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 Chebyshev Polynomials and Economization of Power Series Rational Function Approximation Trigonometric Polynomial Approximation Fast Fourier Transforms

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

Discrete Least Squares Approximation

Discrete Least Squares Approximation Solution

Orthogonal Polynomials and Least Squares Approximation Polynomial Pn(x)

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]

Orthogonal Polynomials and Least Squares Approximation

Orthogonal Polynomials and Least Squares Approximation Gram-Schmidt process

Orthogonal Polynomials and Least Squares Approximation

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

Chebyshev Polynomials and Economization of Power Series

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

Rational Function Approximation Pade Rational Approximation

Rational Function Approximation

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

Result r(x) = p(x)/q(x) n=5, m=0 n=4, m=1 n=3, m=2 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

Graph

Result [f(x)-r(x)] n=5, m=0 n=4, m=1 n=3, m=2 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

Rational Function Approximation Chebyshev Rational Approximation

Rational Function Approximation

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

Graph

Result [f(x)-r(x)] n=5, m=0 n=4, m=1 n=3, m=2 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 -0.2137 0.2684 0.6582 0.8495 0.7804 0.4524 -0.0557 -0.5787 -0.8582 -0.5410 0.8206

Result [f(x)-r(x)] Pade Chebyshev n=5, m=0 n=4, m=1 n=3, m=2 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 -0.4500 -0.3517 -0.1315 0.1357 0.3529 0.4266 0.3010 -0.0020 -0.3362 -0.3792 0.4244 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 -0.2137 0.2684 0.6582 0.8495 0.7804 0.4524 -0.0557 -0.5787 -0.8582 -0.5410 0.8206

Trigonometric Polynomial Approximation All linear combinations of the functions Fourier series