Week08&09_Numerical Integration & Differentiation

WEEK 8 & 9

Numerical Integration and Differentiation Newton-Cotes integration formulas:

Trapezoidal rule

Simpson’s Rule

Integration with unequal segments

Numerical differentiation: High accuracy differentiation formula

Richardson extrapolation 2

At the end of this topic, the students will be able:

To identify and apply the methods outlines

to integrate formulae and data set

To identify and apply the methods outlines to

differentiate formulae and data set

LESSON OUTCOMES

4

Derivative – represents the rate of change of a dependent variable with respect to an independent variable given by mathematical definition begin with a difference approximation: If Δx is allowed to approach zero, the differences becomes a derivative

x

xfxxf

x

y ii

)()(

x

xfxxf

dx

dy ii

x

)()(lim

0

y and f(x) dependent variable

x is dependent variable

dy/dx : first derivative of y with

respect to x evaluated at xi.

derivative slope of the

tangent to the curve at xi

DIFFERENTIATION

5

INTEGRATION

Integrate – to bring together, as parts, into a whole; to unite; to indicate the total amount…. – Mathematical definition represented by which stands for the integral of the function f(x) with respect to the independent variable x, evaluated between the limits x=a to x=b.

b

adxxfI )(

The integral is

equivalent to the

area under the curve

6

• The function to be differentiated or integrated will typically be in one of the following three forms:

– A simple continuous function such as polynomial, an exponential, or a trigonometric function.

– A complicated continuous function that is difficult or impossible to differentiate or integrate directly.

– A tabulated function where values of x and f(x) are given at a number of discrete points, as is often the case with experimental or field data.

NONCOMPUTER METHODS FOR

DIFFERENTIATION AND INTEGRATION

7

A noncomputer method for determining derivatives from data : equal-area graphical differentiation

Centered finite divided differences are used to estimates the derivative for each interval between the data points

Then, these values are plotted as a stepped curve versus x. That is, it is drawn so that visually, the positive and negative area are balance.

Then, values of dy/dx can be read off the smooth curve at given valus of x.

8

A noncomputer method for determining integration from data :

The use of grid to approximate an integral

The use of rectangles or strips to approximate an integral

Application of a numerical integration or quadrature

method

(a) A complicated, continous function

(b) Table of discrete values of f(x) generated

from the function

(c) Use of numerical method to estimate the integral on the basis of

the discrete points.

9

Newton-Cotes Integration Formulas

• The Newton-Cotes formulas are the most common numerical integration schemes.

• They are based on the strategy of replacing a complicated function or tabulated data with an approximating function that is easy to integrate:

n

n

n

nn

n

b

a

n

b

a

xaxaxaaxf

xf

dxxfdxxfI

1

110)(

polynomial of form )( where

)()(

10

(a) First-order polynomial (straight

line) is used as an approximation

(b) A parabola is employed as an approximation

The integral can also be approximated using a series of polynomials applied piecewise to the function or data over segments of constant length.

The approximation of an integral by the area under three straight-

line segments

11

• Closed and open forms of the Newton-Cotes formulas are available.

Closed forms where the data points at the beginning and end of the limits of integration are known.

Open forms where have integration limits that extend beyond the range of the data. This formulas not generally used for difinite integration. However, they are utilized for evaluating improper integrals and for solution of ODE.

12

THE TRAPEZOIDAL RULE

• The Trapezoidal rule is the first of the Newton-Cotes closed integration formulas, corresponding to the case where the polynomial is first order:

• The area under this first order polynomial is an estimate of

the integral of f(x) between the limits of a and b:

b

a

b

a

dxxfdxxfI )()( 1

2

)()()(

bfafabI

Trapezoidal Rule

13

)()()(

)()(

)()()(

)()(

)()()()(

1

0

01

0101

01

01

0

01

axab

afbfafxf

xxxx

xfxfxfxf

xx

xfxf

xx

xfxf

b

a

b

a

dxxfdxxfI )()( 1

b

a

dxaxab

afbfafI )(

)()()(

2

)()()(

bfafabI

Derivation:

Refer Box 21.1

(page 603) text book

14

• Geometrically, the trapezoidal rule is equivalent to approximating the area of the trapezoid under the straight line connecting f(a) and f(b).

(a) Area of trapezoid ≈ height x average bases

(b) I ≈ width x average height

I ≈ (b – a) x average height

Average height = f(a) + f(b)

2

2

)()()(

bfafabI

15

Error of the Trapezoidal Rule

• When we employ the integral under a straight line segment to approximate the integral under a curve, error may be substantial:

where x lies somewhere in the interval from a to b.

• If the function being integrated is linear, the trapezoidal rule will be exact.

• Otherwise, for functions with second and higher order derivatives (curvature), some error can occur.

33 )(''12

1))((''

12

1abfabfEt x

Derivation and Error Estimate:

Refer Box 21.2

(page 606) text book

16

The Multiple-Application Trapezoidal Rule

• One way to improve the accuracy of the trapezoidal rule is to divide the integration interval from a to b into a number of segments and apply the method to each segment.

