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Lecture34

# Lecture34 - Cubic Splines 4 Two additional conditions are...

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Cubic Splines 4. Two additional conditions are needed (arbitrary) The last two equations Total equations: 2(n-1) + (n-2) + (n-2) +2 = 4(n-1) + = = = = - - - - 1 n 1 n 1 n n 1 n 1 1 1 h d 6 c 2 0 x s c 2 0 x s ) ( ) ( = + = = = = - - - - 0 h d 3 c c 0 c 0 x s 0 x s 1 n 1 n 1 n n 1 n 1 n 1 1 ) ( ) (

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Cubic Splines 3 i i 2 i i i i i i x x d x x c x x b a x s ) ( ) ( ) ( ) ( - + - + - + = Solve for ( a i , b i , c i , d i ) – see textbook for details Tridiagonal system with boundary conditions c 1 = c n = 0 ]) , [ ] , [ ( ) ( 1 i i i 1 i 1 i 1 i i i i 1 i 1 i i i i 1 i 1 i 1 i x x f x x f 3 h f f 3 h f f 3 c h c h h 2 c h - + - - + + - - - - = - - - = + - + ( 29 1 i i i i i 1 i i i i 1 i i i i c c 2 3 h h f f b h 3 c c d f a + + + + - - = - = = , ,
Cubic Splines ( 29 ( 29 ( 29 - - - = + + + - - - - - - - - 0 x x f x x f 3 x x f x x f 3 x x f x x f 3 0 c c c c c 1 h h h 2 h h h h 2 h h h h 2 h 1 2 n 1 n 1 n n 2 3 3 4 1 2 2 3 n 1 n 3 2 1 1 n 1 n 2 n 2 n 3 3 2 2 2 2 1 1 ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ) ( ) ( ) ( Tridiagonal matrix

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Hand Calculations f(4) estimate 58 19 2 4 ) x ( f 6 5 2 0 x - 0 f(4) , 4 x 3 x 5 x f(x) : solution Exact 2 3 = + + - = 39 1 19 58 h f f x x f 7 3 2 19 h f f x x f 3 2 4 2 h f f x x f 58 f 19 f 2 f 4 f 1 5 6 x x h 3 2 5 x x h 2 0 2 x x h 3 3 4 3 4 2 2 3 2 3 1 1 2 1 2 4 3 2 1 3 4 3 2 3 2 1 2 1 = - = - = = - - = - = - = - - = - = = = - = = = - = - = = - = - = = - = - = ] , [ ) ( ] , [ ] , [ , , , , ,
Hand Calculations ( 29 ( 29 ( 29 ( 29 = - - - =

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Lecture34 - Cubic Splines 4 Two additional conditions are...

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