AOSC 652: Week 6, Day 2

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Numerical Integration

Week 6, Day 2

8 Oct 2014

Analysis Methods in Atmospheric and Oceanic Science

AOSC 652

1


AOSC 652: Analysis Methods in AOSCNumerical Integration

Also known as “quadrature”

In numerical analysis, a quadrature rule is a method for evaluatinga definite integral of a function, usually stated as a weighted sum of the

function values evaluated a specified points within the domain of integration.

Many of the classic algorithms assume thatthe value of the function is known at equally spaced points.

If a climate model is devised using altitude as the vertical coordinate,Then integrals wrt height could possibly be evaluated using a classic algorithms

(i.e., Simpson’s Rule).

If a climate model is devised using pressure as the vertical coordinate,then integrals wrt height can be must be evaluated using a technique

designed for unequally spaced points (Trapezoidal Rule, Gaussian Integration)

2


AOSC 652: Analysis Methods in AOSC

Simpson’s Rule:

Trapezoidal Rule:

Will work for evenly OR unevenly spaced grid and any N

Classic method will work only for evenly spaced grid and odd N

[ ]

1

1

2

2 11, 3, 5

where = ( ) /

( ) ( ) + 4 ( ) ( )3

−

+ +=

−

≈ +Σ∫N

NNx

i i ix i

h x x N

hf x dx f x f x f x

[ ]1

1

11

1( )( ) ( ) + ( )

2

−

+=

+ −≈ Σ∫

NNx

i ix i

iix xf x dx f x f x

3



Simpson’s Rule:

Trapezoidal Rule:



[ ]

1

1

2

2 11, 3, 5

where = ( ) /

( ) ( ) + 4 ( ) ( )3

−

+ +=

−

≈ +Σ∫N

NNx

i i ix i

h x x N

hf x dx f x f x f x

Why bother with the more complicated Simpson’s rule ?

[ ]1

1

11

1( )( ) ( ) + ( )

2

−

+=

+ −≈ Σ∫

NNx

i ix i

iix xf x dx f x f x

4



[ ]

1

1

2

2 11, 3, 5

where = ( ) /

( ) ( ) + 4 ( ) ( )3

−

+ +=

−

≈ +Σ∫N

NNx

i i ix i

h x x N

hf x dx f x f x f x

1

1

44 3 2 1

1, 5, 9

where = ( ) /

14 ( ) + 64 ( ) 24 ( ) 64 ( ) + 14 ( ) ( )

45

−+ + + +

=

−

+ +≈ Σ∫

N

N

Nxi i i i i

xi

h x x N

f x f x f x f x f xf x dx h

Bode’s Rule: N must be evenly divisible by 5; based on 4th order Polynomials;Error order 6th derivative of f

[ ]

1

1

3

3 2 11, 4, 6

where = ( ) /

3( ) ( ) 3 ( ) 3 ( ) + ( ) 8

−

+ + +=

−

≈ + +Σ∫N

NNx

i i i ix i

h x x N

hf x dx f x f x f x f x

Simpson’s Rule: N must be odd (or need work around); based on 3d order polynomialsError order 4th derivative of f

Simpson’s 3/8 Rule: N must be evenly divisible by 4; also based on 3d order polynomialsError order 4th derivative of f

5


All of these routines require even grid spacing !

[ ]

1

1

2

2 11, 3, 5

where = ( ) /

( ) ( ) + 4 ( ) ( )3

−

+ +=

−

≈ +Σ∫N

NNx

i i ix i

h x x N

hf x dx f x f x f x

1

1

44 3 2 1

1, 5, 9

where = ( ) /

14 ( ) + 64 ( ) 24 ( ) 64 ( ) + 14 ( ) ( )

45

−+ + + +

=

−

+ +≈ Σ∫

N

N

Nxi i i i i

xi

h x x N

f x f x f x f x f xf x dx h

Bode’s Rule: N must be evenly divisible by 5; based on 4th order Polynomials;Error order 6th derivative of f

[ ]

1

1

3

3 2 11, 4, 6

where = ( ) /

3( ) ( ) 3 ( ) 3 ( ) + ( ) 8

−

+ + +=

−

≈ + +Σ∫N

NNx

i i i ix i

h x x N

hf x dx f x f x f x f x

Simpson’s Rule: N must be odd (or need work around); based on 3d order polynomialsError order 4th derivative of f

