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lecture 8

# lecture 8 - CE 30125 Lecture 8 LECTURE 8 INTERPOLATION...

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CE 30125 - Lecture 8 p. 8.1 LECTURE 8 INTERPOLATION USING CHEBYSHEV ROOTS • For both Lagrange and Newton interpolation through data points ( degree poly- nomial which fits through all data points). • Typically over a small interval, won’t change dramatically (although strictly speaking does depend on ) • We can not control the portion of since we don’t know and we can’t specify . N 1 + N th N 1 + ex () xx o 1 2 N N 1 + ! ---------------------------------------------------------------------------------- f N 1 + ξ = x o ξ x N << 1 N 1 + ! ------------------- i f N 1 + ξ i 0 = N = x o ξ x N f N 1 + ξ ξ x f N 1 + ξ f N 1 + x m where x m x o x N + 2 ----------------- = f N 1 + ξ f ξ

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CE 30125 - Lecture 8 p. 8.2 • The error is for the most part controlled by is small in the center of the interval , but large within the end zones. • For example for N=5 , we examine a plot of : • Our objective is to minimize by minimizing by selecting a “special” set of non-equispaced interpolating points xx i () i 1 = N i i 0 = N x o x N , [] i i 0 = 5 14 0 23 5 Error small Error large ex i i 1 = N
CE 30125 - Lecture 8 p. 8.3 Chebyshev Polynomials • Consider only the interval . (We will generalize to any interval at a later point.) • The Chebyshev polynomial is defined as: on • Note that the term restricts the range since is not defined. • Zero th Degree Chebyshev Polynomial: • First Degree Chebyshev Polynomial: 1 x 1 ≤≤ T j x () j 1 cos x cos 1 x 1 j 01 ,, = 1 cos x x 1 T o x 0 1 cos x cos = T o x 1 = T 1 x 1 1 cos x cos = T 1 x x =

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CE 30125 - Lecture 8 p. 8.4 • Second Degree Chebyshev Polynomial: • From the CRC Math Handbook: Double Angle Relation • Third Degree Chebyshev Polynomial: • From the CRC Math Handbook: Multiple Angle Relation T 2 x () 2 1 cos x cos = 2 α cos 2 2 cos α 1 = T 2 x 2 21 cos x cos 1 = T 2 x 2 1 cos x cos [] 1 cos x cos
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lecture 8 - CE 30125 Lecture 8 LECTURE 8 INTERPOLATION...

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