1. NUMERICAL DIFFERENTIATION AND INTEGRATION
ENGR 351
Numerical Methods for Engineers
Southern Illinois University Carbondale
College of Engineering
Dr. L.R. Chevalier
Dr. B.A. DeVantier

2. Copyright © 2003 by Lizette R. Chevalier
Permission is granted to students at Southern Illinois University at Carbondale
to make one copy of this material for use in the class ENGR 351, Numerical
Methods for Engineers. No other permission is granted.
All other rights are reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise, without
the prior written permission of the copyright owner.

4. Specific Study Objectives
Understand the derivation of the Newton-Cotes formulas
Recognize that the trapezoidal and Simpson’s 1/3 and 3/8 rules represent the areas of 1st, 2nd, and 3rd order polynomials
Be able to choose the “best” among these formulas for any particular problem

5. Specific Study Objectives
Recognize the difference between open and closed integration formulas
Understand the theoretical basis of Richardson extrapolation and how it is applied in the Romberg integration algorithm and for numerical differentiation

6. Specific Study Objectives
Recognize why both Romberg integration and Gauss quadrature have utility when integrating equations (as opposed to tabular or discrete data).
Understand basic finite difference approximations
Understand the application of high-accuracy numerical-differentiation.
Recognize data error on the processes of integration and differentiation.

7. Numerical Differentiation and Integration
Calculus is the mathematics of change.
Engineers must continuously deal with systems and processes that change, making calculus an essential tool of our profession.
At the heart of calculus are the related mathematical concepts of differentiation and integration.

8. Differentiation
Dictionary definition of differentiate - “to mark off by differences, distinguish; ..to perceive the difference in or between”
Mathematical definition of derivative - rate of change of a dependent variable with respect to an independent variable

9. f(x) Dy Dx x

10. Integration The inverse process of differentiation
Dictionary definition of integrate - “to bring together, as parts, into a whole; to unite; to indicate the total amount”
Mathematically, it is the total value or summation of f(x)dx over a range of x. In fact the integration symbol is actually a stylized capital S intended to signify the connection between integration and summation.

11. f(x) x

12. Mathematical Background

13. Mathematical Background

14. Overview Newton-Cotes Integration Formulas Integration of Equations
Trapezoidal rule
Simpson’s Rules
Unequal Segments
Open Integration
Integration of Equations
Romberg Integration
Gauss Quadrature
Improper Integrals

15. Overview Numerical Differentiation Applied problems
High accuracy formulas
Richardson’s extrapolation
Unequal spaced data
Uncertain data
Applied problems

16. Newton-Cotes Integration
Common numerical integration scheme
Based on the strategy of replacing a complicated function or tabulated data with some approximating function that is easy to integrate

17. Newton-Cotes Integration
Common numerical integration scheme
Based on the strategy of replacing a complicated function or tabulated data with some approximating function that is easy to integrate
fn(x) is an nth order
polynomial

18. The approximation of an integral by the area under
- a first order polynomial
- a second order polynomial

19. The approximation of an integral by the area under
- a first order polynomial
- a second order polynomial
We can also approximated the integral by using a
series of polynomials applied piece wise.

20. An approximation of an integral by the area under straight line segments.

21. An approximation of an integral by the area under straight line segments.

22. Newton-Cotes Formulas
Closed form - data is at the beginning and end of the limits of integration
Open form - integration limits extend beyond the range of data.

23. Trapezoidal Rule First of the Newton-Cotes closed integration formulas
Corresponds to the case where the polynomial is a first order

24. Trapezoidal Rule
A straight line can be represented as:

25. Trapezoidal Rule
Integrate this equation. Results in the trapezoidal rule.

26. Trapezoidal Rule
Recall the formula for computing the area of a trapezoid:
height x (average of the bases)
base
height
base

27. Trapezoidal Rule
The concept is the same but the trapezoid is on its side.
base
height
height
height
width
base

28. Error of the Trapezoidal Rule
This indicates that is the function being integrated is linear, the trapezoidal rule will be exact.
Otherwise, for section with second and higher order derivatives (that is with curvature) error can occur.
A reasonable estimate of x is the average value of b and a

29. Multiple Application of the Trapezoidal Rule
Improve the accuracy by dividing the integration interval into a number of smaller segments
Apply the method to each segment
Resulting equations are called multiple-application or composite integration formulas

30. Multiple Application of the Trapezoidal Rule
where there are n+1 equally spaced base points.

31. } } Multiple Application of the Trapezoidal Rule
We can group terms to express a general form } }
width
average height

32. } Multiple Application of the Trapezoidal Rule width average height
The average height represents a weighted average
of the function values
Note that the interior points are given twice the weight
of the two end points

