Читать книгу Fundamentals of Numerical Mathematics for Physicists and Engineers - Alvaro Meseguer - Страница 4

1 Chapter 1Figure 1.1 (a) A simple exploration of reveals a change of sign within the...Figure 1.2 (a) Analysis of the order of convergence of the Newton's method: ...Figure 1.3 (a) Graph intercepting ordinates and at abscissas and , ...Figure 1.4 (a) Convergence history of Newton's and secant methods when the s...

2 Chapter 2Figure 2.1 Equispaced grid with .Figure 2.2 (a) Trigonometric function (black solid curve) and the interpol...Figure 2.3 Interpolation of the function (solid black curve) using equispa...Figure 2.4 Lebesgue functions associated with sets of equispaced nodes (2....Figure 2.5 Interpolation of with equispaced nodes. (a) (solid black) and...Figure 2.6 Chebyshev nodes (hollow circles) as the horizontal projection of ...Figure 2.7 Lebesgue functions and constants associated with the Chebyshev no...Figure 2.8 Interpolation of the function (solid black curve) using Chebysh...Figure 2.9 Interpolation of the function using Chebyshev nodes. (a) Pointw...

3 Chapter 3Figure 3.1 Numerical differentiation: starting from the known values of a ...Figure 3.2 From top to bottom, finite difference formulas (3.3), (3.4), (3.5...Figure 3.3 Centered difference formula (3.5) applied on a set of equidistant...

4 Chapter 4Figure 4.1 Geometrical representation of simple interpolatory quadratures fo...Figure 4.2 Areas associated with the composite trapezoidal rule (hatched reg...Figure 4.3 Areas associated with the composite Simpson's rule (hatched regio...Figure 4.4 Absolute quadrature errors of composite trapezoidal and Simpson r...Figure 4.5 Chebyshev polynomials (solid black), (dashed black), (solid...Figure 4.6 Absolute quadrature error in the Clenshaw–Curtis quadrature appro...Figure 4.7 Computation of the length of the ellipse . (a) We compute the ...Figure 4.8 Semi‐logarithmic plot of the absolute quadrature errors correspon...Figure 4.9 Cotangent transformation applied on the abscissas (4.89) for Figure 4.10 Absolute quadrature error of cotangent quadrature formula (4.8...Figure 4.11 Hyperbolic tangent transformation applied on the abscissas ....Figure 4.12 Absolute quadrature errors of the approximation of integral (4...

5 Chapter 5Figure 5.1 Linear fitting. The gray disks are the experimental observations....Figure 5.2 The resulting vector is the reflection of across the line ....Figure 5.3 (a) Properties (5.137). (b) There are two possible reflectors (ac...Figure 5.4 QR‐factorization of a non‐square matrix .Figure 5.5 Classical Gram–Schmidt (CGS) algorithm.Figure 5.6 CGS and QR‐factorization equivalence.Figure 5.7 Near parallelism between and results in a vector with very ...

6 Chapter 6Figure 6.1 Geometrical interpretation of a system of two nonlinear equations...Figure 6.2 (a) Solutions of system (6.21) showing the first three iterates o...Figure 6.3 Parameter‐dependent solution branches of (6.25).Figure 6.4 (a) Rotating pendulum. (b) Trivial solution branch and new solu...Figure 6.5 Continuation methods. (a) Natural continuation fails in the prese...Figure 6.6 Level curve of Himmelblau's function (6.46). (a) Starting the s...

7 Chapter 7Figure 7.1 DFT of . (a) Fourier coefficients provided by Code 22. (b) Dis...Figure 7.2 Sampling process. (a) The ordinates (gray circles) are the samp...Figure 7.3 (a) The signal is sampled using measurements every time uni...Figure 7.4 Fourier differentiation of function . (a) Sampled function at ...Figure 7.5 Exponential convergence of Fourier differentiation of function ....Figure 7.6 Differentiation of function in Fourier space, following Figure ...

8 Chapter 8Figure 8.1 Numerical solution of (8.41) provided by the approximated linear ...Figure 8.2 Numerical solution of (8.42). (a) Graphical representation of the...Figure 8.3 Solutions of BVP (8.47). (a) First solution approximation (whit...Figure 8.4 Eigenvalues and eigenfunctions of Mathieu's equation (8.58). (a) ...Figure 8.5 Eigenvalues and eigenfunctions of Schrödinger's equation for the ...Figure 8.6 Discretization of (8.76) on an equally spaced time grid. Only i...Figure 8.7 Accuracy of Runge–Kutta methods RK1, RK2 and RK4. (a) Time integr...Figure 8.8 Adams method. (a) Previously computed ‐steps . (b) Evaluation o...Figure 8.9 A BDF method is based on differentiating the interpolant (black...Figure 8.10 Absolute global error at of linear multistep formulas (LMSFs...Figure 8.11 Numerical solution of using explicit Euler method (8.83). (a) Figure 8.12 Numerical solution of using implicit Euler method BDF1 with ....Figure 8.13 (a) Stability region of the explicit Euler method (8.158), in gr...Figure 8.14 (a) Region of absolute stability (light gray region) of Heun's R...Figure 8.15 (a) Region of absolute stability (in gray) of ‐step Adams–Bashf...Figure 8.16 (a) Instability regions of ‐step BDF2 (light gray) and ‐step B...Figure 8.17 Solution to the IVP (8.224) starting from (solid black). If Figure 8.18 Nonlinear system (8.236) for and . (a) Phase portrait showing...Figure 8.19 Numerical integration of nonlinear system (8.236) for and . (...