Numerical Algorithms in Computer Science

Aim

In computational mathematics, numerical methods are developed and analyzed for the solution of commonly occurring mathematical problems, mainly from technologyand natural sciences. Important characteristics of the methods are robustness, accuracy and efficiency. As the methods are intended to be implemented on a computer,it is also important to understand how a computer processes numerical information.

After completing the course, the student should be able to:

• M1: explain basic concepts in computational science and know how a computer stores numbers and with what precision calculation operations can be performed.
• M2: use a selection of numerical methods to solve mathematical problems from applications using a pocket calculator or a computer.
• M3: discuss possible sources of error in numerical calculations and estimate the accuracy of calculated results.
• M4: use standard software to solve practical problems from applications.

Content

• Error analysis and number representation: IEEE standard for number representation in computers. The machine precision. Analysis of calculation errors. Cancellation.Error propagation and Error sources.
• Linear algebra: Systems of linear equations. LU factorization. The condition number and error analysis. The least squares problem. The normal equations.Orthogonal bases. Projections. The QR factorization.
• Nonlinear Equations: Interval Bisection. Fixed point iteration. Order of Convergence. Newton-Raphson's method. Error estimation.
• Interpolation: Polynomial and Spline interpolation. B-splines. Description of curves and surfaces with Bezier polynomials.
• Differentiation, Integration and Extrapolation: Difference approximation, order of accuracy, the trapezoidal rule and Simpon's rule. Estimation ofTruncation Errors.
• Ordinary differential equations: Runge-Kutta methods, local and global truncation error, stability and convergence. Implicit methods.