Simpson’s 3/8 Rule: N must be evenly divisible by 4; also based on 3d order polynomialsError order 4th derivative of f

6



Gaussian Quadrature:

y1, y2, … yn nodes are not uniformly spacedc1, c2, … cn Gauss coefficients are determined once n is specified

Nice descriptions of theory at http://en.wikipedia.org/wiki/Gaussian_quadratureand http://www.efunda.com/math/num_integration/num_int_gauss.cfm

( )n1

1i=1

1 ( ) = ( ( )) ( ( )) 2

b

i ia

dxf x dx f g y dy b a c f g ydy−

≈ − ∑∫ ∫

Note:

produces exact result for polynomials of degree 3 or less

1

1

3 3( ) 3 3−

−≈ +

∫ f y dy f f

7

http://en.wikipedia.org/wiki/Gaussian_quadrature

http://www.efunda.com/math/num_integration/num_int_gauss.cfm



Simpson’s Rule:

Trapezoidal Rule:



[ ]

1

1

2

2 11, 3, 5

where = ( ) /

( ) ( ) + 4 ( ) ( )3

−

+ +=

−

≈ +Σ∫N

NNx

i i ix i

h x x N

hf x dx f x f x f x

[ ]1

1

11

1( )( ) ( ) + ( )

2

−

+=

+ −≈ Σ∫

NNx

i ix i

iix xf x dx f x f x

8



Simpson’s Rule:

Trapezoidal Rule:



[ ]

1

1

2

2 11, 3, 5

where = ( ) /

( ) ( ) + 4 ( ) ( )3

−

+ +=

−

≈ +Σ∫N

NNx

i i ix i

h x x N

hf x dx f x f x f x

However, we’ve written a version of Simpson’s Rulethat will work for either even or odd N

[ ]1

1

11

1( )( ) ( ) + ( )

2

−

+=

+ −≈ Σ∫

NNx

i ix i

iix xf x dx f x f x

9



Simpson’s Rule:

Trapezoidal Rule:



[ ]

1

1

2

2 11, 3, 5

where = ( ) /

( ) ( ) + 4 ( ) ( )3

−

+ +=

−

≈ +Σ∫N

NNx

i i ix i

h x x N

hf x dx f x f x f x

We’re now going to examine the Trapezoidal Rule & Simpson’s Rule using“data files” of Y vs X, based on the evaluation of several analytic functions”

[ ]1

1

11

1( )( ) ( ) + ( )

2

−

+=

+ −≈ Σ∫

NNx

i ix i

iix xf x dx f x f x

10



Simpson’s Rule:

Trapezoidal Rule:



[ ]

1

1

2

2 11, 3, 5

where = ( ) /

( ) ( ) + 4 ( ) ( )3

−

+ +=

−

≈ +Σ∫N

NNx

i i ix i

h x x N

hf x dx f x f x f x

We’re now going to examine the Trapezoidal Rule & Simpson’s Rule using“data files” of Y vs X, based on the evaluation of several analytic functions”

Why do you think we are going to use “analytic functions”?

[ ]1

1

11

1( )( ) ( ) + ( )

2

−

+=

+ −≈ Σ∫

NNx

i ix i

iix xf x dx f x f x

11



Please copy files:

~rjs/aosc652/week_06/data_generate.f and~rjs/aosc652/week_06/data_integrate.f

to your work area.

Numerical Integration

12



We’re going to use four “analytic functions” to generate “data files” of Y vs X,which we will then integrate:

1 1 2

0 0 1 = 1 ( ) 3 =∫ ∫Integral Function x dx x dx

1 1 4


1 1 8


1 1 12


13



What do these functions look like?

Compile code data_generate.f, run using “100” for number of intervals,and plot resulting functions using:

Black for Function 1; Blue for Function 2;Green for Function 3; Red for Function 4

We’re going to use four “analytic functions” to generate “data files” of Y vs X,which we will then integrate:

1 1 2


1 1 4


1 1 8


1 1 12


14


AOSC 652: Analysis Methods in AOSC1 2

0Black : 3 ∫ x dx

1 4

0Blue : 5 ∫ x dx

1 8

0Green : 9 ∫ x dx

1 12

0Red : 13 ∫ x dx

NPTS = 101

15



Why did I write NPTS = 101 ?

1 2

0Black : 3 ∫ x dx

1 4

0Blue : 5 ∫ x dx

1 8

0Green : 9 ∫ x dx

1 12

0Red : 13 ∫ x dx

NPTS = 101

16



Why did I write NPTS = 101 ?What is the numerical value of these “definite integrals”?

