# Chapter 4 Numerical Differentiation and Integration

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Chapter 4 Numerical Differentiation and Integration. Numerical Analysis. Overview of Numerical Integration. Midpoint Rule. Midpoint Rule. Midpoint Rule. Midpoint Rule. Midpoint Rule. Trapezoidal Rule. Trapezoidal Rule. Trapezoidal Rule. Simpson Rule. Simpson Rule. Simpson Rule. - PowerPoint PPT Presentation

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5

Chapter 4Numerical Differentiation and IntegrationNumerical AnalysisOverview of Numerical Integration

Midpoint Rule

Midpoint Rule

Midpoint Rule

Midpoint Rule

Midpoint Rule

Trapezoidal Rule

Trapezoidal Rule

Trapezoidal Rule

Simpson Rule

Simpson Rule

Simpson Rule

Numerical Integration: Example

Numerical Integration: Example

Composite Rule Simpson

Composite Rule

Composite Rule

Composite Rule

Composite Rule

Composite Simpson Rule

Composite Simpson Rule

Composite Simpson Rule

Composite Simpson Rule

Composite Simpson Rule

Composite Trapezoidal Rule

Composite Midpoint Rule

Composite Midpoint Rule

Composite Rules: Example

Composite Rules: Example

Composite Rules: Example

Composite Rules: Round-off Error

Composite Rules: Round-off Error

Numerical Differentiation

Two Points Formula

Two Points Formula

46Two Points Formula: Example

Generalize Numerical Differentiation Formula

Generalize Numerical Differentiation Formula

Three Points Formula

Three Points Formula

Three Points Formula

Three Points Formula

Other Numerical Differentiation Formula

Numerical Differentiation: Example

Numerical Differentiation: Example

Round-Off Error Analysis

Round-Off Error Analysis

Effect of Round-Off Error

Effect of Round-Off Error

Effect of Round-Off Error

Richardsons Extrapolation

Richardsons Extrapolation

Richardsons Extrapolation

Richardsons Extrapolation

Richardsons Extrapolation

Richardsons Extrapolation

Richardsons Extrapolation: Example

68Richardsons Extrapolation: Example

Richardsons Extrapolation: Example

Five Points Formula by Extrapolation

Five Points Formula by Extrapolation

Five Points Formula by Extrapolation

Five Points Formula by Extrapolation

Romberg Integral

Romberg Integral

Romberg Integral

Romberg Integral: Example

Romberg Integral

Romberg Integral

Romberg Integral

Romberg Integral: Example

ab

()

..

, 0 , ,

(Simpson Rule)

0 Simpson

0.242670.297660.095310.297420.177940.60184

0.242000.292820.095240.297320.178240.60083

0.244000.307360.095450.296260.177350.60384

Simpson0.242670.297660.095310.297420.177940.60184

2.6676.4001.0992.9581.4166.389

2.0002.0001.0002.8181.6825.436

4.00016.0001.3333.3260.9098.389

Simpson2.6676.6671.1112.9641.4256.421

0.0316717 2 0.0021541 0.0001376 1. 2. 3.

Simpson

Simpson

xyo , , ()

xyo , , ()

Simpson 0.00002

0.00002

, Simpson

Simpson ( )

yoxyxoyxo Gaussian Legendre Legendre ,,,,

20.57735026921.0000000000

-0.57735026921.0000000000

30.77459666920.5555555556

0.00000000000.8888888889

-0.77459666920.5555555556

40.86113631160.3478548451

0.33998104360.6521451549

-0.33998104360.6521451549

-0.86113631160.3478548451

50.90617984590.2369268850

0.53846931010.4786286705

0.00000000000.5688888889

-0.53846931010.4786286705

-0.90617984590.2369268850

Linear Transform

Gaussian

0.1093643 Gaussian

20.57735026921.0000000000

-0.57735026921.0000000000

30.77459666920.5555555556

0.00000000000.8888888889

-0.77459666920.5555555556

Gaussian 3 Simpson

(Forward-difference Formula) (backward-difference Formula) , ,

0.10.641853890.54067220.0154321

0.010.593326850.55401800.0015432

0.0010.588342070.55540130.0001543

,,,,

( )

1.810.889365

1.912.703199

2.014.778112

2.117.148957

2.219.855030

, () () () () ()

0.80000.71736

0.85000.75128

0.88000.77074

0.89000.77707

0.89500.78021

0.89800.78208

0.89900.78270

0.90100.78395

0.90200.78457

0.90500.78643

0.91000.78950

0.92000.79560

0.95000.81342

1.00000.84147

0.0010.625000.00339

0.0020.622500.00089

0.0050.622000.00039

0.0100.62150-0.00011

0.0200.62150-0.00011

0.0500.62140-0.00021

0.1000.62055-0.00106

5

(1) (2)2(2) - (1)

(3) (4) 4(4)-(3)

4 2 ,

(5)(6) (5)-(6) (7)

(7) (8) (8) (9) 4(8)-(9)

3

, (10) (10)

(11) (11)

(12) (13)(12)-4(13)

(14)

0.00000000

1.570796332.09439511

1.896118902.004559761.99857073

1.974231602.000269171.999983132.00000555

1.993570342.000016591.999999752.000000011.99999999

1.998393362.000001032.000000002.000000002.000000002.00000000

2.00000000