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HW4 - compute the condition number by hand However you are...

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EMAE 250.100: Homework 4 February 3, 2011 1. In some problems, it is possible to obtain a complex systems of equations (Text pp. 267), such that C ~x = ~ y, (1) where C M n × n is a matrix containing complex numbers. In this case, each term can be divided into the real number and the complex number terms, such that C = A + i B , ~x = ~a + i ~ b and ~ y = ~ c + i ~ d where A , B R n × n and ~ c, ~ d R n × 1 . Then, Eq. (1) can be rewritten by { A + i B } ( ~a + i ~ b ) = ( ~ c + i ~ d ) . (2) (a) [1pt] Using Eq. (2), show that A - B B A ! ~a ~ b ! = ~ c ~ d ! . (b) [2pt] Find ~x = ( x 1 , x 2 ) T using (a). 3 + 2 i 4 - i 1 ! x 1 x 2 ! = 2 + i 3 ! . 2. Given the following matrix A = 2 - 1 1 - 2 2 1 4 - 1 2 (a) [1pt] Derive an LU decomposition of A . (b) [2pt] Compute A - 1 by using the fact that A - 1 A = I and A = LU . 1
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3. Compute the condition number for the following matrix and determine whether it is ‘ill-conditioned’, ‘well-conditioned’ or ‘perfectly conditioned’. Use one of matrix norms discussed in the lecture (clearly specify which one is used). You can use a calculator or MATLAB to compute the inverse of each matrix, but
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Unformatted text preview: compute the condition number by hand. However, you are still allowed to use a calculator for scalar multiplications. (a) [1pt] Find the condition number of the following matrix: A = 1 1 1 . 01-1-. 99-1-1 . (b) [1pt] Given A ~x = ~ b find ~x = [ x 1 ,x 2 ,x 3 ] T where: i) ~ b = [1 ,-1 , 1] T and ii) ~ b = [1 . 01 ,-. 99 , 1 . 01] T and discuss your results in relation to the condition number computed in (a). (c) [1pt] Find the condition number of the following matrix: A = 3-2 1-1 4 1 2 5 3 . (d) [1pt] Given A ~x = ~ b find ~x = [ x 1 ,x 2 ,x 3 ] T where: i) ~ b = [1 ,-1 , 1] T and ii) ~ b = [1 . 01 ,-. 99 , 1 . 01] T and discuss your results in relation to the condition number computed in (c). 2...
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