LEARNING OUTCOMES

Acquiring the basic notions of linear algebra (vectors, matrices, linear transformations and eigenvalues) and of numerical analysis (complexity and error). Knowing the main computational methods for solving numerical linear algebra problems and some approximation problems.

AIMS AND LEARNING OUTCOMES

At the end of the course, the student will be able to:

•      Know the fundamentals of numerical computation and know how to evaluate the conditioning of simple mathematical problems and computational cost and stability for some basic algorithms, in particular in the case of linear systems solution.
•      Apply matrix theory and vector calculus to numerical analysis problems.
•      Understand the fundamental relationships between linear algebra and geometry, know the tool of orthogonal matrices and how to use them for reduction algorithms, understand the concept of eigenvalues and know how to compute them for small matrices.
•      Understand the concept of approximation in its various forms, know some techniques and how to solve linear least squares problems.
•      Implement some numerical algorithms on the computer and evaluate the reliability of the results.

PREREQUISITES

• Basics of algebraic structures
• Differential and integral calculus
• Programming in C or C++

TEACHING METHODS

Traditional.

Lectures are mainly given in classroom (apart pandemics restrictions), except for 2 lab sessions in the official timetable.

In addition, in the second half of semester 2 hours a week are scheduled outside the official timetable, under the assistance of a tutor.

SYLLABUS/CONTENT

• Error analysis 
  • Floating-point numbers and machine precision.
  • Inerent error. Estimate for rational functions.
  • Algorithmic error.
  • Total error.
• Basics of linear algebra and solution of nonsingular linear systems 
  • Matrix operations and inversion.
  • Solution of linear systems by Gaussian elimination.
  • Determinants and rank of matrices. Theorems of Laplace, Cramer and Rouché-Capelli.
  • Conditioning of matrices.
  • Complexity and algorithmic error for the solution of linear systems.
• Other topics in linear algebra: geometric interpretation of vectors and matrices
  • Scalar product and orthonormal bases.
  • Matrices as geometric linear transformations.
  • Null space, range and rank.
  • Orthogonal matrices: rotations, reflections, QR factorization.
• Approximated solution of linear systems in the least-squares sense
  • Geometric formulation of the problem.
  • Normal equations.
  • Solution through orthogonalization.
• Interpolation by spline functions 
  • Definition of interpolating spline.
  • Computational procedure.
  • Survey of mathematical and numerical properties.
• Other topics in linear algebra: eigenvalues
  • Eigenvalues, eigenvectors, eigenspaces.
  • Characteristic polynomial.
  • Similarity relations e diagonalization.
  • Applications.
• SVD and applications to least-squares
  • Singular values decomposition (SVD) and relations with eigenvalues.
  • Geometric properties of SVD and numerical rank.
  • Generalized inverse and conditioning.
  • Solution of the least-squares problem via SVD.
  • Application to discrete data approximation (smoothing).
• Numerical treatment of eigenvalues
  • Numerical properties: conditioning and localization.
  • Iterative power method and variants.
  • Other numerical methods: similarity reduction to a simplified form, QR method.

Computer experiences in C and Matlab languages are planned.

RECOMMENDED READING/BIBLIOGRAPHY

For the parts of the program concerning linear algebra basics, any classic textbook of linear algebra and geometry can help; for instance,

Serge Lang, Linear Algebra, Third Edition. Springer-Verlag New York, 1987.

Concerning the numerical analysis content, the use of lesson afternotes is recommended. Also available on Aulaweb are the notes of the course (in italian) taken by student Stefano Sabatini in the academic year 2010-11 and supervised by the teacher. Common textbooks are generally oversized with respect to the course. Just for reference, we suggest

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis. Springer-Verlag New York, 2002.