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Amath 352 useful formulas

# Amath 352 useful formulas - C k 1 on this interval •...

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Summary of some useful formulas Taylor polynomial of degree k (Note typo in (7.7), (7.8) — sums start at j = 0): P k ( x ) = k j =0 1 j ! f ( j ) ( x 0 )( x - x 0 ) . Error in Taylor polynomial P k ( x ): If f C k +1 (Int[ x 0 , x ]), then | f ( x ) - P k ( x ) | = | f ( k +1) ( ξ ) || x - x 0 | k +1 for some ξ in this interval, and | f ( x ) - P k ( x ) | ≤ 1 ( k + 1)! max z Int [ x 0 ,x ] | f ( k +1) ( z ) | | x - x 0 | k +1 . Divided differences: f [ x i , x i +1 , . . . , x i + p ] = f [ x i +1 , . . . , x i + p ] - f [ x i , . . . , x i + p - 1 ] x i + p - x i Special case of divided difference used in Hermite interpolation: f [ x i , x i ] = f ( x i ) if an interpolation point is repeated. “Newton form” of the interpolating polynomial of degree k : p k ( x ) = f [ x 1 ] + f [ x 1 , x 2 ]( x - x 1 ) + · · · + f [ x 1 , x 2 , . . . , x k +1 ]( x - x 1 )( x - x 2 ) · · · ( x - x k ) . The error in polynomial interpolation of a function f ( x ): f x ) - p k x ) = 1 ( k + 1)! f ( k +1) ( ξ )(¯ x - x 1 )(¯ x - x 2 ) · · · x - x k +1 ) for some point ξ Int( x 1 , . . . , x
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Unformatted text preview: C ( k +1) on this interval. • Error in piecewise linear interpolation when subintervals all have length h : k f-p h 1 k ∞ ≤ h 2 8 k f 00 k ∞ . • Quadrature formulas on a single subinterval [ x i , x i +1 ]: – Midpoint: hf ( x i +1 / 2 ) – Trapozoid: h 2 ( f ( x i ) + f ( x i +1 )) – Simpson on [ x 2 i-1 , x 2 i +1 ]: h 3 ( f ( x 2 i-1 ) + 4 f ( x 2 i ) + f ( x 2 i +1 )) • Error bounds for quadrature rules: | Q-Q mid ( h ) | ≤ h 2 ( b-a ) 24 k f 00 k ∞ , | Q-Q trap ( h ) | ≤ h 2 ( b-a ) 12 k f 00 k ∞ , | Q-Q simp ( h ) | ≤ h 4 ( b-a ) 180 k f (4) k ∞ ....
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