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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

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CHAPTER 8. Approximation Theory

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  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

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