aimz3 - A n for general n Do the entries in the factors...

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AIMS Exercise Set # 3 Peter J. Olver 1. Which of the following matrices are regular? If reguolar, write down its L U factorization. ( a ) µ 2 1 1 4 , ( b ) µ 0 - 1 3 - 2 , ( c ) 1 - 2 3 - 2 4 - 1 3 - 1 2 . 2. In each of the following problems, Fnd the A = L U factorization of the coe±cient matrix, and then use ²orward and Back Substitution to solve the corresponding linear systems A x = b for each of the indicated right hand side: ( a ) A = µ - 1 3 3 2 , b = µ 1 - 1 ; ( b ) A = 1 0 - 1 0 0 2 3 - 1 - 1 3 2 2 0 - 1 2 1 , b = 1 0 - 1 1 . 3. ²ind the L D L T factorization of the matrix 1 - 1 - 1 - 1 3 2 - 1 2 0 . 4. ( a ) ²ind the L U factorization of the n × n tridiagonal matrix A n with all 2’s along the diagonal and all - 1’s along the sub- and super-diagonals for n = 3 , 4 and 5. ( b ) Use your factorizations to solve the system A n x = b , where b = (1 , 1 , 1 , . . . , 1) T . ( c ) Can you write down the L U factorization of
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Unformatted text preview: A n for general n ? Do the entries in the factors approach a limit as n gets larger and larger? 5. True or false : ( a ) The product of two tridiagonal matrices is tridiagonal. ( b ) The inverse of a tridiagonal matrix is tridiagonal. 6. ( a ) ²ind the exact solution to the linear system x-5 y-z = 1 , 1 6 x-5 6 y + z = 0 , 2 x-y = 3 . ( b ) Solve the system using Gaussian Elimination with 4 digit rounding. ( c ) Solve the system using Partial Pivoting and 4 digit rounding. Compare your answers. 7. Implement the computer experiment with Hilbert matrices outlined in the last paragraph of the section. 1 c ° 2006 Peter J. Olver...
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