Assignment 2 Solution

# Assignment 2 Solution - A = x x . . . x x x . . . x x x 0 0...

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CS3251, Homework 2 Solution 1. Problem # 23 A = 2 1 0 1 2 1 0 2 1 . We apply Gauss-Jordan elimination, i.e., [ A, I ] [ I, A - 1 ]. [ A, I ] = 2 1 0 1 0 0 1 2 1 0 1 0 0 1 2 0 0 1 2 1 0 1 0 0 0 3 / 2 1 - 1 / 2 1 0 0 1 2 0 0 1 2 1 0 1 0 0 0 3 / 2 1 - 1 / 2 1 0 0 0 4 / 3 1 / 3 - 2 / 3 1 2 1 0 1 0 0 0 3 / 2 0 - 3 / 4 3 / 2 - 3 / 4 0 0 4 / 3 1 / 3 - 2 / 3 1 2 0 0 3 / 2 - 1 1 / 2 0 3 / 2 0 - 3 / 4 3 / 2 - 3 / 4 0 0 4 / 3 1 / 3 - 2 / 3 1 1 0 0 3 / 4 - 1 / 2 1 / 4 0 1 0 - 1 / 2 1 - 1 / 2 0 0 1 1 / 4 - 1 / 2 3 / 4 = [ I, A - 1 ] . 2. Problem # 3 A = ± 1 1 1 2 ² , x = ± x 1 x 2 ² , b = ± 5 7 ² We have the system Ax = b . A = LU = ± 1 0 1 1 ²± 1 1 0 1 ² . ± 1 0 1 1 ² c = ± 5 7 ² , ± 1 1 0 1 ² x = c We check that c = (5 , 2) T satisﬁes equation Lc = b and we solve the second equation Ux = c . This gives x = (3 , 2) T . Problem # 5 We want to ﬁnd the elimination matrix E such that EA = U and compute L = E - 1 . A = 2 1 0 0 4 2 6 3 5 we have E = 1 0 0 0 1 0 - 3 0 1 and L = E - 1 = 1 0 0 0 1 0 3 0 1 1

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3. Problem # 7 (a) false, (b) false, (c) true, (d) true Problem # 16 The symmetric ones are (a) and (c). 4. We want to solve a linear system where the coeﬃcient matrix is tridiagonal, i.e.,
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Unformatted text preview: A = x x . . . x x x . . . x x x 0 0 . . . 0 0 . . . x x Consider elimination applied to a column of the matrix. There is only one row below that column and this row has at most two nonzero elements. Thus, elimination takes constant time for this column. The total cost of elimination is proportional to n because there are n columns. Back-substitution also has cost proportional to n because the rows have at most two non-zero elements. Hence, the total cost to solve a system where the coeﬃcient matrix is tridiagonal is proportional to n . 2...
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## This note was uploaded on 01/31/2012 for the course COMS 3251 taught by Professor Anargyrospapageorgiou during the Fall '11 term at Columbia.

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Assignment 2 Solution - A = x x . . . x x x . . . x x x 0 0...

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