1 2

0Black : 3 ∫ x dx

1 4

0Blue : 5 ∫ x dx

1 8

0Green : 9 ∫ x dx

1 12

0Red : 13 ∫ x dx

NPTS = 101

17


AOSC 652: Analysis Methods in AOSCNumerical Integration: Let’s fill in the following tables:

a) Run program data_generate.e to generate “fn_for_integration*.dat”files, for 25, 10, and 6 intervals (in addition to the file for 100 intervalsyou should have already generated)

b) Compile code data_integrate.f

c) Run program data_integrate.e to complete table

18



Function 100 intervals

25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

Numerical Integration: Let’s fill in the following tables:Trapezoidal Rule:

Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

19


AOSC 652: Analysis Methods in AOSCNumerical Integration: Let’s fill in the following tables:

a) Run program data_generate.e to generate “fn_for_integration*.dat”files, for 25, 10, and 6 intervals (in addition to the file for 100 intervalsyou should have already generated)

b) Compile code data_integrate.f

c) Run program data_integrate.e to complete table

20




25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

21




25 intervals

10 intervals

6 intervals

3 x2 1.00005

5 x4

9 x8

13 x12


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

22




25 intervals

10 intervals

6 intervals

3 x2 1.00005

5 x4 1.00017

9 x8

13 x12


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

23




25 intervals

10 intervals

6 intervals

3 x2 1.00005

5 x4 1.00017

9 x8 1.00060

13 x12


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

24




25 intervals

10 intervals

6 intervals

3 x2 1.00005

5 x4 1.00017

9 x8 1.00060

13 x12 1.00130


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

25




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080

5 x4 1.00017

9 x8 1.00060

13 x12 1.00130


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

26




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080

5 x4 1.00017 1.00267

9 x8 1.00060

13 x12 1.00130


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

27




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080

5 x4 1.00017 1.00267

9 x8 1.00060 1.00959

13 x12 1.00130


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

28




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080

5 x4 1.00017 1.00267

9 x8 1.00060 1.00959

13 x12 1.00130 1.02074


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

29




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500

5 x4 1.00017 1.00267

9 x8 1.00060 1.00959

13 x12 1.00130 1.02074


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

30




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500

5 x4 1.00017 1.00267 1.01665

9 x8 1.00060 1.00959

13 x12 1.00130 1.02074


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

31




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500

5 x4 1.00017 1.00267 1.01665

9 x8 1.00060 1.00959 1.05958

13 x12 1.00130 1.02074


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

32




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500

5 x4 1.00017 1.00267 1.01665

9 x8 1.00060 1.00959 1.05958

13 x12 1.00130 1.02074 1.12766


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

33




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665

9 x8 1.00060 1.00959 1.05958

13 x12 1.00130 1.02074 1.12766


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

34




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958

13 x12 1.00130 1.02074 1.12766


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

35




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

36




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

37




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2

5 x4

9 x8

13 x12

38




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1.

5 x4

9 x8

13 x12

39




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1.

5 x4 1.

9 x8

13 x12

40




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1.

5 x4 1.

9 x8 1.

13 x12

41




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1.

5 x4 1.

9 x8 1.

13 x12 1.

42




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003

5 x4 1.

9 x8 1.

13 x12 1.

43




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003

5 x4 1. 1.

9 x8 1.

13 x12 1.

44




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003

5 x4 1. 1.

9 x8 1. 1.00004

13 x12 1.

45




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003

5 x4 1. 1.

9 x8 1. 1.00004

13 x12 1. 1.00024

46




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003 1.

5 x4 1. 1.

9 x8 1. 1.00004

13 x12 1. 1.00024

47




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003 1.

5 x4 1. 1. 1.

9 x8 1. 1.00004

13 x12 1. 1.00024

48




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003 1.

5 x4 1. 1. 1.

9 x8 1. 1.00004 1.00164

13 x12 1. 1.00024

49




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003 1.

5 x4 1. 1. 1.

9 x8 1. 1.00004 1.00164

13 x12 1. 1.00024 1.00875

50




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003 1. 1.

5 x4 1. 1. 1.

9 x8 1. 1.00004 1.00164

13 x12 1. 1.00024 1.00875

51




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003 1. 1.

