AE4B01NUM NUMERICAL ANALYSIS

Annotation: The course introduces to basic numerical methods of interpolation and approximation of functions, numerical differentiation and integration, solution of transcendent and ordinary differential equations and systems of linear equations. Emphasis is put on estimation of errors, practical skills with the methods and demonstration of their properties using Maple, and computer graphics. All topics are supported by worksheets for verification of knowledge from lectures; students apply them during seminars and work on exercises for the assessment. Preliminary knowledge of Maple is not supposed, the necessary minimum is taught at the first two seminars.

Curricula of lectures:

Overview of the subject of Numerical Analysis.

• Approximation of functions, polynomial interpolation. Errors of polynomial interpolation and their estimation. Hermite interpolating polynomial. Splines.
• [NR 3.0, 3.1, 3.3, 3.5] [KJD 2.1-2.4]
• Least squares approximation.
• [NR 15.0, 15.1, 15.4]
• Basic root-finding methods. Iteration method, fixed point theorem. Basic theorem of algebra, root separation and finding roots of polynomials.
• [NR 9.0, 9.1, 9.2, 9.4, 9.5] [KJD 3.1]
• Solution of systems of linear equations.
• [NR 2.0-5] [KJD 1.1-1.2.2, 1.3.1]
• Numerical integration (quadrature). [KJD 2.6.1]
• Error estimates and stepsize control, Richardson's extrapolation in integration. [KJD 2.5.3]
• Gaussian and Romberg integration.
• [NR 4.0-5]
• One-step methods of solution of ODE's. Multistep methods of solution of ODE's. Richardson's extrapolation in ODE's.
• [NR 16.0, 16.1, 16.3., 16.4] [KJD 4.1-4.4]

References:

• [NR] Press, W. H., Flannery, B. P., Teukolsky, S. A., Vetterling, W. T.: Numerical Recipes (The Art of Scientific Computing). 2nd ed., Cambridge University Press, Cambridge, 1992.
• [ACP] Knuth, D. E.: The Art of Computer Programming. Addison Wesley, Boston, 1997.
• [KJD] Kubíček, M., Janovská, D., Dubcová, M.: Numerical Methods and Algorithms. Institute of Chemical Technology, Prague, 2005.

Lectures:

• Approximation of functions
• Solving nonlinear equations
• Solving systems of linear equations
• Solving differential equations

Instructions for seminars:

• List of skills in Maple, required after the first two weeks of seminars
  • Symbolical and numerical computation, control of precision
  • Basic constants: Pi, exp(1), I
  • Conditional statements, piecewise defined functions, cycles
  • Chains, names of variables
  • Unassignment of variables
  • Rules of evaluation, delayed evaluation (f('variable'))
  • Data structures: sequences, lists, sets, vectors, matrices
  • Solving equations symbolically and numerically (solve, fsolve)
  • Limits
  • The difference between a function and an expression, how to evaluate, integrate and differentiate both of them
  • Plotting, basic options
• Interpolation
• Least squares approximation
• Solving nonlinear equations
• The instructions for seminars on solving systems of linear equations are included in the Maple document numlinrovE.mw
• Numerical integration
• Solving differential equations

Programs in Maple:

• Interpolation, least squares approximation
• Solving nonlinear equations, its Czech update
• Iterative methods of solving systems of linear equations
• Numerical integration
• Solving differential equations