301_F10_hw2

301_F10_hw2 - x from Ux = c . Note that only 10 elementary...

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AME 301 Homework #2 | Due Fri Sep 10, 2010 1. Solve the following system of equations by using Gauss elimination, with partial piv- oting if required. u + v + w = ± 2 3 u + 3 v ± w = 6 u ± v + w = ± 1 2. The matrix equations that arise in ±nite-di²erence numerical methods are often banded. Consider the matrix equation 2 6 6 6 6 4 2 ± 1 0 0 0 ± 1 2 ± 1 0 0 0 ± 1 2 ± 1 0 0 0 ± 1 2 ± 1 0 0 0 ± 1 2 3 7 7 7 7 5 2 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 3 7 7 7 7 5 = 2 6 6 6 6 4 1 2 3 2 1 3 7 7 7 7 5 where we note that the coe³cient matrix A is tridiagonal. Verify that A has the factorization LU = A where L = 2 6 6 6 6 4 1 0 0 0 0 ± 1 = 2 1 0 0 0 0 ± 2 = 3 1 0 0 0 0 ± 3 = 4 1 0 0 0 0 ± 4 = 5 1 3 7 7 7 7 5 and U = 2 6 6 6 6 4 2 ± 1 0 0 0 0 3 = 2 ± 1 0 0 0 0 4 = 3 ± 1 0 0 0 0 5 = 4 ± 1 0 0 0 0 6 = 5 3 7 7 7 7 5 by multiplying L and U to recover A . Solve the problem Ax = b by ±rst using forward substitution to obtain c from Lc = b , then using back substitution to ±nd
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Unformatted text preview: x from Ux = c . Note that only 10 elementary steps are required to nd x in this case. 3. Problem set 7.3: #14 4. Problem set 7.4: #2 nd the rank, a basis for the row space, a basis for the null space, and a basis for the column space. 5. Problem set 7.4: #7 nd the rank, a basis for the row space, a basis for the null space, and a basis for the column space. 6. Problem set 7.4: #20 1...
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