5 x4 1. 1. 1. 1.00051

9 x8 1. 1.00004 1.00164

13 x12 1. 1.00024 1.00875

52




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003 1. 1.

5 x4 1. 1. 1. 1.00051

9 x8 1. 1.00004 1.00164 1.01212

13 x12 1. 1.00024 1.00875

53




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003 1. 1.

5 x4 1. 1. 1. 1.00051

9 x8 1. 1.00004 1.00164 1.01212

13 x12 1. 1.00024 1.00875 1.05807

54




25 intervals

10 intervals

6 intervals

3 x2 1.00005 1.00080 1.00500 1.01389

5 x4 1.00017 1.00267 1.01665 1.04617

9 x8 1.00060 1.00959 1.05958 1.16347

13 x12 1.00130 1.02074 1.12766 1.34357


Simpson’s Rule:


25 intervals

10 intervals

6 intervals

3 x2 1. 1.00003 1. 1.

5 x4 1. 1. 1. 1.00051

9 x8 1. 1.00004 1.00164 1.01212

13 x12 1. 1.00024 1.00875 1.05807

55


Trapezoidal Rule:

Simpson’s Rule:

1.163471.059581.009591.000609 x8

1.12766

1.01665

1.00500

10 intervals

1.343571.020741.0013013 x12

1.046171.002671.000175 x4

1.013891.000801.000053 x2

6 intervals

25 intervals

100 intervals

Function

1.163471.059581.009591.000609 x8

1.12766

1.01665

1.00500

10 intervals

1.343571.020741.0013013 x12

1.046171.002671.000175 x4

1.013891.000801.000053 x2

6 intervals

25 intervals

100 intervals

Function

1.012121.001641.000041.9 x8

1.00875

1.

1.

10 intervals

1.058071.000241.13 x12

1.000511.1.5 x4

1.1.000031.3 x2

6 intervals

25 intervals

100 intervals

Function

1.012121.001641.000041.9 x8

1.00875

1.

1.

10 intervals

1.058071.000241.13 x12

1.000511.1.5 x4

1.1.000031.3 x2

6 intervals

25 intervals

100 intervals

Function

Let’s conduct a graphical analysisto investigate behavior of integrationby Trapezoidal rule and Simpson’s rule:

a) focus first on the 6 interval calculationof function = 13x12

b) what should we plot ?!?

56


Trapezoidal Rule:

Simpson’s Rule:

1.163471.059581.009591.000609 x8

1.12766

1.01665

1.00500

10 intervals

1.343571.020741.0013013 x12

1.046171.002671.000175 x4

1.013891.000801.000053 x2

6 intervals

25 intervals

100 intervals

Function

1.163471.059581.009591.000609 x8

1.12766

1.01665

1.00500

10 intervals

1.343571.020741.0013013 x12

1.046171.002671.000175 x4

1.013891.000801.000053 x2

6 intervals

25 intervals

100 intervals

Function

1.012121.001641.000041.9 x8

1.00875

1.

1.

10 intervals

1.058071.000241.13 x12

1.000511.1.5 x4

1.1.000031.3 x2

6 intervals

25 intervals

100 intervals

Function

1.012121.001641.000041.9 x8

1.00875

1.

1.

10 intervals

1.058071.000241.13 x12

1.000511.1.5 x4

1.1.000031.3 x2

6 intervals

25 intervals

100 intervals

Function




57


Trapezoidal Rule:

Simpson’s Rule:

1.163471.059581.009591.000609 x8

1.12766

1.01665

1.00500

10 intervals

1.343571.020741.0013013 x12

1.046171.002671.000175 x4

1.013891.000801.000053 x2

6 intervals

25 intervals

100 intervals

Function

1.163471.059581.009591.000609 x8

1.12766

1.01665

1.00500

10 intervals

1.343571.020741.0013013 x12

1.046171.002671.000175 x4

1.013891.000801.000053 x2

6 intervals

25 intervals

100 intervals

Function

1.012121.001641.000041.9 x8

1.00875

1.

1.

10 intervals

1.058071.000241.13 x12

1.000511.1.5 x4

1.1.000031.3 x2

6 intervals

25 intervals

100 intervals

Function

1.012121.001641.000041.9 x8

1.00875

1.

1.

