Linear Algebra, 1st semester 1st year, Differential Calculus, 1st semester 1st year, Integral Calculus, 2nd semester 1st year, Programming, 1st semester 1st year

Familiarizing with the basic concepts of numerical analysis: algorithms, stability, accuracy, and efficiency. Interpolating data with different techniques: Lagrange, Hermite, piecewise, splines. Calculating numerical approximations, choosing the most adequate algorithm for each application, in each of the following problems: quadrature and derivation, systems of linear and non-linear equations, linear least-squares. Programming the studied algorithms and use other ready-made algorithms, available in Python. Relating real problems and their mathematical models.

S03: Apply mathematical language and abstract-rigorous reasoning in the enunciation and demonstration of results in various areas of mathematics. S12: Develop numerical calculation techniques, selecting suitable algorithms and programming them on a computer to solve mathematical problems. S13: Formulate real-world problems by means of mathematical models for their subsequent analysis and resolution. S14: Apply appropriate analytical or numerical techniques to solve mathematical models associated with real-world problems and interpret the results obtained. C02: Design programs that solve mathematical problems, applying algorithmic procedures with special attention to performance. C08: Solve mathematical problems arising from new developments in computer science using advanced mathematical tools and techniques.

1. Introduction: errors, algorithms and estimates Sources of error, roundoff, truncation, propagation. Machine numbers, floating-point arithmetics. Taylor polynomials and error. Estimating and bounding errors. 2. Nonlinear equations and nonlinear systems Nonlinear equations: Mean-value theorem, number of roots in an interval. Bisection, Secant, Newton-Raphson. Fixed-point methods. Convergence order. Error analysis. Nonlinear systems. Accelerated, Taylor and interpolation methods. 3. Methods for linear systems of equations Linear systems, stability: condition number. Triangular systems. Gaussian elimination. Pivoting. Computing determinants and matrix inverses. Orthogonalization methods and improved methods. Least-squares problems. Regression. Normal equations and QR method. Overdetermined systems. Fast Fourier Transform. Applications. 4. Polynomial interpolation: Lagrange, Hermite, piecewise, splines Newton/Lagrange Interpolation, errors. Equispaced (or not) nodes. Runge's phenomenon. Hermite interpolation. Richardson's extrapolation. Splines. Natural cubic splines. 5. Numerical quadrature and differentiation Numerical differentiation: back/forward, central, general, higher order. Errors. Numerical Integration: Newton-Côtes formulae. Errors. Adaptive integration.

LEARNING ACTIVITIES AND METHODOLOGY THEORETICAL-PRACTICAL CLASSES. [43 hours with 100% classroom instruction, 1.67 ECTS] Knowledge and concepts students must acquire. Student receive course notes and will have basic reference texts to facilitate following the classes and carrying out follow up work. Students partake in exercises to resolve practical problems and participate in workshops and evaluation tests, all geared towards acquiring the necessary capabilities. TUTORING SESSIONS. [4 hours of tutoring with 100% on-site attendance, 0.15 ECTS] Individualized attendance (individual tutoring) or in-group (group tutoring) for students with a teacher. STUDENT INDIVIDUAL WORK OR GROUP WORK [98 hours with 0 % on-site, 3.72 ECTS] WORKSHOPS AND LABORATORY SESSIONS [8 hours with 100% on site, 0.3 ECTS] FINAL EXAM. [4 hours with 100% on site, 0.15 ECTS] Global assessment of knowledge, skills and capacities acquired throughout the course. METHODOLOGIES THEORY CLASS. Classroom presentations by the teacher with IT and audiovisual support in which the subject's main concepts are developed, while providing material and bibliography to complement student learning. PRACTICAL CLASS. Resolution of practical cases and problem, posed by the teacher, and carried out individually or in a group. TUTORING SESSIONS. Individualized attendance (individual tutoring sessions) or in-group (group tutoring sessions) for students with a teacher as tutor. LABORATORY PRACTICAL SESSIONS. Applied/experimental learning/teaching in workshops and laboratories under the tutor's supervision.