Post on 12-May-2022
Numerical Differentiation and Integration
(4-1) Numerical Differentiation
MaSc 352 - PNU(4-1) Numerical Differentiation
(4-1) Numerical Differentiation
1
Forward and Backward Difference Formulas:
MaSc 352 - PNU(4-1) Numerical Differentiation2
MaSc 352 - PNU(4-1) Numerical Differentiation3
Example (1)
MaSc 352 - PNU(4-1) Numerical Differentiation4
Obtaining General Derivative Approximation Formula:
108
MaSc 352 - PNU(4-1) Numerical Differentiation5
MaSc 352 - PNU(4-1) Numerical Differentiation6
Three-Point Formula:
Since
MaSc 352 - PNU(4-1) Numerical Differentiation7
From Eq. (4-3) we get:
Three-Point Formula (When nodes are equally spaced):
MaSc 352 - PNU(4-1) Numerical Differentiation8
MaSc 352 - PNU(4-1) Numerical Differentiation9
MaSc 352 - PNU(4-1) Numerical Differentiation10
Five-Point Formula (nodes are equally spaced):
MaSc 352 - PNU(4-1) Numerical Differentiation11
Example (2) [Exercise 9 Page 177]:
2.62.52.42.32.22.1
-0.6-0.75-0.92-1.12-1.37-1.7
x
)( xf
)1.2(.1 f ′3-Point Endpoint
MaSc 352 - PNU(4-1) Numerical Differentiation
5-Point Endpoint
12
)3.2(.2 f ′3-Point Endpoint
3-Point Midpoint
MaSc 352 - PNU(4-1) Numerical Differentiation
5-Point Midpoint
13
Example (3) [Exercise 11 Page 177]:
2. The error bound of using 3-Point Midpoint formula
1. Absolute error:)2.2(f ′
MaSc 352 - PNU(4-1) Numerical Differentiation
2. The error bound of using 3-Point Midpoint formula
14
The error bound of using 3-Point Endpoint formula
MaSc 352 - PNU(4-1) Numerical Differentiation15
High Derivative Formula:
Second Derivative Midpoint Formula:
MaSc 352 - PNU(4-1) Numerical Differentiation16
MaSc 352 - PNU(4-1) Numerical Differentiation17
Example (4):
2.22.12.01.91.8
19.8617.1514.7812.710.89
x
)( xf
1. h=
Use The second derivative formula (4.9) to approximate using the following data:
)2(f ′′
MaSc 352 - PNU(4-1) Numerical Differentiation18
2. h=
Round-Off Error Instability:
MaSc 352 - PNU(4-1) Numerical Differentiation19
Example (5):Consider using the values in Table 4.3 to approximate where
Compute the error and the optimal choice of h to reduce the error.
)9.0(f ′ xxf sin)( =
MaSc 352 - PNU(4-1) Numerical Differentiation
We can notice that the best approximations are at:
20
Optimal Choice of h:
MaSc 352 - PNU(4-1) Numerical Differentiation21
Homework:
1 (b) +3 (b)
Exercise Set (4-1)
MaSc 352 - PNU(4-1) Numerical Differentiation
5 ( + ) +7 ( + )
9 ( + )
20
22
)2.1(f ′ )4.1(f ′ )2.1(f ′ )4.1(f ′
)4.2(f ′ )6.2(f ′