10 intervals

1.058071.000241.13 x12

1.000511.1.5 x4

1.1.000031.3 x2

6 intervals

25 intervals

100 intervals

Function




58


Trapezoidal Rule:

Simpson’s Rule:

1.163471.059581.009591.000609 x8

1.12766

1.01665

1.00500

10 intervals

1.343571.020741.0013013 x12

1.046171.002671.000175 x4

1.013891.000801.000053 x2

6 intervals

25 intervals

100 intervals

Function

1.163471.059581.009591.000609 x8

1.12766

1.01665

1.00500

10 intervals

1.343571.020741.0013013 x12

1.046171.002671.000175 x4

1.013891.000801.000053 x2

6 intervals

25 intervals

100 intervals

Function

1.012121.001641.000041.9 x8

1.00875

1.

1.

10 intervals

1.058071.000241.13 x12

1.000511.1.5 x4

1.1.000031.3 x2

6 intervals

25 intervals

100 intervals

Function

1.012121.001641.000041.9 x8

1.00875

1.

1.

10 intervals

1.058071.000241.13 x12

1.000511.1.5 x4

1.1.000031.3 x2

6 intervals

25 intervals

100 intervals

Function




59


Trapezoidal Rule:

Simpson’s Rule:

1.163471.059581.009591.000609 x8

1.12766

1.01665

1.00500

10 intervals

1.343571.020741.0013013 x12

1.046171.002671.000175 x4

1.013891.000801.000053 x2

6 intervals

25 intervals

100 intervals

Function

1.163471.059581.009591.000609 x8

1.12766

1.01665

1.00500

10 intervals

1.343571.020741.0013013 x12

1.046171.002671.000175 x4

1.013891.000801.000053 x2

6 intervals

25 intervals

100 intervals

Function

1.012121.001641.000041.9 x8

1.00875

1.

1.

10 intervals

1.058071.000241.13 x12

1.000511.1.5 x4

1.1.000031.3 x2

6 intervals

25 intervals

100 intervals

Function

1.012121.001641.000041.9 x8

1.00875

1.

1.

10 intervals

1.058071.000241.13 x12

1.000511.1.5 x4

1.1.000031.3 x2

6 intervals

25 intervals

100 intervals

Function




60


Air Quality Model Satellite Data

For which model NO2 profile did Tim check the details of his integration scheme ?

61


Air Quality Model Satellite Data

“If the function you propose to integrate is sharply concentrated in one or more peaks, or if the shape is not readily characterized by a single length-scale, then” you should be concerned about the accuracy of your integration scheme.

62




n1

1i=1

( ) ( )i if y dy c f y−

≈ ∑∫

Nice description of theory at http://en.wikipedia.org/wiki/Gaussian_quadrature

63





n1

1i=1

( ) ( )i if y dy c f y−

≈ ∑∫

Theory based on limits always being from −1 to 1


64





n1

1i=1

( ) ( )i if y dy c f y−

≈ ∑∫


Problem of course is limits we care about are usually not −1 to 1 !


65





n1

1i=1

( ) ( )i if y dy c f y−

≈ ∑∫



Who we gonna call ?


66





n1

1i=1

( ) ( )i if y dy c f y−

≈ ∑∫




Who we gonna call ?

67





n1

1i=1

( ) = ( ( )) ( ( )) b

i ia

dx dxf x dx f g y dy c f g ydy dy−

≈

∑∫ ∫

To use Gaussian Quadrature for the evaluation of integrals with arbitrary limits,

must execute a Variable Transformation

( ) is a linear function: i.e., g(y)=Intercept + Slope y, that relates to 1 and to 1= = − = =

g yx a y x b y

68




n1

1i=1

( ) = ( ( )) ( ( )) b

i ia

dx dxf x dx f g y dy c f g ydy dy−

≈

∑∫ ∫

To use Gaussian Quadrature for the evaluation of integrals with arbitrary limits,

must execute a Variable Transformation

( )

( )

Can show:1 Slope = 2

1 Intercept = 2

b a

b a

−

+

( ) is a linear function: i.e., g(y)=Intercept + Slope y, that relates to 1 and to 1= = − = =

g yx a y x b y

69




( ) ( )1 1 where ( ) = + y 2 2

g y b a b a+ −

( )n1

1i=1

1 ( ) = ( ( )) ( ( )) 2

b

i ia


≈ − ∑∫ ∫

See http://en.wikipedia.org/wiki/Gaussian_quadrature

70





y1, y2, … yn nodes are not uniformly spacedc1, c2, … cn Gauss coefficients are determined once n is specified

