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Numerical Differentiation & IntegrationNumerical Differentiation INumerical Analysis (9th Edition)R L Burden & J D FairesBeamer Presentation Slidesprepared byJohn CarrollDublin City Universityc 2011 Brooks/Cole, Cengage Learning
IntroductionGeneral Formulas3-pt FormulasOutline1Introduction to Numerical DifferentiationNumerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires2 / 33
IntroductionGeneral Formulas3-pt FormulasOutline1Introduction to Numerical Differentiation2General Derivative Approximation FormulasNumerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires2 / 33
IntroductionGeneral Formulas3-pt FormulasOutline1Introduction to Numerical Differentiation2General Derivative Approximation Formulas3Some useful three-point formulasNumerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires2 / 33
IntroductionGeneral Formulas3-pt FormulasOutline1Introduction to Numerical Differentiation2General Derivative Approximation Formulas3Some useful three-point formulasNumerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires3 / 33
IntroductionGeneral Formulas3-pt FormulasIntroduction to Numerical DifferentiationApproximating a DerivativeNumerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires4 / 33
IntroductionGeneral Formulas3-pt FormulasIntroduction to Numerical DifferentiationApproximating a DerivativeThe derivative of the function f at x0 isf (x0 h) f (x0 ).hh 0f ′ (x0 ) limNumerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires4 / 33
IntroductionGeneral Formulas3-pt FormulasIntroduction to Numerical DifferentiationApproximating a DerivativeThe derivative of the function f at x0 isf (x0 h) f (x0 ).hh 0f ′ (x0 ) limThis formula gives an obvious way to generate an approximationto f ′ (x0 ); simply computef (x0 h) f (x0 )hfor small values of h. Although this may be obvious, it is not verysuccessful, due to our old nemesis round-off error.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires4 / 33
IntroductionGeneral Formulas3-pt FormulasIntroduction to Numerical DifferentiationApproximating a DerivativeThe derivative of the function f at x0 isf (x0 h) f (x0 ).hh 0f ′ (x0 ) limThis formula gives an obvious way to generate an approximationto f ′ (x0 ); simply computef (x0 h) f (x0 )hfor small values of h. Although this may be obvious, it is not verysuccessful, due to our old nemesis round-off error.But it is certainly a place to start.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires4 / 33
IntroductionGeneral Formulas3-pt FormulasIntroduction to Numerical DifferentiationApproximating a Derivative (Cont’d)To approximate f ′ (x0 ), suppose first that x0 (a, b), wheref C 2 [a, b], and that x1 x0 h for some h 6 0 that is sufficientlysmall to ensure that x1 [a, b].Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires5 / 33
IntroductionGeneral Formulas3-pt FormulasIntroduction to Numerical DifferentiationApproximating a Derivative (Cont’d)To approximate f ′ (x0 ), suppose first that x0 (a, b), wheref C 2 [a, b], and that x1 x0 h for some h 6 0 that is sufficientlysmall to ensure that x1 [a, b].We construct the first Lagrange polynomial P0,1 (x) for fdetermined by x0 and x1 , with its error term:f (x ) P0,1 (x ) (x x0 )(x x1 ) ′′f (ξ(x ))2!f (x0 )(x x0 h) f (x0 h)(x x0 ) (x x0 )(x x0 h) ′′ f (ξ(x )) hh2for some ξ(x) between x0 and x1 .Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires5 / 33
IntroductionGeneral Formulas3-pt FormulasNumerical Differentiationf (x ) f (x0 )(x x0 h) f (x0 h)(x x0 ) (x x0 )(x x0 h) ′′ f (ξ(x )) hh2Differentiating gives′f (x) (x x0 )(x x0 h) ′′f (x0 h) f (x0 ) Dxf (ξ(x))h2Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires6 / 33
