Show that computing 2x by x x is backward stable. See the example in the notes to show backward stable.

Def : We say that an algorithm f is backward Stable if for each xe x f ( x ) = f ( x ) for some X such that 1/x - All = 0 (Emo chive 11x11 Bosically , a backward stable algorithm gives exactly the right answer to Mearly the right question . Remark: Backward Stability Stability EXAMPLE : Stability of floating pointWe conclude that floating point Susshaction is Bachward Stable , and there fore Stable Theoum : Set algorithm f (x ) be backward Stable , and solves f (x ) with relative Condition number k ( x ) . Then, 11 8 ( x ) - fall S O ( K (x), Emachine ) I B (x ) ( the algorithm is accurate Proof : Since f ( x ) is backward stable B ( x ) = $ ( x ) where 1/x-811 =0 ( Emoa ) 11 x11( the algorithm is accurate Proof : Since f ( x ) is backward stable B ( x ) = $ ( x ) where 1/X- 8/1 = 0 ( Emoce ) 11 x 11 1 8 ( x )- 8()11 - 11 8(x) - 8(x)/1 11 8 ( x ) - 8(x) /1 11x11 11 8 11 |1x-x 1