( )n1

1i=1

1 ( ) = ( ( )) ( ( )) 2

b

i ia


≈ − ∑∫ ∫

See http://en.wikipedia.org/wiki/Gaussian_quadrature

Location of nodes yi (places where the function is evaluated)and weights ci both vary as a function of n

71





http://www.efunda.com/math/num_integration/findgausslaguerre.cfmprovides Gaussian nodes and weights as a function of n

72

http://www.efunda.com/math/num_integration/findgausslaguerre.cfm


73


AOSC 652: Analysis Methods in AOSCGaussian Quadrature:

1

1

3 3( ) 3 3−

−≈ +

∫ f y dy f f

1 2 1 23 3Where do weights of 1 ; 1 and nodes of ; come from ?

3 3

c c y y −

= = = =

74



1 2 1 21

1 1 2 21

12 3 2 3

1 0 1 1 2 1 3 1 2 0 1 2 2 2 3 21

0

Four unknowns: , , ,

Assume: ( ) c ( ) ( ) is exact for polynomials up to order 3

Then: ( ) = ( ) + ( )

Or:

c c y y

f y dy f y c f y

f y dy c a a y a y a y c a a y a y a y

a dy

−

−

−

≈ +

+ + + + + +

∫

∫

1 1

0 1 2 0 1 1 1 2 2 11 1

1 12 2 2 3 3 3

2 2 1 1 2 2 2 3 1 1 2 2 31 1

2 ( ) 0 ( )

2 ( ) 0 ( ) 3

a c c a a y dy c y c y a

a y dy a c y c y a a y dy c y c y a

−

− −

= = + = = +

= = + = = +

∫ ∫

∫ ∫

75



1 2 1 21

1 1 2 21

12 3 2 3

1 0 1 1 2 1 3 1 2 0 1 2 2 2 3 21

0



Then: ( ) = ( ) + ( )

Or:

c c y y

f y dy f y c f y


a dy

−

−

−

≈ +

+ + + + + +

∫

∫

1 1

1 2 1 1 1 2 21 1

1 12 2 2 3 3 3

2 1 1 2 2 3 1 1 2 21 1

0 0 1

2 2 3

2 ( ) 0 ( )

2 ( ) 0 ( ) 3

c c a y dy c y c y

a y dy c y c y a y dy c y

a a a

a a c y a

−

− −

= = + = = +

= = + = = +

∫ ∫

∫ ∫

1 2 1 2Four equations, four unknowns: , , , c c y y

76



1 2 1 21

1 1 2 21

12 3 2 3

1 0 1 1 2 1 3 1 2 0 1 2 2 2 3 21

0



Then: ( ) = ( ) + ( )

Or:

c c y y

f y dy f y c f y


a dy

−

−

−

≈ +

+ + + + + +

∫

∫

1 1

1 2 1 1 1 2 21 1

1 12 2 2 3 3 3

2 1 1 2 2 3 1 1 2 21 1

0 0 1

2 2 3

2 ( ) 0 ( )

2 ( ) 0 ( ) 3

c c a y dy c y c y

a y dy c y c y a y dy c y

a a a

a a c y a

−

− −

= = + = = +

= = + = = +

∫ ∫

∫ ∫

1 2 1 23 3

3 3Solution: =1 =1 =, , , c c y y− =

77



Gaussian Quadrature: we would like to fill in the following table:Function 2 nodes 10 nodes Error,

10 nodes3 x2

5 x4

9 x8

13 x12

a) Assignment #6, part 1: Start with ~rjs/aosc/week_06/gauss_integrate.f

b) A few lines in subroutines fn1, fn2, fn3, and fn4 need to be completed

These lines relate variables x and y, as well as dx/dy, where x is thedependent variable used for our functions so far today (limits of 0 to 1) &y is the dependent variable for Gaussian integration (limits of −1 to 1)

Once these lines are filled in, the cells in the table can be populated

78



Calculation of multi-dimensional integrals adds additional level of complexity

IDL and Matlab have various tools for the evaluation of integrals of data:• good for you, the user, to understand how these tools work !• pay attention to:

− grid spacing: are grid pts evenly spaced?should they be evenly spaced?should they be “linear”?how does integrand vary relative to grid points?

− number of points:some methods place strict limits on “mod N”may need to modify IDL or MATLAB integration toolcode to properly handle end points

79

AOSC 652: Week 6, Day 2

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Transcript of AOSC 652: Week 6, Day 2