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# Numerical Calculus | LEB

 Course: Biomedical Engineering Curricular Unit (UC) Numerical Calculus Mandatory x Optional Scientific Area MAT Category

Course category: B - Basic; C - Core Engineering; E - Specialization; P - Complementary.

 Year: 2nd Semester: 1st ECTS: 5 Total Hours: 140 Contact Hours T: 30 TP: 30 PL: S: OT:3 Professor in charge Sérgio Lopes

T - Lectures; TP - Theory and practice; PL - Lab Work; S - Seminar; OT - Tutorial Guidance.

• Learning outcomes of the curricular unit

After the student receives approval on the curricular unit, he should be able to:

Look at examples involving propagation of errors that occur in applications of numerical techniques.

Understand approximation techniques; explain how, why and when it is expected they are accurate.

Identify typical problems that require the use of numerical techniques in order to obtain its solution.

Implement computationally the numerical methods that have been studied.

Develop structural thinking and demonstrate analytical and critical capacity solving engineering problems.

• Syllabus

Introduction. The importance of Numerical Calculus in Engineering.

Introduction to the theory of errors. Majorants of errors. Fundamental formula of error calculation. Reference to the intervalar analysis.

Least square method. Discret and continuous cases.

Polynomial interpolation. Lagrange interpolating polynomial. Interpolating error. Inverse interpolation.

Numerical integration. Closed Newton-Cotes formulas (simple and composite rules).

Numerical solution of nonlinear equations. Bisection method and Newton-Raphson method.

Numerical integration of ordinary differential equations. First order ordinary differential equations (Euler method; Heun method; Runge-Kutta method). First order systems of ordinary differential equatons (Runge-Kutta method).

• Demonstration of the syllabus coherence with the curricular unit's objectives

The goals are met with the presentation of the chapters of the syllabus, in which analysis, algebra and deductive reasoning skills are widely developed.

In addition to the theory studied in each chapter, the systematic use of problems that illustrate the different given concepts, yields increase of motivation, efficiency and spectrum of learning by the students. In particular, the concret applications enable:

to convey the fact that the concepts of calculus constitute an indispensable tool in the study of engineering;

to practice the mathematical formulation of problems, their solution and criticism of the obtained results;

to help students to recognize the concepts and techniques studied when they appear in the study of other courses of their academic trajectory.

• Teaching methodologies

Teaching methodologies:

Theoretical lectures, where the fundamental concepts and definitions are presented in a clear way using the teaching supporting materials available.

Theoretical-pratical classes,  where exercices that illustrate the theoretical concepts are solved. Exercises sheets are available for an effective monitoring and strengthen of the knowledge presented.

Assessment: two alternative components - continuous assessment or exam assessment.

Continuous evaluation:

One global test (TG) (80% of the final classification)

Practical computational assignment (TP) (20% of the final classification)

NF = 0,8 TG + 0,2 TP;

NF >= 10

Final exam evaluation:

One final written examination (EF). The student is approved with a final grade of at least 10 values.

NF = EF >= 10

Rounded to units. By defect, beneath five tenths, per excess, from five tenths.

• Demonstration of the coherence between the teaching methodologies and the learning outcomes

The theoretical lectures are essential to a correct and comprehensive coverage of all topcis of the syllabus, while the in-class solution of exercices allows the illustration of the practical application of the concepts and the tools studied, enhancing the theoretical knowledge.

By their organization, contents and diversity in the degree of difficulty, the exercises sheets allow students to closely monitor all topics of the syllabus and are the main tool regarding

individual study. The exercises that constitute them are suited for the development of calculus skills and deductive reasoning.

Since the success in mathematics is not compatible with pre-assessment study on its own, it is essential to implement processes to avoid this inclination.

• Main Bibliography

Santos, F. C., “Fundamentos de Análise Numérica”, Edições Sílabo, 2002.

Gilat, A., Subramaniam, V., “Métodos Numéricos para Engenheiros e Cientistas”, 2008.

Burden, R. L., Faires, J. D., “Numerical Analysis”, Books/Cole, 1997.

Chapra, S.C., Canale, R.P. “Numerical Methods for Engineers”, McGraw-Hill, 2006.

Kharab, A. Guenther, R. B., “An introduction to numerical methods: A Matlab Approach”, Chapman & Hall /CRC, 2002.