• The areas of individual segments can then be added to yield the integral for the entire interval.

• The resulting equations are called multiple-application or composite integration formulas.

17

n

n

x

x

x

x

x

x

n

dxxfdxxfdxxfI

n

xbxan

abh

1

2

1

1

0

)()()(

equally segments

0

2

)()(

2

)()(

2

)()( 12110 nn xfxfh

xfxfh

xfxfhI

1

1

0 )()(2)(2

n

i

ni xfxfxfh

I

n

xfxfxf

abI

n

i

ni

2

)()(2)(

)(

1

1

0

General format for multiple

application integrals

18

Error of the multiple-application Trapezoidal Rule

• An error for multiple-application trapezoidal rule can be obtained by summing the individual errors for each segment:

fn

abE

fnf

fn

abE

a

i

n

i

it

2

3

13

3

12

)(

)(

)(12

)(

x

x

19

Example

Given f(x)=0.2+25x–200x2+675x3–900x4+400x5

Use following method to estimate the integral of given equation from a = 0 to b = 0.8. a) Single application of the Trapezoidal Rule b) Multiple application of the Trapezoidal Rule (2 segments)

Given that the exact value of the integral that determined analytically is 1.640533. Find true percent relative error and estimated error of the Trapezoidal Rule.

20

THE SIMPSON’S RULES

• More accurate estimate of an integral is obtained if a high-order polynomial is used to connect the points. The formulas that result from taking the integrals under such polynomials are called Simpson’s rules.

(a) Graphical depiction of Simpson’s 1/3 rule: It consists of taking the area under a parabola connecting three points.

(b) Graphical depiction of Simpson’s 3/8 rule: It consists of taking the area under a cubic equation connecting four points.

21

Simpson’s 1/3 Rule

• Results when a second-order interpolating polynomial is used.

2

a)(b xb; and abetween midway point the xand xb ,xa where

6

)()(4)()(

or

2)()(4)(

3

)())((

))(()(

))((

))(()(

))((

))((

,polynomial Lagrangeorder -second using drepresente is )(

)()(

1120

210

210

2

1202

101

2101

200

2010

21

2

202

2

0

xfxfxfabI

ab

n

abhxfxfxf

hI

dxxfxxxx

xxxxxf

xxxx

xxxxxf

xxxx

xxxxI

xf

xbxadxxfdxxfI

x

x

b

a

b

a

• Single segment application of Simpson’s 1/3 rule has a truncation error of:

• Simpson’s 1/3 rule is more accurate than trapezoidal rule.

bafab

Et

xx )(2880

)( )4(5

22

• Just as the trapezoidal rule, Simpson’s rule can be improved by dividing the integration interval into a number of segments of equal width.

• Yields accurate results and considered superior to trapezoidal rule for most applications.

• However, it is limited to cases where values are equispaced. • Further, it is limited to situations where there are an even

number of segments and odd number of points.

The Multiple-Application Simpson’s 1/3 Rule

)4(

4

5

1

5,3,1

2

6,4,2

0

180

)(

3

)()(2)(4)(

)(

fn

abE

n

xfxfxfxf

abI

a

n

i

n

n

j

ji

23

Example

Given f(x)=0.2+25x–200x2+675x3–900x4+400x5

Use following method to estimate the integral of given equation from a = 0 to b = 0.8. a) Single application of the Simpson’s 1/3 Rule b) Multiple application of the Simpson’s 1/3 Rule with n=4.

Given that the exact value of the integral that determined analytically is 1.640533. Find true percent relative error and estimated error of the Simpson’s Rule.

24

Simpson’s 3/8 Rule

• An odd-segment-even-point formula used in conjunction with the 1/3 rule to permit evaluation of both even and odd numbers of segments.

• The equation called Simpson’s 3/8 rule because h is multiplied by 3/8.

)(6480

)(

8

)()(3)(3)(

)( )()(3)(3)(

8

3

)()(

)4(5

3210

3210

3

xfab

E

xfxfxfxfabI

n

abhxfxfxfxf

hI

dxxfdxxfI

t

b

a

b

a

More accurate

25

Illustration of how Simpson’s 1/3 and 3/8 rules can be applied in tandem to handle multiple applications with odd numbers of intervals.

26

Example

Given f(x)=0.2+25x–200x2+675x3–900x4+400x5

Use following method to estimate the integral of given equation from a = 0 to b = 0.8. a) Simpson’s 3/8 Rule b) Use Simpson’s 3/8 Rule in conjuction with Simpson’s 1/3 Rule

to integrate the same function for five segments.

Given that the exact value of the integral that determined analytically is 1.640533. Find true percent relative error.

27

INTEGRATION WITH UNEQUAL SEGMENTS

• In practice, many situations deal with unequal-sized segments.

• In that case, one method is apply the trapezoidal rule to each segment and sum the results.

