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# set11 - \$ Set 11 Orthogonality and ApproximationPart 2 Kyle...

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a39 a38 a36 a37 Set 11: Orthogonality and Approximation- Part 2 Kyle A. Gallivan Department of Mathematics Florida State University Foundations of Computational Math 2 Spring 2011 1

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a39 a38 a36 a37 Practical Situations In L 2 ω , γ i = ( f, φ i ) ω must be computed either analytically or numerically. Various numerical quadrature methods will be discussed later and we will return to the Generalized Fourier Series (GFS) power series are often used and truncated to finite degree polynomials α 0 + α 1 x + α 2 x 2 + α 3 x 3 + · · · → p 2 ( x ) = α 0 + α 1 x + α 2 x 2 For each orthogonal polynomial family GFS yields optimality with respect to a specific norm. For a particular function, economization reduces the number of terms used to achieving similar accuracy in some norm independent of the various families of polynomials. Economization: change of basis plus truncation 2
a39 a38 a36 a37 Basis Change The GFS can be used to change the basis from monomials to an orthogonal basis or from one orthogonal to another. All inner products involve weighted integration of polynomials – analytically tractable. Coefficients can also be created via incremental algebraic manipulation When approximating f ( x ) on a x b , a change of variables to the interval related to the inner product is done before economization and undone to get the final form of the approximation. 3

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Basis Change Consider monomials, Chebyshev and Legendre bases: p ( x ) = α 0 + α 1 x + α 2 x 2 + α 3 x 3 = ( p, T 0 ) ω ( T 0 , T 0 ) ω T 0 ( x ) + ( p, T 1 ) ω ( T 1 , T 1 ) ω T 1 ( x ) + ( p, T 2 ) ω ( T 2 , T 2 ) ω T 2 ( x ) + ( p, T 3 ) ω ( T 3 , T 3 ) ω T 3 ( x ) = ( p, P 0 ) 1 ( P 0 , P 0 ) 1 P 0 ( x ) + ( p, P 1 ) 1 ( P 1 , P 1 ) 1 P 1 ( x ) + ( p, P 2 ) 1 ( P 2 , P 2 ) 1 P 2 ( x ) + ( p, P 3 ) 1 ( P 3 , P 3 ) 1 P 3 ( x ) 4
a39 a38 a36 a37 Monomial to Legendre: inner products P 0 ( x ) = 1 , P 1 ( x ) = x, P 2 ( x ) = 1 2 (3 x 2 1) , P 3 ( x ) = 1 2 (5 x 3 3 x ) ( P 0 , P 0 ) = 2 , ( P 1 , P 1 ) = 2 3 , ( P 2 , P 2 ) = 2 5 , ( P 3 , P 3 ) = 2 7 (1 , P 0 ) = 2 , (1 , P 1 ) = (1 , P 2 ) = (1 , P 3 ) = 0 ( x, P 0 ) = 0 , ( x, P 1 ) = 2 3 , ( x, P 2 ) = ( x, P 3 ) = 0 ( x 2 , P 0 ) = 2 3 , ( x 2 , P 1 ) = 0 , ( x 2 , P 2 ) = 4 15 , ( x 2 , P 3 ) = 0 ( x 3 , P 0 ) = 0 , ( x 3 , P 1 ) = 2 5 , ( x 3 , P 2 ) = 0 , ( x 3 , P 3 ) = 4 35 5

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a39 a38 a36 a37 Monomial to Legendre: inner products Note j > k ( x k , P j ) = 0 . The other 0 ’s come from even/odd structure.
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set11 - \$ Set 11 Orthogonality and ApproximationPart 2 Kyle...

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