{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# set4 - \$ Set 4 Polynomial Interpolation Part 4 Kyle A...

This preview shows pages 1–8. Sign up to view the full content.

a39 a38 a36 a37 Set 4: Polynomial Interpolation – Part 4 Kyle A. Gallivan Department of Mathematics Florida State University Foundations of Computational Math 2 Spring 2011 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
a39 a38 a36 a37 Hermite Interpolation and Osculatory Polynomials -5 -4 -3 -2 -1 0 1 2 3 4 5 -0.5 0 0.5 1 1.5 2 Note. function values OK at points, derivatives are not, sometimes even wrong sign 2
a39 a38 a36 a37 Approach Solution: specify function values f ( x i ) = y i specify derivative values f ( x i ) = y i Repeat approaches: power basis: p n ( x ) = n i =0 α i x i Lagrange form: p n ( x ) = n i =0 bracketleftBig y i ψ i ( x ) + y i Ψ i ( x ) bracketrightBig Newton form: p n ( x ) = n i =0 α i Ω i ( x ) 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
a39 a38 a36 a37 Constrain Derivatives For example, given 4 constraints construct p 3 ( x ) = 3 i =0 α i x i : y (0) = p 3 (0) y (0) = p 3 (0) y (1) = p 3 (1) y (1) = p 3 (1) α 0 = y (0) α 1 = y (0) α 0 + α 1 + α 2 + α 3 = y (1) α 1 + 2 α 2 + 3 α 3 = y (1) 4
a39 a38 a36 a37 Example α 0 = y (0) α 1 = y (0) α 0 + α 1 + α 2 + α 3 = y (1) α 1 + 2 α 2 + 3 α 3 = y (1) 1 0 0 0 0 1 0 0 1 1 1 1 0 1 2 3 α 0 α 1 α 2 α 3 = y (0) y (0) y (1) y (1) 5

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
a39 a38 a36 a37 Example y (0) = 3 , y (0) = 2 y (1) = 6 , y (1) = 1 α 0 α 1 α 2 α 3 = 3 2 4 3 p 3 ( x ) = 3 x 3 + 4 x 2 + 2 x + 3 , p 3 ( x ) = 9 x 2 + 8 x + 2 p 3 (0) = 3 , p 3 (0) = 2 , p 3 (1) = 6 , p 3 (1) = 1 6
a39 a38 a36 a37 Monomial Form – Hermite Interpolation p d ( x i ) = y i and p d ( x i ) = y i 0 i n 1 x 0 x 2 0 ... x d 0 0 1 2 x 0 ... dx d 1 0 1 x 1 x 2 1 ... x d 1 0 1 2 x 1 ... dx d 1 1 . . . . . . . . . 1 x n x 2 n ... x d n 0 1 2 x n ... dx d 1 n α 0 α 1 . . . α d 1 α d = y 0 y 0 y 1 y 1 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 26

set4 - \$ Set 4 Polynomial Interpolation Part 4 Kyle A...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online