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Lecture+21--Tridiagona.+_+LU+decomposition

Lecture+21--Tridiagona.+_+LU+decomposition - Hand...

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Unformatted text preview: Hand Calculations: Tridiagonal Matrix (a) Forward elimination 1-2-3 1-1-1 1-0.5 1 1 4 - = ------ 5 3 2 5 3 x x x x 25 1 5 5 2 1 1 5 2 2 1 4 3 2 1 . . . . 2 2 2 1 1 2 2 2 1 1 2 5 ( 2) 1 1 2 5 ( 3) 1 1 e f f g f e r r r f- =- =-- = - =- =-- = - 3 3 3 2 2 2 3 2 1 1 1 2 ( 1) 1 1 1 2 ( 1) 1 1 e f f g f e r r r f- =- =-- = - =- =-- = 4 4 4 3 3 4 4 4 3 3 0.5 1.25 ( 0.5) 1 1 0.5 3.5 (1) 4 1 e f f g f e r r r f- =- =-- = - =- =- = Hand Calculations: Tridiagonal Matrix (b) Back substitution 1 2 3 4 1 2 3 1 1 1 1 0.5 1 1 4 x x x x-- -- = - 4 4 4 4 4 1 r x f = = = 1 1 2 1 1 3 ( 2)(2) 1 1 r g x x f-- - - = = = 3 3 4 3 3 1 ( 0.5)(4) 3 1 r g x x f-- - = = = 2 2 3 2 2 1 ( 1)(3) 2 1 r g x x f-- - - = = = MATLAB M-file: Tridiag function x = Tridiag(e,f,g,r) % Tridiag: Tridiagonal equation solver banded system % x = Tridiag(e,f,g,r): Tridiagonal system solver. % input: % e = subdiagonal vector % f = diagonal vector % g = superdiagonal vector % r = right hand side vector % output: % x = solution vector n=length(f); % forward elimination for k = 2:n factor = e(k)/f(k-1); f(k) = f(k) - factor*g(k-1); r(k) = r(k) - factor*r(k-1); end % back substitution x(n) = r(n)/f(n); for k = n-1:-1:1 x(k) = (r(k)-g(k)*x(k+1))/f(k); end One loop; Operation count: ~8n NOT 2 n3/3 No modification on g » [e,f,g,r] = example...
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Lecture+21--Tridiagona.+_+LU+decomposition - Hand...

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