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Unformatted text preview: MIT 18.335, Fall 2007: Homework 4, Solutions 1. (Trefethen/Bau 20.3a) With block matrix multiplication, the four blocks become IA 11 + 0 A 21 = A 11 IA 12 + 0 A 22 = A 12 A 21 A 1 11 A 11 + IA 21 = 0 A 21 A 1 11 A 12 + IA 22 = A 22 A 21 A 1 11 A 12 (Trefethen/Bau 20.3b) After n steps of Gaussian elimination, A has been factorized as parenleftbigg L 11 L 21 I parenrightbiggparenleftbigg U 11 U 12 tildewide U 22 parenrightbigg = parenleftbigg A 11 A 12 A 21 A 22 parenrightbigg or, in expanded form, L 11 U 11 = A 11 L 11 U 12 = A 12 L 21 U 11 = A 21 L 21 U 12 + tildewide U 22 = A 22 Solving for tildewide U 22 gives tildewide U 22 = A 22 L 21 U 12 = A 22 A 21 U 1 11 L 1 11 A 12 = A 22 A 21 A 1 11 A 12 2. (Trefethen/Bau 21.4a) First, the matrix is factorized as A = LU with Gaussian elimination, which requires about 2 m 3 / 3 flops. To compute the inverse, the following system is solved for A 1 : AA 1 = LUA 1 = I This is done with m backsubstitutions with right hand sides e i . Each backsubstitution requires about m summationdisplay k =1 2 k = m 2 flops and the total number of flops is mm 2 + mm 2 + 2 3 m 3 = 8 3 m 3 1 (Trefethen/Bau 21.4b) When doing the backsubstitution for the right hand side e i , the first i 1 elements will be zero. The number of flops can then be reduced to m summationdisplay i =1 m summationdisplay k = i 2( m k ) m 3 / 3 The second backsubstitution still requires m 2 flops for each right hand side, and the total number of flops is 1 3 m 3 + m 3 + 2 3 m 3 = 2 m 3 (Trefethen/Bau 21.4c) 1. According to the previous part, each right hand side requires 2 m 2 flops, so the total operation count is 2 m 3 / 3 + 2 nm 2 . 2. Producing the inverse requires 2 m 3 flops, and each right hand side requires 2 m 2 flops, so the total operation count is 2 m 3 + 2 nm 2 . 3. (Trefethen/Bau 22.2) The following Matlab code creates the matrix, computes the LU factorization and the norm of error. m=60; A=tril(ones(m),1)+diag(ones(m,1)); A(:,m)=1; [L,U,P]=lu(A); max(abs(U(:))) norm(L*UP*A,inf) The maximum element in U is 2 59 , as expected, but the error in A is zero, since there are never and calculations resulting in cancellation. However, if the last row of A has all elements equal to 1: A(m,:)=1; the norm will be 2, or more exactly, the element A mm is 1 instead of 1, due to the rounding...
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 Fall '08
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 Linear Algebra, Algebra, Multiplication, Unitary matrix, Matrix decomposition, total operation count, A21 A−1 A12, Trefethen/Bau

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