The course is addressed to second year students. It is an introduction to numerical analysis and covers numerical techniques and algorithms for the solution of mathematical problems which are encountered in physics.
Numeral systems. IEEE Standards for integer and floating point numbers.Computer representation of numbers.
Numerical solution of a nonlimear equation. Definitions,useful theorems. Methods: bisection, regula falsi, secant, Muller,fixed point, Ηοuseholder (Newton-Raphson, Halley).
Systems of linear equations. Direct methods (Gauss elimination,Gauss-Jordan, LU).Iterative methods (Gauss-Seidel, Jacobi, SOR). Other methods.Applications: calculation of the determinant of a matrix,inverse matrix, matrix eigenvalues and eigenvectors.Numerical solution of systems of nonlinear equations.
Function/set of points approximation: Interpolation of polynomial, rational, piecewise polynomial, spline. Runge phenomenon.Numerical differentiation.
Least squares approximation: line, polynomial, logarithmic and exponential. Correlation coefficient.
Numerical quadrature. Trapezoid and Simpson rules. Newton-Cotesformulas. Gauss quadrature methods (Legendre, Hermite, Laguerre,Chebyshev). Clenshaw–Curtis method. Other methods.
Numerical solution of initial value problems of first orderordinary differential equations (ODE).Methods: Euler (explicit/implicit), Taylor, Runge-Kutta 2ndand 4th orders. Systems of ODEs. Higher order ODEs.
Other topics (FFT, optimization, etc).
Grammatikakis M., Kopidakis G., Papadakis N., Stamatiadis S.- Introduction to Numerical Analysis, Lecture and Lab Notes (in Greek) http://www.edu.physics.uoc.gr/~tety213/notes.pdf
Forsythe G.E., Malcom M.A., Moler C.B.- Computer Methods for Mathematical Computations.
Akrivis G.D., Dougalis V.A.- Introduction to Numerical Analysis (in Greek)
