An Introduction to Numerical Analysis

Transcription

1 Weimin Han AMCS & Dept of Math University of Iowa MATH:38

2 Example 1 Question: What is the area of the region between y = e x 2 and the x-axis for x 1? Answer: Z 1 e x 2 dx = : Figure: The function y = e x 2 for x 2 [ 1; 2]

3 Example 1 (cont.) Running Matlab: >> quad(inline('exp(-x.^2)'),,1,1e-5) ans =.7468 >> format long e >> ans ans = e-1 Numerical values: (error tolerance 1 5 ) (error tolerance 1 1 ) (error tolerance 1 16 ) What are behind the Matlab built-in function quad?

4 Example 2 Fresnel spiral. The curve dened by the parametric equations x(t) = y(t) = Z t Z t cos(s 2 ) ds ; sin(s 2 ) ds : Figure: Fresnel spiral

5 Example 3 Question: What are the roots of x 1 x 1:1 =? >> p=[ ]; >> z = roots(p) z = e e e-1i e e-1i e e+i e e+i e e-1i e e-1i e e-1i e e-1i e-1 >> format short e >> polyval(p,z) ans = e e e-14i

6 Example 4 Sensitivity of roots w.r.t. perturbation Theorem. Roots of a polynomial are continuous functions of the coecients. However, for some polynomials, their roots are very sensitive w.r.t. small changes in the coecients. Let p(x) = (x 1) (x 2) (x 3) (x 4) (x 5) (x 6) (x 7) = x 7 28x x 5 196x x x x 54

7 >>roots([1,-28,322,-196,6769,-13132,1368,-54]) ans = 7.446e e e e e e+ 1.18e+ >>roots([1, ,322,-196,6769,-13132,1368,-54]) ans = e e-1i e e-1i e e-1i e e-1i e e e-1

8 Example 5 A fractal is an object that displays self-similarity under magnication and can be constructed using a simple motif (an image repeated on ever-reduced scales). Fractals have generated a great deal of interest since the advent of the computer. Many shops now sell colorful posters and T-shirts displaying fractals. Many objects in nature display self-similarity at dierent scales: ferns, trees, clouds, etc. Helge Von Koch rst imagined the Koch curve in 194. It is constructed by replacing a unit line segment with a motif consisting of four line segments each of length 1=3. At the kth stage, there are 4 k line segments each of length 3 k, and the total length is (4=3) k. So the mathematical Koch curve is innitely long.

9 Example 5 (cont.) stage stage stage stage stage stage Figure: Construction of the Koch curve up to stage 5

10 Denition Numerical analysis is the area of mathematics and computer science that creates, analyzes, and implements algorithms for numerically solving problems of continuous mathematics. (Ken Atkinson) Numerical analysis is the study of algorithms for the problems of continuous mathematics. (L.N. Trefethen) Numerical analysis is the study of approximate solutions to mathematical problems, taking into account the extent of possible errors. Numerical analysis is the study of methods of approximation and their accuracy, etc. Numerical analysis is the study of quantitative approximations to the solutions of mathematical problems including consideration of the errors and bounds to the errors involved.

11 Closely related names Computational Mathematics (Computational Physics, Computational Chemistry, Computational Biology, Computational Mechanics, Computational Engineering, Computational Sciences, Computational Topology (?)) Scientic Computing

12 General Areas of NA Approximation Theory Fact: a computer can only do basic arithmetic operations (,, ) and comparison operations. Moreover, a computer can represent a number with a nite number of (binary) digits; i.e., most numbers (including.1!) used in a computer are only approximations (though very accurate ones). Evaluating a general function is usually through approximations by polynomials or rational functions. Topics include: interpolation by polynomials, rational functions, trigonometric polynomials, splines; least-squares approximation (when the number of data points is large); best approximations with respect to various norms and various approximation function sets; numerical integrations, numerical dierentiations; Fast Fourier transform; wavelets;

13 General Areas of NA (cont.) Numerical Linear/Nonlinear Algebra, Optimization Linear system Ax = b with matrix A of the order n; n = O(1 6 ) O(1 9 ) not uncommon in engineering applications. Direct methods: Gaussian elimination and its variants. Iterative methods: Jacobi, GS, SOR, CG,. Eigenvalue problem Ax = x; need some or all of the eigenvalues of a large size matrix A. Nonlinear system of the form f (x) = for some f : R n! R d. Newton's method: choose initial guess x (), and for k = ; 1;, compute f (x (k) ) h (k) = f (x (k) ); x (k+1) = x (k) + h (k) : Optimization: minff (x) j x 2 R n g. Constrained: if is described by some equalities and/or inequalities. Unconstrained: if is a \natural" domain.

14 General Areas of NA (cont.) Numerical Dierential and Integral Equations This is a huge area. Numerical ODE (IVP): Runge-Kutta methods, multistep methods. Numerical PDE: Finite dierence method, nite element methods, spectral method, nite volume method,. Numerical integral equations, boundary element method.