solved07 - S 7.1(i m x1 x2 x3 b 2-4 2 1 1/2-1 5 2 0-1/2 1-1...

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S 7.1 _________ (i) m x1 x2 x3 b ________________________________________ 2* -4 2 1 1/2 -1 5 2 0 -1/2 1 -1 3 0 ________________________________________ 2 -4 2 1 0 3* 3 1/2 -1/3 1 1 2 -1/2 ________________________________________ 2 -4 2 1 0 3 3 1/2 0 0 1 -2/3 ________________________________________ End of forward elimination. Proceed with scaling and backward substitution: ________________________________________ -1 1 -2 1 1/2 -1 0 1 1 1/6 0 0 1* -2/3 ________________________________________ 2 1 -2 0 7/6 0 1* 0 5/6 0 0 1 -2/3 ________________________________________ 1 0 0 17/6 0 1 0 5/6 0 0 1 -2/3 (ii) A = L*U, where L = [ 1 0 0 -1/2 1 0 1/2 1/3 1 ] U = [ 2 -4 2 0 3 3 0 0 1 ] (iii) Given the LU factorization of A, A*x=b can be solved in two steps: solving L*y = b to obtain y, followed by solving U*x = b. This can be done for e2 and e3 at the same time using two b columns. L*y = b: m y1 y2 y3 b b __________________________________________ 1* 0 0 0 0 1/2 -1/2 1 0 1 0 -1/2 1/2 1/3 1 0 1 __________________________________________ 1 0 0 0 0 0 1* 0 1 0 -1/3 0 0 1 0 1 ___________________________________________ 1 0 0 0 0
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0 1 0 1 0 0 0 1 -1/3 1 U*x = y: m x1 x2 x3 y y __________________________________________ 2 -4 2 0 0 0 3 3 1 0 0 0 1 -1/3 1 ___________________________________________ -1 1 -2 1 0 0 -1 0 1 1 1/3 0 0 0 1* -1/3 1 ___________________________________________ 2 1 -2 0 1/3 -1 0 1* 0 2/3 -1 0 0 1 -1/3 1 ___________________________________________ 1 0 0 5/3 -3 0 1 0 2/3 -1 0 0 1 -1/3 1 Thus A*x = e2 has solution x = [5/3 ; 2/3 ; -1/3].' A*x = e3 has solution x = [-3 ; -1 ; 1].' (iv) Let X be the matrix consisting of the three solutions (for b = e1, e2 and e3, in that order): X = [ 17/6 5/3 -3 5/6 2/3 =1 -2/3 -1/3 1 ] Then A*X = [e1 e2 e3] = I (3x3 identity), and thus X = inv(A).
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This note was uploaded on 10/18/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.

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solved07 - S 7.1(i m x1 x2 x3 b 2-4 2 1 1/2-1 5 2 0-1/2 1-1...

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