• In other method, inclusion of trapezoidal rule and Simpson's Rule for unequal segments.

2

)()(

2

)()(

2

)()( 12110 nn xfxfh

xfxfh

xfxfhI

28

Example Given data for f(x)=0.2+25x–200x2+675x3–900x4+400x5.

Given that the exact value of the integral that determined

analytically is 1.640533. Find true percent relative error.

Use following method to determine the integral for this data. a) Trapezoidal rule with unequal segments b) Inclusion of Trapezoidal and Simpson’s rules

x f(x) x f(x)

0.0 0.200000 0.44 2.842985

0.12 1.309729 0.54 3.507297

0.22 1.305241 0.64 3.181929

0.32 1.743393 0.70 2.363000

0.36 2.074903 0.80 0.232000

0.40 2.456000

Forward finite divided difference

Backward finite divided difference

Centered finite divided difference

29

Numerical Differentiation

30

• High-accuracy divided difference formulas can be generated by including additional terms from the Taylor series expansion. • For example, the forward Taylor series expansion can be written as

)()()(

)('

: terms)derivativehigher and second (excluding

difference forwardfirst So,

)(2

)(")()()('

for solved becan which

...2

)(")(')()(

1

21

2

1

hOh

xfxfxf

hOhxf

h

xfxfxf

hxf

hxfxfxf

iii

iiii

iiii

HIGH-ACCURACY DIFFERENTIATION FORMULAS

31

)(2

)(3)(4)()('

terms,collectingby or,

)(2

)()(2)()()()('

yield todifference forwardfirst into

)()()(2)(

)("

,derivative second theof

ion approximat thengsubstitutiby termderivative secondFor

212

2

2

121

2

12

hOh

xfxfxfxf

hOhh

xfxfxf

h

xfxfxf

hOh

xfxfxfxf

iiii

iiiiii

iiii

* This equation incorporates more terms of Taylor series expansion and more accurate.

Refer Text book

Page 92 & 93

32

Forward finite-divided difference formulas: two versions are presented for each derivative. The latter version incorporates more terms of Taylor Series expansion and consequently more accurate.

h4

33

Backward finite-divided difference formulas: two versions are presented for each derivative. The latter version incorporates more terms of Taylor Series expansion and consequently more accurate.

34

Centered finite-divided difference formulas: two versions are presented for each derivative. The latter version incorporates more terms of Taylor Series expansion and consequently more accurate.

35

Example

Estimated the first derivative of f(x) = – 0.1x4 – 0.15x3 – 0.5x2 – 0.25x + 1.2 at x = 0.5 using finite divided differences and a step size of h = 0.25.

a) Use forward and backward differences approximations

of O(h) and centered difference approximation of O(h2).

b) Use high-accuracy formulas of finite divided differences.

36

2.0 1

6363.0 75.0

925.0 5.0

1035.1 25.0

2.1 0

n,calculatio From

22

11

11

22

ii

ii

ii

ii

ii

xfx

xfx

xfx

xfx

xfx

Forward

O(h)

Backward

O(h)

Centered

O(h2)

Estimate -1.155 -0.714 -0.934

ε t (%) -26.5 21.7 -2.4

Forward

O(h2)

Backward

O(h2)

Centered

O(h4)

Estimate -0.8594 -0.8781 -0.9125

ε t (%) 5.82 3.77 0

Solution

37

RICHARDSON EXTRAPOLATION

• Richardson extrapolation is a method uses two estimates of an integral to compute a third, more accurate approximation.

• Two ways to improve derivative estimates when employing finite divided difference:

1) decrease the step size 2) use a higher-order formula that employs

more point.

)](3

1)(

3

4

s,derivativefor itten fashion wrSimilar

)](3

1)(

3

4

)]()([1)2(

1)(

)]()([1))/2((

1)(

,2/ wherecase for the written formula This

.h and h sizes step twousing estimates integral are )( and )( where

)]()([1)/(

1)(

estimate, integral oft improvemen Formula

12

12

1222

122

11

2

12

2121

122

21

2

hDhDD

hIhII

hIhIhII

hIhIhh

hII

hh

hIhI

hIhIhh

hII

I

For centered difference

approximations with O(h2),

the application of this

formula will yield a new

derivative estimate of O(h4).

39

Example

Estimated the first derivative of f(x) = – 0.1x4 – 0.15x3 – 0.5x2 – 0.25x + 1.2 at x = 0.5 employing step sizes of h1 = 0.5 and h2 = 0.25. Compute an improved estimate with Richardson extrapolation. The true value is -0.9125.

Solution The first derivative estimates can be computed with centered differences

%0 9125.0)0.1(3

1)9344.0(

3

4)(

3

1)(

3

4

%4.2 9344.0)25.0(2

1035.16363.0

2

)()()(')25.0(

%6.9 0.1)5.0(2

2.12.0

2

)()()(')5.0(

t12

t11

2

t11

1

hDhDD

h

xfxfxfhD

h

xfxfxfhD

iii

iii