Syllabus
1. Real data in real computers: Storing integer and real numbers (format IEEE). Errors: classification, sources and spreading.
2. Introduction to Fortran 95: Syntactical elements, specification statements, commands and constructs. Arrays, array sections, array-valued functions. Program units and structuring source codes. Parallelization. Examples: tabulating Earth models.
3. Libraries of numerical methods: Numerical Recipes, LAPACK, MKL, IMSL, NAG. Orientation in libraries, calling library routines. Examples: spherical Bessel functions, algebraic matrix operations.
4. Mini-algorithms: Horner scheme. Recurrence relations and their properties. Difference schemes. Numerical derivatives. FFT. Random numbers. Searching, sorting. What not to compute. Examples: Legendre polynomials and functions.
5. Systems of linear algebraic equations. Matrix conditionality. Direct methods - Gauss elimination and factorization methods (LU factorization). Methods for systems with tridiagonal, band diagonal matrices and special matrices. Iterative methods, conjugate gradient method. Overdetermined and underdetermined systems, singular value decomposition. Matrix eigenvalues and eigenvectors - real symmetric vs. nonsymmetric matrices.
6. Approximation, basic applications: Interpolation of functions and derivatives (polynomial and rational interpolation, splines). Method of least squares. Chebyshev approximation, economization. Solution of nonlinear equations (classic methods, Newton method, combined methods). Examples: Fornberg schemes.
7. Numerical integration: Newton-Cotes formulas. Romberg integration. Gaussian quadratures. Examples: multiplication of spherical harmonic functions.
8. Systems of nonlinear algebraic equations: Linearization (Newton method). Minimalization (simplex and Powell's method, conjugate gradient methods and variable metric methods).
9. Ordinary differential equations: Initial value problems: Properties of numerical solution (local and global accuracy, convergence, stability, stiff systems). Properties of explicit and implicit schemes (Euler scheme). Runge-Kutta methods (of 2nd and 4th order, variants with adaptive stepping and for stiff systems). Extrapolation methods. Multistep methods. Boundary value problems: reduction to initial value problems, shooting methods, finite difference method, variational methods. Systems of differential and algebraic equations. Examples: Adams-Williamson equation, free oscillations of the Earth.
10. Partial differential equations: Discretization, difference schemes, properties, classification of methods. Finite difference method (difference equations, boundary conditions). Semi-discrete methods (method of lines, Rothe's method). Examples: Laplace equation, heat convection equation.
Annotation
Course of numerical methods with the emphasis on the implementation in Fortran. From numerical libraries through standard methods of algebra and analysis to solution of ordinary and partial differential equations.
Less theory, more practice. Examples of geophysical applications.