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
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
Gaussian Quadrature
Gaussian Quadrature
Gaussian Quadrature
Gaussian Quadrature
Gaussian Quadrature
Gaussian Quadrature
Gaussian Quadrature: Example
Gaussian Quadrature: Example
Gaussian Quadrature: Example
Gaussian Quadrature: Example
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
