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Exercise 2.3.3

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We have $Ax = b$ and $A Δ x = Δ b$ , thus $| Ax | = | b |$ and $| A Δ x | = | Δ b |$ .

We also have $| Δ x | = | A^{- 1} A Δ x | \leq | A^{- 1} | | A Δ x | = | A^{- 1} | | Δ b |$ , divide both sides by $| Ax |$ , we have $\frac{| Δ x |}{| Ax |} \leq \frac{| A^{- 1} | | Δ b |}{| Ax |} = \frac{| A^{- 1} | | Δ b |}{| b |}$ , i.e. $| Δ x | \leq | Ax | \frac{| A^{- 1} | | Δ b |}{| b |} \leq | A | | x | \frac{| A^{- 1} | | Δ b |}{| b |}$ , divide both sides by $| x |$ , we now have

$\frac{| Δ x |}{| x |} \leq | A | | A^{- 1} | \frac{| Δ b |}{| b |}$

niuers

2020-03-20 00:00

Exercise 2.3.3

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