Numerische Mathematik für Informatiker

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The content covers a comprehensive range of topics in numerical analysis and computational mathematics. It begins with an introduction to error analysis, discussing types of errors, error propagation, stability, and rounding errors in floating-point arithmetic. The exploration continues with polynomials and rational functions, emphasizing polynomial evaluation, numerical stability, and linear recursions. Interpolation and quadrature are examined next, detailing algebraic interpolation, Newton's representation of interpolation polynomials, and the integration of divided differences with B-splines. The section also addresses interpolation and quadrature errors, including Gaussian quadrature formulas. The text then delves into splines and graphics, covering mathematical filters, Bernstein polynomials, Bézier representations, piecewise polynomial functions, and cubic B-splines, highlighting their minimal curvature properties and applications in curve generation. Further topics include periodicity and the fast Fourier transform (FFT), with discussions on exponential and trigonometric functions, trigonometric interpolation, and the FFT algorithm's efficiency in multiplying large numbers. Approximations are also explored, focusing on normed vector spaces, best approximations, and Fourier series development. The book addresses elimination and linear systems, detailing Gaussian elimination, Cholesky methods, and fast matrix algorit