Linear Algebra and Its Applications · Problem 7.2.26

Computations with Matrices — Gilbert Strang

Mathematics · lu-factorization-roundoff

Problem

Find the LULU factorization of A=[ε111]A = \begin{bmatrix} \varepsilon & 1 \\ 1 & 1 \end{bmatrix}. On your computer, solve by elimination when ε=10−3,10−6,10−9,10−12,10−15\varepsilon = 10^{-3}, 10^{-6}, 10^{-9}, 10^{-12}, 10^{-15}: [ε111][x1x2]=[1+ε2].\begin{bmatrix} \varepsilon & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 1+\varepsilon \\ 2 \end{bmatrix}. The true xx is (1,1)(1,1). Make a table to show the error for each ε\varepsilon. Exchange the two equations and solve again---the errors should almost disappear.

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