1. Numerical Methods Lecture 14 Differentiation-Continuous Functions
Dr. S. M. Lutful Kabir Visiting Professor, BRAC University
& Professor (On-leave), BUET

2. Forward Difference Approximation
For a finite

3. Forward Difference Approximation
Figure 1 Graphical Representation of forward difference approximation of first derivative.

4. Derive the forward difference approximation from Taylor series
Taylor’s theorem says that if you know the value of a function ‘f’ at a point xi and all its derivatives at that point, provided the derivatives are continuous between xi and xi+1, then
Substituting for convenience

5. Derive the forward difference approximation from Taylor series (cont.)
The term shows that the error in the approximation is of the order of
Can you now derive from Taylor series the formula for backward divided difference approximation of the first derivative?
It can be shown that both forward and backward divided difference approximation of the first derivative are accurate on the order of
Can we get better approximation?
Yes, another method to approximate the first derivative is called the Central difference approximation of the first derivative.

6. Backward Difference Approximation of the First Derivative
We know
For a finite
If is chosen as a negative number,

7. Backward Difference Approximation of the First Derivative (continued)
This is a backward difference approximation as you are taking a point backward from x. To find the value of at we may choose another point behind as
This gives
where,

8. Backward Difference Approximation of the First Derivative (continued)
x-Δx
f(x)
Figure 2 Graphical Representation of backward difference approximation of first derivative

9. Derive the backward difference approximation from Taylor series
Taylor’s theorem says that if you know the value of a function ‘f’ at a point xi and all its derivatives at that point, provided the derivatives are continuous between xi and xi-1, then
Substituting for convenience

10. Derive the Central difference approximation from Taylor series (cont.)
Subtracting equation (2) from the first (1),

11. Central Divided Difference
Hence showing that we have obtained a more accurate formula as the
error is of the order of x
f(x)
x-Δx x x+Δx
Figure 3 Graphical Representation of central difference approximation of first derivative

12. Example 1 The velocity of a rocket is given by where
is given in m/s and
is given in seconds.
Use forward, backward and central divided difference approximation of the first derivative of to calculate the acceleration at Use a step size of
Find the absolute relative true error for part (a).

13. Comparison of FDD, BDD, CDD
The results from the three difference approximations are given in Table 1.
Table 1 Summary of a (16) using different divided difference approximations
Type of Difference
Approximation
Forward
Backward
Central
30.475
28.915
29.695
2.6967
2.5584

14. Effect of step size on the accuracy
%Error 2 1
0.5
0.25
0.125
28.915
29.289
29.480
29.577
29.625
1.2792

15. Finite Difference Approximation of Higher Derivatives
One can use Taylor series to approximate a higher order derivative.
For example, to approximate
, the Taylor series for
(3)
where
(4)
where

16. Finite Difference Approximation of Higher Derivatives (cont.)
Subtracting 2 times equation (4) from equation (3) gives
(5)

17. Higher order accuracy of higher order derivatives
The formula given by equation (5) is a forward difference
approximation of 2nd derivative and has error
of the order of
Can we get a formula that has a better accuracy? We can get the
Central difference approximation of the second derivative.
The Taylor series for
(6)
where

18. Higher order accuracy of higher order derivatives (cont.)
(7)
where
Adding equations (6) and (7), gives

19. Example 2 The velocity of a rocket is given by
Use central difference approximation of second derivative of at Use a step size of

20. Program
clear all; clc; t=1; delta=0.2; decrement=0.04; disp(' First Derivative') disp(' ') disp( ' Delta Actual Forw Back Central %Err_F %Err_B %Err_C') disp( ' ')

21. Program (continued)
for i=1:5 F1F=(f(t+delta)-f(t))/delta; F1B=(f(t)-f(t-delta))/delta; F1C=(f(t+delta)-f(t-delta))/(2*delta); F1A=f1(t); Y(1)=delta; Y(2)=F1A; Y(3)=F1F; Y(4)=F1B; Y(5)=F1C; Y(6)=abs((F1F-F1A)*100/F1A); Y(7)=abs((F1B-F1A)*100/F1A); Y(8)=abs((F1C-F1A)*100/F1A); disp(Y); delta=delta-decrement; end

22. Program (continued) delta=0.2; disp(' ‘); disp(' Second Derivative')
disp( ' Delta Actual Forw Back Central %Err_F %Err_B %Err_C')
disp( ' ')

23. Program (continued) for i=1:5
F2F=(f(t)-2*f(t+delta)+f(t+2*delta))/(delta^2);
F2B=(f(t-2*delta)-2*f(t-delta)+f(t))/(delta^2);
F2C=(f(t+delta)-2*f(t)+f(t-delta))/(delta^2);
F2A=f2(t);
Y(1)=delta;
Y(2)=F2A;
Y(3)=F2F;
Y(4)=F2B;
Y(5)=F2C;
Y(6)=abs((F2F-F2A)*100/F2A);
Y(7)=abs((F2B-F2A)*100/F2A);
Y(8)=abs((F2C-F2A)*100/F2A);
disp(Y);
delta=delta-decrement;
end

24. Thanks