33. Example
Evaluate the following integral using the trapezoidal rule and h = 0.1

34. Simpson’s 1/3 Rule
Corresponds to the case where the function is a second order polynomial

35. Simpson’s 1/3 Rule
Designate a and b as x0 and x2, and estimate f2(x) as a second order Lagrange polynomial

36. Simpson’s 1/3 Rule
After integration and algebraic manipulation, we get the following equations }
width
average height

37. Error Single application of Trapezoidal Rule.
Single application of Simpson’s 1/3 Rule

38. Multiple Application of Simpson’s 1/3 Rule

39. The odd points represent the middle term for each application
The odd points represent the middle term for each application. Hence carry the weight 4.
The even points are common to adjacent applications and are counted twice.
i=1 (odd) weight of 4
i=2 (even) weight of 2
f(x) x

40. Simpson’s 3/8 Rule
Corresponds to the case where the function is a third order polynomial

41. Integration of Unequal Segments
Experimental and field study data is often unevenly spaced
In previous equations we grouped the term (i.e. hi) which represented segment width.

42. Integration of Unequal Segments
We should also consider alternately using higher order equations if we can find data in consecutively even segments
trapezoidal
rule

43. Integration of Unequal Segments
We should also consider alternately using higher order equations if we can find data in consecutively even segments
1/3
rule
trapezoidal
rule

44. Integration of Unequal Segments
We should also consider alternately using higher order equations if we can find data in consecutively even segments
1/3
rule
trapezoidal
rule
3/8
rule

45. Integration of Unequal Segments
We should also consider alternately using higher order equations if we can find data in consecutively even segments
trapezoidal
rule
1/3
rule
trapezoidal
rule
3/8
rule

46. Example
Integrate the following using the trapezoidal rule, Simpson’s 1/3 Rule, a multiple application of the trapezoidal rule with n=2 and Simpson’s 3/8 Rule. Compare results with the analytical solution.

47. Integration of Equations
Integration of analytical as opposed to tabular functions
Romberg Integration
Richardson’s Extrapolation
Romberg Integration Algorithm
Gauss Quadrature
Improper Integrals

48. Richardson’s Extrapolation
Use two estimates of an integral to compute a third more accurate approximation
The estimate and error associated with a multiple application trapezoidal rule can be represented generally as:
I = I(h) + E(h)
where I is the exact value of the integral
I(h) is the approximation from an n-segment application
E(h) is the truncation error
h is the step size (b-a)/n

49. Make two separate estimates using step sizes of h1 and h2 .
I(h1) + E(h1) = I(h2) + E(h2)
Recall the error of the multiple-application of the trapezoidal
rule
Assume that is constant regardless of the step size

50. Substitute into previous equation:
I(h1) + E(h1) = I(h2) + E(h2)

51. Thus we have developed an estimate of the truncation
error in terms of the integral estimates and their step
sizes. This estimate can then be substituted into:
I = I(h2) + E(h2)
to yield an improved estimate of the integral:

52. What is the equation for the special case where the interval is halved?
i.e. h2 = h1 / 2

54. Example
Use Richardson’s extrapolation to evaluate:

55. Romberg Integration
We can continue to improve the estimate by successive
halving of the step size to yield a general formula:
k = 2; j = 1
Note:
the subscripts
m and l refer to
more and less
accurate estimates

56. Gauss Quadrature
Extend the area
under the straight
line
f(x) x
f(x) x

57. Method of Undetermined Coefficients
Recall the trapezoidal rule
Before analyzing
this method,
answer this
question.
What are two
functions that
should be evaluated
exactly
by the trapezoidal
rule?
This can also be expressed as
where the c’s are constant

58. The two cases that should be evaluated exactly
by the trapezoidal rule: 1) y = constant
2) a straight line
f(x)
y = 1
f(x)
y = x
-(b-a)/2 x
(b-a)/2
-(b-a)/2 x
(b-a)/2

59. Thus, the following equalities should hold.
FOR y=1
since f(a) = f(b) =1
FOR y =x
since f(a) = x =-(b-a)/2
and
f(b) = x =(b-a)/2

60. Evaluating both integrals
For y = 1
For y = x
Now we have two equations and two unknowns, c0 and c1.
Solving simultaneously, we get :
c0 = c1 = (b-a)/2
Substitute this back into:

61. We get the equivalent of the trapezoidal rule.
DERIVATION OF THE TWO-POINT
GAUSS-LEGENDRE FORMULA
Lets raise the level of sophistication by:
- considering two points between -1 and 1
- i.e. “open integration”

62. Previously ,we assumed that the equation
f(x)
-1 x x1 1 x
Previously ,we assumed that the equation
fit the integrals of a constant and linear function.
Extend the reasoning by assuming that it also fits the integral of a parabolic and a cubic function.

