the following educational objectives:

- knowledge of the numerical-mathematical aspects and of the main algorithmic methodologies that deal with the numerical solution of differential problems described by ordinary differential equations;

- ability to solve real problems of interest in engineering by using or developing numerical methods and writing the corresponding algorithms in MATLAB.

The end-of-course exam (the evaluation of which is in thirtieths) will take place in the computer laboratory. The student has 120 minutes to solve two exercises that include both the development of MATLAB codes for the numerical solution of differential problems, and the written answer to theoretical questions on the topics covered in the lessons.

During the exam, the use of support material such as textbooks, notes and computer supports is not allowed.

To access the exam it is necessary to show an identification document.

The dates of the exams can be consulted on the AlmaEsami web platform of the University of Bologna.

 

Clarifications on the recording of the exam grade for those who have the NUMERICAL ANALYSIS course (6 CFU) as a module of the integrated course of NUMERICAL AND MATHEMATICAL METHODS FOR ENGINEERING (12 CFU)

If the NUMERICAL ANALYSIS course (6 CFU) is one of the two modules that, together with MATHEMATICAL METHODS FOR ENGINEERING (6 CFU), constitutes the integrated course of NUMERICAL AND MATHEMATICAL METHODS FOR ENGINEERING (12 CFU), the grade that will be recorded will be calculated with the arithmetic average of the single grades that the student has obtained in the two modules. It should be noted that the result of the average will be rounded to the nearest integer. Only if the resulting average is exactly equidistant between two integers, the grade will be obtained by rounding up to the next highest integer. Lastly, in order to obtain the "30 cum laude" final evaluation, the student must be in one of the two following cases:

- receiving "30 cum laude" in both modules;

- obtaining "30 cum laude" in one module and 30 in the other one.

Teaching tools

The course includes a laboratory activity in which the MATLAB software will be used. The corresponding teaching material will be made available to the student in electronic format and will be downloadable at https://virtuale.unibo.it/.

Office hours

See the website of Lucia Romani