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The analysis leading to 9119 cannot be used on this

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Unformatted text preview: tions x i; i i i i G( x)Pr = G( G( i i 1 )r ( 1 ) 2 )r ( 2 ) i if x 1 x < if x<x i (9.1.20) i; i i i as shown in Figure 9.1.3. The error in piecewise constant interpolation is GLy ; GPr = (x ; (x ; i i 1 )(GLy ) ( 1 ) 2 )(GLy ) ( 2 ) 0 0 i i if x 1 x < if x<x : i; i A similar expression applies for GLY ; GPr. Combining these results in the manner used to obtain (9.1.18a) yields e( )= Z i ;1 xi (x ; 1)^( x)dx + g i Z xi (x ; 2)^( x)dx g i 7 2 (x 1 i; x) i (9.1.21a) where x< g( x) = (GLy) ( 1 ) ; (GLY ) ( 1) iif x 1 x < x : ^ (GLy) ( 2 ) ; (GLY ) ( 2) f 0 0 0 i i 0 i i We bound (9.1.21a) as Z je ( )j i xi ;1 (9.1.21b) i; i jx ; 1jjg( x)jdx + ^ Z xi i jx ; 2jjg( x)jdx ^ i or je ( )j h2 kg( )k ^ i 2 (x i1 i i; 1 x ): (9.1.21c) i We'll use the symbol kg( )k with the understanding that the maximum is computed ^ on (x 1 ) ( x ). Finally, substituting (9.1.19a) and (9.1.21c) into (9.1.15) yields i1 i; i h2jjg( ^ je( )j )jj j j1 X N + i or je( )j h2 + j X N i6 i6 1 j; x) j 2 (x 1 j; 1 i x) j j kg( )k = max(kg( )k kg( )k ) ~ ^ ^ 1 2 (x i1 i j h3 ]kg( )k ~ =1 = i where =1 = h3kg( )k 1 kf ( )k...
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