set6 - \$ Set 6 Splines Part 1 Kyle A Gallivan Department of...

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a39 a38 a36 a37 Set 6: Splines – Part 1 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 Interpolation and Smoothness The piecewise Hermite interpolant is cubic locally but still only C (1) globally. Derivative values may not be available. To get C (2) globally and maintain piecewise cubic polynomial we must give up something. Give up interpolating f i . Interpolate f i at nodes. Require piecewise cubic polynomial. Use continuity of first and second derivatives as constraints but do not specify values. Family of interpolatory cubic splines. 2
a39 a38 a36 a37 Polynomial Splines Polynomial splines are the subject of a large body of literature. In addition to the text the following have useful discussions at the appropriate level for this class. and have been used as source material: P. M. Prenter, Splines and Variational Methods, Wiley. C. W. Ueberhuber, Numerical Computation, Springer An excellent more advanced reference is: Carl de Boor, A Practical Guide to Splines, Springer-Verlag, 1978. 3

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a39 a38 a36 a37 Polynomial Splines Definition 6.1. Given [ a,b ] let the distinct points a = x 0 <x 1 < · · · <x n define a partition into intervals [ x i 1 ,x i ) denoted π . A polynomial spline, s ( t ) , of degree d is a piecewise polynomial of degree d , s ( t ) = p i,d ( t ) on [ x i 1 ,x i ) for 1 i n . Further, the polynomials are such that their values and the values of their first to ( d 1) -st derivatives match at x i for 1 i n 1 . 4
a39 a38 a36 a37 Polynomial Splines Definition 6.2. A subspline of degree d is a piecewise polynomial that satisfies all of the conditions of a spline but is only continuous to the m -th derivatives with m<d 1 . Note. Piecewise Hermite interpolating polynomials are cubic subsplines since they are only C (1) . 5

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a39 a38 a36 a37 Cubic Splines Lemma. Given a partition, π , the set of cubic splines, S 3 ( π ) is a linear space with dimension n + 3 . Informal Argument: n intervals each with a cubic polynomial require 4 n parameters. continuity of s ( t ) at x i , 1 i n 1 , imposes n 1 constraints continuity of s ( t ) at x i , 1 i n 1 , imposes n 1 constraints continuity of s ′′ ( t ) at x i , 1 i n 1 , imposes n 1 constraints 4 n 3( n 1) = n + 3 degrees of freedom Note. A proof requires exhibiting a basis with n + 3 linearly independent functions. 6
a39 a38 a36 a37 Interpolating Cubic Spline Definition 6.3. An interpolating cubic spline is a cubic spline that satisfies s ( x i ) = f i 0 i n where a = x 0 <x 1 < · · · <x n = b are distinct points. Interpolation imposes n + 1 constraints. 2 degrees of freedom remain Typically two boundary conditions are specified.

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