IntroductionGeneral Formulas3-pt FormulasNumerical Differentiationf (x ) f (x0 )(x x0 h) f (x0 h)(x x0 ) (x x0 )(x x0 h) ′′ f (ξ(x )) hh2Differentiating gives′f (x) (x x0 )(x x0 h) ′′f (x0 h) f (x0 ) Dxf (ξ(x))h2f (x0 h) f (x0 ) 2(x x0 ) h ′′ f (ξ(x))h2(x x0 )(x x0 h) Dx (f ′′ (ξ(x)))2Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires6 / 33
IntroductionGeneral Formulas3-pt FormulasNumerical Differentiationf (x ) f (x0 )(x x0 h) f (x0 h)(x x0 ) (x x0 )(x x0 h) ′′ f (ξ(x )) hh2Differentiating gives′f (x) (x x0 )(x x0 h) ′′f (x0 h) f (x0 ) Dxf (ξ(x))h2f (x0 h) f (x0 ) 2(x x0 ) h ′′ f (ξ(x))h2(x x0 )(x x0 h) Dx (f ′′ (ξ(x)))2Deleting the terms involving ξ(x) givesf ′ (x) Numerical Analysis (Chapter 4)f (x0 h) f (x0 )hNumerical Differentiation IR L Burden & J D Faires6 / 33
IntroductionGeneral Formulas3-pt FormulasNumerical Differentiationf ′ (x) f (x0 h) f (x0 )hApproximating a Derivative (Cont’d)One difficulty with this formula is that we have no informationabout Dx f ′′ (ξ(x)), so the truncation error cannot be estimated.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires7 / 33
IntroductionGeneral Formulas3-pt FormulasNumerical Differentiationf ′ (x) f (x0 h) f (x0 )hApproximating a Derivative (Cont’d)One difficulty with this formula is that we have no informationabout Dx f ′′ (ξ(x)), so the truncation error cannot be estimated.When x is x0 , however, the coefficient of Dx f ′′ (ξ(x)) is 0, and theformula simplifies tof ′ (x0 ) Numerical Analysis (Chapter 4)f (x0 h) f (x0 ) h ′′ f (ξ)h2Numerical Differentiation IR L Burden & J D Faires7 / 33
IntroductionGeneral Formulas3-pt FormulasNumerical Differentiationf ′ (x0 ) f (x0 h) f (x0 ) h ′′ f (ξ)h2Forward-Difference and Backward-Difference FormulaeFor small values of h, the difference quotientf (x0 h) f (x0 )hcan be used to approximate f ′ (x0 ) with an error bounded byM h /2, where M is a bound on f ′′ (x) for x between x0 andx0 h.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires8 / 33
IntroductionGeneral Formulas3-pt FormulasNumerical Differentiationf ′ (x0 ) f (x0 h) f (x0 ) h ′′ f (ξ)h2Forward-Difference and Backward-Difference FormulaeFor small values of h, the difference quotientf (x0 h) f (x0 )hcan be used to approximate f ′ (x0 ) with an error bounded byM h /2, where M is a bound on f ′′ (x) for x between x0 andx0 h.This formula is known as the forward-difference formula if h 0and the backward-difference formula if h 0.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires8 / 33
IntroductionGeneral Formulas3-pt FormulasForward-Difference Formula to Approximate f ′ (x0 )ySlope f 9(x 0)Slopex0Numerical Analysis (Chapter 4)f (x0 1 h) 2 f (x 0)hxx0 1 hNumerical Differentiation IR L Burden & J D Faires9 / 33
IntroductionGeneral Formulas3-pt FormulasNumerical DifferentiationExample 1: f (x) ln xUse the forward-difference formula to approximate the derivative off (x) ln x at x0 1.8 using h 0.1, h 0.05, and h 0.01, anddetermine bounds for the approximation errors.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires1n 1)f(ξ(x))(n 1!)(x x0 ) · · · (x xn )Dx [f (n 1) (ξ(x))](n 1)!Method of Construction (Cont’d)We again have a problem estimating the truncation error unless x isone of the numbers xj . In this case, the term multiplyingDx [f (n 1) (ξ(x))] is 0, and the formula becomesf ′ (xj ) nXk 0f (xk )L′k (xj ) nf (n 1) (ξ(xj )) Y(xj xk )(n 1)!k 0k 6 jwhich is called an (n 1)-point formula to approximate f ′ (xj ).Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires16 / 33
