EE3561_Unit 6(c)AL-DHAIFALLAH14351 EE 3561 : Computational Methods Unit 6 Numerical Differentiation Dr. Mujahed AlDhaifallah ( Term 342)

EE3561_Unit 6(c)AL-DHAIFALLAH14352 Lecture 17 Numerical Differentiation  First order derivatives  High order derivatives  Richardson Extrapolation  Examples

EE3561_Unit 6(c)AL-DHAIFALLAH14353 Motivation How do you evaluate the derivative of a tabulated function. How do we determine the velocity and acceleration from tabulated measurements. Time (second) Displacemen t (meters) Calculus is the mathematics of change. Because engineers must continuously deal with systems and processes that change, they always need to estimate the value of f '(x) for a given function f(x).. Standing in the heart of calculus are the mathematical concepts of differentiation and integration:

EE3561_Unit 6(c)AL-DHAIFALLAH14354 Recall Numerical differentiation and integration The derivative represents the rate of change of a dependent variable with respect to an independent variable. The difference approximation If x is allowed to approach zero, the difference becomes a derivative

EE3561_Unit 6(c)AL-DHAIFALLAH14355 Difference Formulas

EE3561_Unit 6(c)AL-DHAIFALLAH14356 Recall

EE3561_Unit 6(c)AL-DHAIFALLAH14357 Three formula

EE3561_Unit 6(c)AL-DHAIFALLAH14358 Forward Difference formula

EE3561_Unit 6(c)AL-DHAIFALLAH14359 Central Difference formula

EE3561_Unit 6(c)AL-DHAIFALLAH The Three formula (revisited)

EE3561_Unit 6(c)AL-DHAIFALLAH Higher Order Formulas

EE3561_Unit 6(c)AL-DHAIFALLAH Other Higher Order Formulas

EE3561_Unit 6(c)AL-DHAIFALLAH HIGH-ACCURACY DIFFERENTIATION FORMULAS The forward Taylor series expansion is: From this, we can write Substitute the second derivative approximation into the formula to yield: By collecting terms

EE3561_Unit 6(c)AL-DHAIFALLAH HIGH-ACCURACY DIFFERENTIATION FORMULAS  Inclusion of the 2nd derivative term has improved the accuracy to O(h 2 ).  This is the forward divided difference formula for the first derivative.

EE3561_Unit 6(c)AL-DHAIFALLAH Forward Formulas

EE3561_Unit 6(c)AL-DHAIFALLAH Backward Formulas

EE3561_Unit 6(c)AL-DHAIFALLAH Centered Formulas

EE3561_Unit 6(c)AL-DHAIFALLAH Example Estimate f '(1) for f(x) = e x + x using the centered formula of O(h 4 ) with h = Solution From Table 23.3:

EE3561_Unit 6(c)AL-DHAIFALLAH substituting the values results in

EE3561_Unit 6(c)AL-DHAIFALLAH Numerical Differentiation  Richardson Extrapolation  Examples

EE3561_Unit 6(c)AL-DHAIFALLAH Richardson Extrapolation

EE3561_Unit 6(c)AL-DHAIFALLAH Richardson Extrapolation

EE3561_Unit 6(c)AL-DHAIFALLAH Richardson Extrapolation Table D(0,0)=Φ(h) D(1,0)=Φ(h/2)D(1,1) D(2,0)=Φ(h/4)D(2,1)D(2,2) D(3,0)=Φ(h/8)D(3,1)D(3,2)D(3,3)

EE3561_Unit 6(c)AL-DHAIFALLAH Richardson Extrapolation Table

EE3561_Unit 6(c)AL-DHAIFALLAH Example

EE3561_Unit 6(c)AL-DHAIFALLAH Example First Column

EE3561_Unit 6(c)AL-DHAIFALLAH Example Richardson Table

EE3561_Unit 6(c)AL-DHAIFALLAH Example Richardson Table This is the best estimate of the derivative of the function All entries of the Richardson table are estimates of the derivative of the function. The first column are estimates using the central difference formula with different h.

EE3561_Unit 6(c)AL-DHAIFALLAH Summary  Several formulas are available to determine first order, second order or higher order derivatives  Richardson Extrapolation provides high accuracy estimates of the first order derivative