Lecture 21 - Tridiagonal Matrix Forward elimination Back...

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Unformatted text preview: Tridiagonal Matrix Forward elimination Back substitution n 3 2 k r f e r r g f e f f 1 k 1 k k k k 1 k 1 k k k k , , , = - =- =---- 1 2 3 2 n 1 n k f x g r x f r x k 1 k k k k n n n , , , , , -- =- = = + Use factor = e k / f k- 1 to eliminate subdiagonal element Apply the same matrix operations to right hand side 1 1 1 1 ' ' 2 2 2 2 ' ' 3 3 3 3 ' ' ' ' 1 1 1 1 ' ' i i i i n n n n n n n f g x r f g x r f g x r f g x r f g x r f x r---- = O O O M M O O O M M Tridiagonal Matrix Special case of banded matrix with bandwidth = 3 Save storage, 3 n instead of n n Hand Calculations: Tridiagonal Matrix - = ------ 5 3 2 5 3 x x x x 25 1 5 5 2 1 1 5 2 2 1 4 3 2 1 . . . . =-- =- = =--- =- = =--- =- = =--- =- = - =--- =- = =--- =- = 4 1 1 5 5 3 r f e r r 1 5 1 5 25 1 g f e f f 1 1 1 1 2 r f e r r 1 1 1 1 2 g f e f f 1 3 1 2 5 r f e r r 1 2 1 2 5 g f e f f 3 3 4 4 4 3 3 4 4 4 1 1 2 2 3 2 2 3 3 3 1 1 2 2 2 1 1 2 2 2 ) ( . . ) . ( . . ) ( ) ( ) ( ) ( 1 1 2 2 3 f x g r x 2 1 3 1 1 f x g r x 3 1 4 5 1 f x g r x 4 1 4 f r x 1 2 1 1 1 2 3 2 2 2 3 4 3 3 3 4 4 4 =--- =- = =--- =- = =-- =- = = = = ) )( ( ) )( ( ) )( . ( (a) Forward elimination (b) Back substitution MATLAB M-file: Tridiag One loop; Operation count: ~8n NOT 2 n 3 /3 No modification on g [e,f,g,r] = example...
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Lecture 21 - Tridiagonal Matrix Forward elimination Back...

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