IntroductionGeneral Formulas3-pt FormulasGeneral Derivative Approximation Formulasf ′ (xj ) nXk 0f (xk )L′k (xj ) nf (n 1) (ξ(xj )) Y(xj xk )(n 1)!k 0k 6 jComment on the (n 1)-point formulaIn general, using more evaluation points produces greateraccuracy, although the number of functional evaluations andgrowth of round-off error discourages this somewhat.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires17 / 33
IntroductionGeneral Formulas3-pt FormulasGeneral Derivative Approximation Formulasf ′ (xj ) nXk 0f (xk )L′k (xj ) nf (n 1) (ξ(xj )) Y(xj xk )(n 1)!k 0k 6 jComment on the (n 1)-point formulaIn general, using more evaluation points produces greateraccuracy, although the number of functional evaluations andgrowth of round-off error discourages this somewhat.The most common formulas are those involving three and fiveevaluation points.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires17 / 33
IntroductionGeneral Formulas3-pt FormulasGeneral Derivative Approximation Formulasf ′ (xj ) nXk 0f (xk )L′k (xj ) nf (n 1) (ξ(xj )) Y(xj xk )(n 1)!k 0k 6 jComment on the (n 1)-point formulaIn general, using more evaluation points produces greateraccuracy, although the number of functional evaluations andgrowth of round-off error discourages this somewhat.The most common formulas are those involving three and fiveevaluation points.We first derive some useful three-point formulas and consider aspectsof their errors.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires17 / 33
IntroductionGeneral Formulas3-pt FormulasOutline1Introduction to Numerical Differentiation2General Derivative Approximation Formulas3Some useful three-point formulasNumerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires18 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasImportant Building BlocksSinceL0 (x) Numerical Analysis (Chapter 4)(x x1 )(x x2 )(x0 x1 )(x0 x2 )Numerical Differentiation IR L Burden & J D Faires19 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasImportant Building BlocksSinceL0 (x) (x x1 )(x x2 )(x0 x1 )(x0 x2 )L′0 (x) 2x x1 x2(x0 x1 )(x0 x2 )we obtainNumerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires19 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasImportant Building BlocksSinceL0 (x) (x x1 )(x x2 )(x0 x1 )(x0 x2 )L′0 (x) 2x x1 x2(x0 x1 )(x0 x2 )we obtainIn a similar way, we find thatL′1 (x) L′2 (x) Numerical Analysis (Chapter 4)2x x0 x2(x1 x0 )(x1 x2 )2x x0 x1(x2 x0 )(x2 x1 )Numerical Differentiation IR L Burden & J D Faires19 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasImportant Building Blocks (Cont’d)Using these expressions for L′j (x), 1 j 2, the n 1-point formulaf ′ (xj ) nXk 0Numerical Analysis (Chapter 4)f (xk )L′k (xj ) nf (n 1) (ξ(xj )) Y(xj xk )(n 1)!k 0k 6 jNumerical Differentiation IR L Burden & J D Faires20 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasImportant Building Blocks (Cont’d)Using these expressions for L′j (x), 1 j 2, the n 1-point formulaf ′ (xj ) nXk 0f (xk )L′k (xj ) nf (n 1) (ξ(xj )) Y(xj xk )(n 1)!k 0k 6 jbecomes for n 2: f ′ (xj ) f (x0 ) 2xj x1 x22xj x0 x2 f (x1 )(x0 x1 )(x0 x2 )(x1 x0 )(x1 x2 ) 2Y2xj x0 x11 f (x2 ) f (3) (ξj )(xj xk )(x2 x0 )(x2 x1 )6k 0k 6 jfor each j 0, 1, 2, where ξj ξj (x).Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires20 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulas 2xj x1 x22xj x0 x2f (xj ) f (x0 ) f (x1 )(x0 x1 )(x0 x2 )(x1 x0 )(x1 x2 ) 2Y2xj x0 x11 f (3) (ξj )(xj xk ) f (x2 )(x2 x0 )(x2 x1 )6′ k 0k 6 jAssumptionNumerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires21 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulas 2xj x1 x22xj x0 x2f (xj ) f (x0 ) f (x1 )(x0 x1 )(x0 x2 )(x1 x0 )(x1 x2 ) 2Y2xj x0 x11 f (3) (ξj )(xj xk ) f (x2 )(x2 x0 )(x2 x1 )6 ′k 0k 6 jAssumptionThe 3-point formulas become especially useful if the nodes are equallyspaced, that is, whenx1 x0 hNumerical Analysis (Chapter 4)and x2 x0 2h,Numerical Differentiation Ifor some h 6 0R L Burden & J D Faires21 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulas 2xj x1 x22xj x0 x2f (xj ) f (x0 ) f (x1 )(x0 x1 )(x0 x2 )(x1 x0 )(x1 x2 ) 2Y2xj x0 x11 f (3) (ξj )(xj xk ) f (x2 )(x2 x0 )(x2 x1 )6 ′k 0k 6 jAssumptionThe 3-point formulas become especially useful if the nodes are equallyspaced, that is, whenx1 x0 hand x2 x0 2h,for some h 6 0We will assume equally-spaced nodes throughout the remainder ofthis section.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires21 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulas 2xj x1 x22xj x0 x2f (xj ) f (x0 ) f (x1 )(x0 x1 )(x0 x2 )(x1 x0 )(x1 x2 ) 2Y2xj x0 x11 (3) f (x2 ) f (ξj )(xj xk )(x2 x0 )(x2 x1 )6′ k 0k 6 jThree-Point Formulas (1/3)With xj x0 , x1 x0 h, and x2 x0 2h, the general 3-point formulabecomes 131h2′f (x0 ) f (x0 ) 2f (x1 ) f (x2 ) f (3) (ξ0 )h223Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires22 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulas 2xj x1 x22xj x0 x2f (xj ) f (x0 ) f (x1 )(x0 x1 )(x0 x2 )(x1 x0 )(x1 x2 ) 2Y2xj x0 x11 (3) f (x2 ) f (ξj )(xj xk )(x2 x0 )(x2 x1 )6 ′k 0k 6 jThree-Point Formulas (2/3)Doing the same for xj x1 gives 111h2f (x1 ) f (x0 ) f (x2 ) f (3) (ξ1 )h226′Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires23 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulas 2xj x1 x22xj x0 x2f (xj ) f (x0 ) f (x1 )(x0 x1 )(x0 x2 )(x1 x0 )(x1 x2 ) 2Y2xj x0 x11 (3) f (x2 ) f (ξj )(xj xk )(x2 x0 )(x2 x1 )6 ′k 0k 6 jThree-Point Formulas (3/3). . . and for xj x2 , we obtain 1 13h2f (x2 ) f (x0 ) 2f (x1 ) f (x2 ) f (3) (ξ2 )h 223′Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires24 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasThree-Point Formulas: Further SimplificationNumerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires25 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasThree-Point Formulas: Further SimplificationSince x1 x0 h and x2 x0 2h, these formulas can also beexpressed as 131h2′f (x0 ) f (x0 ) 2f (x0 h) f (x0 2h) f (3) (ξ0 )h223 2111hf ′ (x0 h) f (x0 ) f (x0 2h) f (3) (ξ1 )h226 1 13h2′f (x0 2h) f (x0 ) 2f (x0 h) f (x0 2h) f (3) (ξ2 )h 223Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires25 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasThree-Point Formulas: Further SimplificationSince x1 x0 h and x2 x0 2h, these formulas can also beexpressed as 131h2′f (x0 ) f (x0 ) 2f (x0 h) f (x0 2h) f (3) (ξ0 )h223 2111hf ′ (x0 h) f (x0 ) f (x0 2h) f (3) (ξ1 )h226 1 13h2′f (x0 2h) f (x0 ) 2f (x0 h) f (x0 2h) f (3) (ξ2 )h 223As a matter of convenience, the variable substitution x0 for x0 h isused in the middle equation to change this formula to an approximationfor f ′ (x0 ). A similar change, x0 for x0 2h, is used in the last equation.