AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to introduce fundamental concepts of numerical analysis (complexity, errors);  we present the main computational methods for solving the most relevant problems of numerical linear algebra and some interpolation and minimization problems.

SYLLABUS/CONTENT

• Error analysis 
  • Floating-point numbers and machine precision.
  • Inerent error. Estimate for rational functions.
  • Algorithmic error.
  • Total error.
• Solution of nonsingular linear systems 
  • Numerical solution of linear systems (direct method of Gaussian elimination).
  • 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.

TEACHERS AND EXAM BOARD

Exam Board

FABIO DI BENEDETTO (President)

FEDERICO BENVENUTO

CLAUDIO ESTATICO

CLAUDIA FASSINO

EXAMS

Exam schedule