1 / 27

CHAPTER 8. Approximation Theory

CHAPTER 8. Approximation Theory. Dongshin Kim Computer Networks Research Lab. Dept. of Computer Science and Engineering Korea University dongshin@korea.ac.kr 9 november 2005. Contents. Discrete Least Squares Approximation Orthogonal Polynomials and Least Squares Approximation

willem
Download Presentation

CHAPTER 8. Approximation Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CHAPTER 8.Approximation Theory Dongshin Kim Computer Networks Research Lab. Dept. of Computer Science and Engineering Korea University dongshin@korea.ac.kr 9 november 2005.

  2. Contents • Discrete Least Squares Approximation • Orthogonal Polynomials and Least Squares Approximation • Chebyshev Polynomials and Economization of Power Series • Rational Function Approximation • Trigonometric Polynomial Approximation • Fast Fourier Transforms

  3. Discrete Least Squares Approximation • Finding best equation • minimize a0 and a1 • Another approach (absolute deviation): • minimize a0 and a1

  4. Discrete Least Squares Approximation • Least square

  5. Discrete Least Squares Approximation • Solution

  6. Orthogonal Polynomials and Least Squares Approximation • Polynomial Pn(x)

  7. Orthogonal Polynomials and Least Squares Approximation • Definition 8.1: linearly independent • If is a polynomial of degree j, for each j=0,1,…,n, then is linearly independent on any interval [a, b]

  8. Orthogonal Polynomials and Least Squares Approximation

  9. Orthogonal Polynomials and Least Squares Approximation • Gram-Schmidt process

  10. Orthogonal Polynomials and Least Squares Approximation

  11. Chebyshev Polynomials and Economization of Power Series • Chebyshev Polynomials • Orthogonal on (-1,1) with respect to the weight function

  12. Chebyshev Polynomials and Economization of Power Series

  13. Chebyshev Polynomials and Economization of Power Series • Approximating an arbitrary nth-degree polynomial

  14. Rational Function Approximation • Pade Rational Approximation

  15. Rational Function Approximation

  16. Result • n=5, m=0 • p=1.00000000 -1.00000000 0.50000000 -0.16666667 0.04166667 -0.00833333 • q=1.00000000 • n=4, m=1 • p=1.00000000 -0.80000000 0.30000000 -0.06666667 0.00833333 • q=1.00000000 0.20000000 • n=3, m=2 • p=1.00000000 -0.60000000 0.15000000 -0.01666667 • q=1.00000000 0.40000000 0.05000000

  17. Result • r(x) = p(x)/q(x) • n=5, m=0 • p(x)=1.00000000-1.00000000*x+0.50000000*x^2 -0.16666667*x^3+ 0.04166667*x^4 -0.00833333*x^5 • q(x)=1.00000000 • n=4, m=1 • p(x)=1.00000000-0.80000000*x+0.30000000*x^2 -0.06666667*x^3 + 0.00833333*x^4 • q(x)=1.00000000+0.20000000*x • n=3, m=2 • p(x)=1.00000000 -0.60000000*x+0.15000000*x^2 -0.01666667*x^3 • q(x)=1.00000000+0.40000000*x+0.05000000*x^2

  18. Graph

  19. Result [f(x)-r(x)] • n=5, m=0 • 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0003 0.0007 0.0012 • n=4, m=1 • 1.0e-003 * • 0 -0.0000 -0.0000 -0.0002 -0.0009 -0.0034 -0.0098 -0.0236 -0.0503 -0.0977 -0.1761 • n=3, m=2 • 1.0e-004 * • 0 0.0000 0.0001 0.0008 0.0041 0.0145 0.0401 0.0935 0.1930 0.3628 0.6335

  20. Rational Function Approximation • Chebyshev Rational Approximation

  21. Rational Function Approximation

  22. Result • n=5, m=0 • p=1.26606600 -1.13031800 0.27149500 -0.04433700 0.00547400 -0.00054300 • q=1.00000000 • n=4, m=1 • p=1.15394275 -0.85220885 0.15497369 -0.01686273 0.00102207 • q=1.00000000 0.19839240 • n=3, m=2 • p=1.05526480 -0.61301701 0.07747850 -0.00450556 • q=1.00000000 0.37833060 0.02221579

  23. Graph

  24. Result [f(x)-r(x)] • n=5, m=0 • 1.0e-004 * • -0.4500 -0.3517 -0.1315 0.1357 0.3529 0.4266 0.3010 -0.0020 -0.3362 -0.3792 0.4244 • n=4, m=1 • 1.0e-005 * • 0.8870 0.8399 0.5231 0.0633 -0.3753 -0.6349 -0.6114 -0.3045 0.1338 0.3510 -0.2506 • n=3, m=2 • 1.0e-005 * • -0.2137 0.2684 0.6582 0.8495 0.7804 0.4524 -0.0557 -0.5787 -0.8582 -0.5410 0.8206

  25. Result [f(x)-r(x)] • Pade • n=5, m=0 • 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0003 0.0007 0.0012 • n=4, m=1 • 1.0e-003 * • 0 -0.0000 -0.0000 -0.0002 -0.0009 -0.0034 -0.0098 -0.0236 -0.0503 -0.0977 -0.1761 • n=3, m=2 • 1.0e-004 * • 0 0.0000 0.0001 0.0008 0.0041 0.0145 0.0401 0.0935 0.1930 0.3628 0.6335 • Chebyshev • n=5, m=0 • 1.0e-004 * • -0.4500 -0.3517 -0.1315 0.1357 0.3529 0.4266 0.3010 -0.0020 -0.3362 -0.3792 0.4244 • n=4, m=1 • 1.0e-005 * • 0.8870 0.8399 0.5231 0.0633 -0.3753 -0.6349 -0.6114 -0.3045 0.1338 0.3510 -0.2506 • n=3, m=2 • 1.0e-005 * • -0.2137 0.2684 0.6582 0.8495 0.7804 0.4524 -0.0557 -0.5787 -0.8582 -0.5410 0.8206

  26. Trigonometric Polynomial Approximation • Trigonometric Polynomial • All linear combinations of the functions • Fourier series

More Related