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires25 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasThree-Point Formulas: Further Simplification (Cont’d)This gives three formulas for approximating f ′ (x0 ):Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires26 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasThree-Point Formulas: Further Simplification (Cont’d)This gives three formulas for approximating f ′ (x0 ):1h2[ 3f (x0 ) 4f (x0 h) f (x0 2h)] f (3) (ξ0 )2h321hf ′ (x0 ) [ f (x0 h) f (x0 h)] f (3) (ξ1 ), and2h61h2f ′ (x0 ) [f (x0 2h) 4f (x0 h) 3f (x0 )] f (3) (ξ2 )2h3f ′ (x0 ) Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires26 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasThree-Point Formulas: Further Simplification (Cont’d)This gives three formulas for approximating f ′ (x0 ):1h2[ 3f (x0 ) 4f (x0 h) f (x0 2h)] f (3) (ξ0 )2h321hf ′ (x0 ) [ f (x0 h) f (x0 h)] f (3) (ξ1 ), and2h61h2f ′ (x0 ) [f (x0 2h) 4f (x0 h) 3f (x0 )] f (3) (ξ2 )2h3f ′ (x0 ) Finally, note that the last of these equations can be obtained from thefirst by simply replacing h with h, so there are actually only twoformulas.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires26 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasThree-Point Endpoint Formula1h2[ 3f (x0 ) 4f (x0 h) f (x0 2h)] f (3) (ξ0 )2h3where ξ0 lies between x0 and x0 2h.f ′ (x0 ) Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires27 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulasThree-Point Endpoint Formula1h2[ 3f (x0 ) 4f (x0 h) f (x0 2h)] f (3) (ξ0 )2h3where ξ0 lies between x0 and x0 2h.f ′ (x0 ) Three-Point Midpoint Formula1h2[f (x0 h) f (x0 h)] f (3) (ξ1 )2h6where ξ1 lies between x0 h and x0 h.f ′ (x0 ) Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires27 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulas(1)f ′ (x0 ) (2)f ′ (x0 ) 1h2[ 3f (x0 ) 4f (x0 h) f (x0 2h)] f (3) (ξ0 )2h31h2 (3)[f (x0 h) f (x0 h)] f (ξ1 )2h6CommentsNumerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires28 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulas(1)f ′ (x0 ) (2)f ′ (x0 ) 1h2[ 3f (x0 ) 4f (x0 h) f (x0 2h)] f (3) (ξ0 )2h31h2 (3)[f (x0 h) f (x0 h)] f (ξ1 )2h6CommentsAlthough the errors in both Eq. (1) and Eq. (2) are O(h2 ), the errorin Eq. (2) is approximately half the error in Eq. (1).Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires28 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulas(1)f ′ (x0 ) (2)f ′ (x0 ) 1h2[ 3f (x0 ) 4f (x0 h) f (x0 2h)] f (3) (ξ0 )2h31h2 (3)[f (x0 h) f (x0 h)] f (ξ1 )2h6CommentsAlthough the errors in both Eq. (1) and Eq. (2) are O(h2 ), the errorin Eq. (2) is approximately half the error in Eq. (1).This is because Eq. (2) uses data on both sides of x0 and Eq. (1)uses data on only one side.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires28 / 33
IntroductionGeneral Formulas3-pt FormulasSome useful three-point formulas(1)f ′ (x0 ) (2)f ′ (x0 ) 1h2[ 3f (x0 ) 4f (x0 h) f (x0 2h)] f (3) (ξ0 )2h31h2 (3)[f (x0 h) f (x0 h)] f (ξ1 )2h6CommentsAlthough the errors in both Eq. (1) and Eq. (2) are O(h2 ), the errorin Eq. (2) is approximately half the error in Eq. (1).This is because Eq. (2) uses data on both sides of x0 and Eq. (1)uses data on only one side.Note also that f needs to be evaluated at only two points inEq. (2), whereas in Eq. (1) three evaluations are needed.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires28 / 33
IntroductionGeneral Formulas3-pt FormulasThree-Point Midpoint Formula1h2[f (x0 h) f (x0 h)] f (3) (ξ1 )2h6where ξ1 lies between x0 h and x0 h.f ′ (x0 ) ySlope f 9(x 0)Slopex0 2 hNumerical Analysis (Chapter 4)x01[ f (x0 1 h) 2 f (x 0 2 h)]2hx0 1 hNumerical Differentiation IxR L Burden & J D Faires29 / 33
IntroductionGeneral Formulas3-pt FormulasExamples of five-point formulasFive-Point Midpoint Formulaf ′ (x0 ) 1[f (x0 2h) 8f (x0 h) 8f (x0 h) f (x0 2h)]12hh4 f (5) (ξ)30where ξ lies between x0 2h and x0 2h.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires30 / 33