63. We now have four
equations and four
unknowns
c0 c1 x0 and x1
What equations are
you solving?

64. f(xi) is either 1, xi, xi2 or xi3
Solve these equations simultaneously

65. This results in the following
The interesting result is that the integral can be estimated by the simple addition of the function values at

66. What if we aren’t integrating from –1 to 1?
A simple change in variables can be use to translate other limits.
Assume that the new variable xd is related to the
original variable x in a linear fashion.
x = a0 + a1xd
Let the lower limit x = a correspond to xd = -1 and the upper limit x=b correspond to xd=1
a = a0 + a1(-1) b = a0 + a1(1)

67. a = a0 + a1(-1) b = a0 + a1(1) SOLVE THESE EQUATIONS SIMULTANEOUSLY
substitute

68. These equations are substituted for x and dx respectively.
Let’s do an example to appreciate the theory
behind this numerical method.

69. Example
Estimate the following using two-point Gauss Legendre:

70. Higher-Point Formulas
For two point, we determined that c0 = c1= 1
For three point:
c0 = x0=-0.775
c1= x1=0.0
c2= x2=0.775

71. Higher-Point Formulas
Your text goes on to provide additional weighting
factors (ci’s) and function arguments (xi’s)
in Table 8.5 p. 593

72. Numerical Differentiation
Forward finite divided difference
Backward finite divided difference
Center finite divided difference
All based on the Taylor Series

73. Forward Finite Difference

74. Forward Divided Difference
f(x)
What is derivative at this
point?
(xi, yi) x

75. Forward Divided Difference
f(x)
(x i+1,y i+1)
(xi, yi)
Determine a second point
base on Dx (h) x

76. Forward Divided Difference
f(x)
(x i+1,y i+1)
How does
this compare
to the actual
first derivative
at xi?
estimate
(xi, yi) x

77. Forward Divided Difference
f(x)
(x i+1,y i+1)
actual
estimate
(xi, yi) x

78. Forward Divided Difference
Error is proportional to
the step size
first forward divided difference
O(h2) error is proportional to the square of the step size
O(h3) error is proportional to the cube of the step size

79. f(x)
actual
(xi,yi)
estimate
(xi-1,yi-1) x

80. Backward Difference Approximation of the
First Derivative
Expand the Taylor series backwards
The error is still O(h)

81. Centered Difference Approximation of the
First Derivative
Subtract backward difference approximation
from forward Taylor series expansion

82. f(x)
actual
(xi+1,yi+1)
(xi,yi)
estimate
(xi-1,yi-1) x

83. f(x) f(x) f(x) x x f(x) x x forward finite divided difference approx.
true derivative x x
f(x)
f(x)
backward
finite divided
difference approx.
centered
finite divided
difference approx. x x

84. Numerical Differentiation
You should be familiar with the following Tables in your text
Table 7.1: Common Finite Difference Formulas
Table 7.2: Higher order finite difference formulas

85. Richardson Extrapolation
Two ways to improve derivative estimates
decrease step size
use a higher order formula that employs more points
Third approach, based on Richardson extrapolation, uses two derivatives estimates to compute a third, more accurate approximation

86. Richardson Extrapolation
For a centered difference
approximation with
O(h2) the application of
this formula will yield
a new derivative estimate
of O(h4)

87. Example
Given the following function, use Richardson’s extrapolation to determine the derivative at 0.5.
f(x) = -0.1x x x x +1.2
Note:
f(0) = 1.2
f(0.25) =1.1035
f(0.75) = 0.636
f(1) = 0.2

88. Derivatives of Unequally Spaced Data
Common in data from experiments or field studies
Fit a second order Lagrange interpolating polynomial to each set of three adjacent points, since this polynomial does not require that the points be equi-spaced
Differentiate analytically

89. Derivative and Integral Estimates for Data with Errors
In addition to unequal spacing, the other problem related to differentiating empirical data is measurement error
Differentiation amplifies error
Integration tends to be more forgiving
Primary approach for determining derivatives of imprecise data is to use least squares regression to fit a smooth, differentiable function to the data
In absence of other information, a lower order polynomial regression is a good first choice

91. Specific Study Objectives
Understand the derivation of the Newton-Cotes formulas
Recognize that the trapezoidal and Simpson’s 1/3 and 3/8 rules represent the areas of 1st, 2nd, and 3rd order polynomials
Be able to choose the “best” among these formulas for any particular problem

92. Specific Study Objectives
Recognize the difference between open and closed integration formulas
Understand the theoretical basis of Richardson extrapolation and how it is applied in the Romberg integration algorithm and for numerical differentiation

93. Specific Study Objectives
Recognize why both Romberg integration and Gauss quadrature have utility when integrating equations (as opposed to tabular or discrete data).
Understand the application of high-accuracy numerical-differentiation.
Recognize data error on the processes of integration and differentiation.

94. …..end of lecture