1. Introduction to Numerical Methods Mathematical Procedures

2. Mathematical Procedures
Nonlinear Equations
Differentiation
Simultaneous Linear Equations
Curve Fitting
Interpolation
Regression
Integration
Ordinary Differential Equations
Other Advanced Mathematical Procedures:
Partial Differential Equations
Optimization
Fast Fourier Transforms

3. How much of the floating ball is under water?
Nonlinear Equations
How much of the floating ball is under water?
Diameter=0.11m
Specific Gravity=0.6

4. Nonlinear Equations
How much of the floating ball is under the water?

5. Differentiation
What is the acceleration at t=7 seconds?

6. Differentiation What is the acceleration at t=7 seconds? Time (s) 5 812
Vel (m/s)
106
177
600

7. Simultaneous Linear Equations
Find the velocity profile, given
Time (s) 5 8 12
Vel (m/s)
106
177
600
Three simultaneous linear equations

8. Interpolation What is the velocity of the rocket at t=7 seconds?
Time (s) 5 8 12
Vel (m/s)
106
177
600

9. What is Interpolation ?
Given (x0,y0), (x1,y1), …… (xn,yn), find the value of ‘y’ at a value of ‘x’ that is not given.

10. Interpolants Evaluate Differentiate, and Integrate.
Polynomials are the most common choice of interpolants because they are easy to:
Evaluate
Differentiate, and
Integrate.

11. Newton’s Divided Difference Method
Linear interpolation: Given pass a linear interpolant through the data
where

12. Figure 2: Velocity vs. time data
Example
The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for linear interpolation. t
v(t) s
m/s 10
227.04 15
362.78 20
517.35
22.5
602.97 30
901.67
Figure 2: Velocity vs. time data
for the rocket example
Table 1: Velocity as a
function of time

13. Linear Interpolation

14. Linear Interpolation (contd)

15. Quadratic Interpolation

16. Figure 2: Velocity vs. time data
Example
The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for quadratic interpolation. t
v(t) s
m/s 10
227.04 15
362.78 20
517.35
22.5
602.97 30
901.67
Figure 2: Velocity vs. time data
for the rocket example
Table 1: Velocity as a
function of time

17. Quadratic Interpolation (contd)

18. Quadratic Interpolation (contd)

19. Quadratic Interpolation (contd)

20. General Form
where
Rewriting

21. General Form

22. General form

23. Figure 2: Velocity vs. time data
Example
The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for cubic interpolation. t
v(t) s
m/s 10
227.04 15
362.78 20
517.35
22.5
602.97 30
901.67
Figure 2: Velocity vs. time data
for the rocket example
Table 1: Velocity as a
function of time

24. Example The velocity profile is chosen as
we need to choose four data points that are closest to

25. Example

26. Example

27. Comparison Table

28. Distance from Velocity Profile
Find the distance covered by the rocket from t=11s to
t=16s ?

29. Acceleration from Velocity Profile
Find the acceleration of the rocket at t=16s given that

30. Regression
Thermal expansion coefficient data for cast steel

31. Regression (cont)

32. Integration
Finding the diametric contraction in a steel shaft when dipped in liquid nitrogen.

33. Ordinary Differential Equations
How long does it take a trunnion to cool down?