IntroductionGeneral Formulas3-pt FormulasExamples of five-point formulasFive-Point Midpoint Formulaf ′ (x0 ) 1[f (x0 2h) 8f (x0 h) 8f (x0 h) f (x0 2h)]12hh4 f (5) (ξ)30where ξ lies between x0 2h and x0 2h.Five-Point Endpoint Formulaf ′ (x0 ) 1[ 25f (x0 ) 48f (x0 h) 36f (x0 2h)12hh4 16f (x0 3h) 3f (x0 4h)] f (5) (ξ)5where ξ lies between x0 and x0 4h.Numerical Analysis (Chapter 4)Numerical Differentiation IR L Burden & J D Faires30 / 33
Questions?
Reference Material
The Lagrange Polynomial: Theoretical Error BoundSuppose x0 , x1 , . . . , xn are distinct numbers in the interval [a, b] andf C n 1 [a, b]. Then, for each x in [a, b], a number ξ(x) (generallyunknown) between x0 , x1 , . . . , xn , and hence in (a, b), exists withf (x) P(x) f (n 1) (ξ(x))(x x0 )(x x1 ) · · · (x xn )(n 1)!where P(x) is the interpolating polynomial given byP(x) f (x0 )Ln,0 (x) · · · f (xn )Ln,n (x) nXf (xk )Ln,k (x)k 0Return to General Derivative Approximations1
Introduction to Numerical Differentiation Approximating a Derivative (Cont’d) To approximate f′(x0), suppose first that x0 (a,b), where f C2[a,b], and that x1 x0 h for some h 6 0 that is sufficiently small to ensure that x1 [a,b]. Numerical Analysis (Chapter 4) Numerical Diff
Automatic Differentiation Introductions Automatic Differentiation What is Automatic Differentiation? Algorithmic, or automatic, differentiation (AD) is concerned with the accurate and efficient evaluation of derivatives for functions defined by computer programs. No truncation errors are incurred, and the resulting numerical derivative
theory of four aspects: differentiation, functionalization, added value, and empathy. The purpose of differentiation is a strategy to distinguish oneself from competitors through technology or services, etc. It is mainly divided into three aspects: market differentiation, product differentiation and image differentiation.
Categories and Subject Descriptors: G.1.4 [Numerical Analysis]: Automatic Differentiation General Terms: Automatic Differentiation, Numerical Methods, MATLAB Additional Key Words and Phrases: algorithmic differentiation, scientific computation, applied mathemat-ics, chain rule, forward mode, overloading, source transformation ACM Reference Format:
These relationships yield the two-point Gauss-Legendre formula Because Gauss quadrature requires function evalutions at nonuniformly spaced points within the integration interval, it is not appropriate for . variable that transforms the infinite range to one that is finite. Numerical Differentiation Improve derivative estimates zDecrease the .
MATLAB has many tools that make this package well suited for numerical computations. This tutorial deals with the rootfinding, interpolation, numerical differentiation and integration and numerical solutions of the ordinary differential equations. Numerical methods
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simplifies automatic differentiation. There are other automatic differentiation tools, such as ADMAT. In 1998, Arun Verma introduced an automatic differentiation tool, which can compute the derivative accurately and fast [12]. This tool used object oriented MATLAB
This booklet offers practice for BEC Higher exam with a strong tie-in to Market Leader Advanced. It provides Reading and Writing tasks that will help familiarise students with the different task types and give practice for the exam, as well as act as revision practice for non-exam students. Each unit has one Reading and one Writing task, which are closely aligned to the vocabulary, grammar and .
