ch4 - Chapter 4 Numerical Dierentiation and Integration...

Chapter 4
Numerical Differentiation and Integration
Per-Olof Persson
[email protected]
Department of Mathematics
University of California, Berkeley
Math 128A Numerical Analysis

Numerical Differentiation
Forward and Backward Differences
Inspired by the definition of derivative:
f
0
( x
0
) = lim
h → 0
f ( x
0
+ h ) - f ( x
0
)
h
,
choose a small h and approximate
f
0
( x
0
) ≈
f ( x
0
+ h ) - f ( x
0
)
h
The error term for the linear Lagrange polynomial gives:
f
0
( x
0
) =
f ( x
0
+ h ) - f ( x
0
)
h
-
h
2
f
00
( ξ )
Also known as the forward-difference formula if h > 0 and the
backward-difference formula if h < 0

General Derivative Approximations
Differentiation of Lagrange Polynomials
Differentiate
f ( x ) =
n X
k =0
f ( x
k
) L
k
( x ) +
( x - x
0
) · · · ( x - x
n
)
( n + 1)!
f
( n +1)
( ξ ( x ))
to get
f
0
( x
j
) =
n
X
k =0
f ( x
k
) L
0
k
( x
j
) +
f
( n +1)
( ξ ( x
j
))
( n + 1)!
Y
k 6 = j
( x
j
- x
k
)
This is the ( n + 1) -point formula for approximating f
0
( x
j
) .

Commonly Used Formulas
Using equally spaced points with h = x
j +1
- x
j
, we have the
three-point formulas
f
0
( x
0
) =
1
2 h
[ - 3 f ( x
0
) + 4 f ( x
0
+ h ) - f ( x
0
+ 2 h )] +
h
2
3
f
(3)
( ξ
0
)
f
0
( x
0
) =
1
2 h
[ - f ( x
0
- h ) + f ( x
0
+ h )] -
h
2
6
f
(3)
( ξ
1
)
f
0
( x
0
) =
1
2 h
[ f ( x
0
- 2 h ) - 4 f ( x
0
- h ) + 3 f ( x
0
)] +
h
2
3
f
(3)
( ξ
2
)
f
00
( x
0
) =
1
h 2
[ f ( x
0
- h ) - 2 f ( x
0
) + f ( x
0
+ h )] -
h
2
12
f
(4)
( ξ )
and the five-point formula
f
0
( x
0
) =
1
12 h
[ f ( x
0
- 2 h ) - 8 f ( x
0
- h ) + 8 f ( x
0
+ h ) - f ( x
0
+ 2 h )]
+
h
4
30
f
(5)
( ξ )

Optimal h
Consider the three-point central difference formula:
f
0
( x
0
) =
1
2 h
[ f ( x
0
+ h ) - f ( x
0
- h )] -
h
2
6
f
(3)
( ξ
1
)
Suppose that round-off errors ε are introduced when
computing f . Then the approximation error is
f
0
( x
0
) -
˜
f ( x
0
+ h ) -
˜
f ( x
0
- h )
2 h
≤
ε
h
+
h
2
6
M = e ( h )
where
˜
f is the computed function and | f
(3)
( x ) | ≤ M
Sum of truncation error h
2
M/ 6 and round-off error ε/h
Minimize e ( h ) to find the optimal h =
3
p

Richardson’s Extrapolation
Suppose N ( h ) approximates an unknown M with error
M - N ( h ) = K
1
h + K
2
h
2
+ K
3
h
3
+ · · ·
then an O ( h
j
) approximation is given for j = 2 , 3 , . . . by
N
j
( h ) = N
j - 1
h
2
+
N
j - 1
( h/ 2) - N
j - 1
( h )
2 j - 1 - 1
The results can be written in a table:
O ( h ) O ( h
2
) O ( h
3
) O ( h
4
)
1: N
1
( h ) ≡ N ( h )
2: N
1
(
h
2
) ≡ N (
h
2
) 3: N
2
( h )
4: N
1
(
h
4
) ≡ N (
h
4
) 5: N
2
(
h
2
) 6: N
3
( h )
7: N
1
(
h
8
) ≡ N (
h
8
) 8: N
2
(
h
4
) 9: N
3
(
h
2
) 10: N
4
( h )

Richardson’s Extrapolation
If some error terms are zero, different and more efficient
formulas can be derived
Example: If
M - N ( h ) = K
2
h
2
+ K
4
h
4
+ · · ·
then an O ( h
2 j
) approximation is given for j = 2 , 3 , . . . by
N
j
( h ) = N
j - 1
h
2
+
N
j - 1
( h/ 2) - N
j - 1
( h )
4
j - 1
- 1

Numerical Quadrature
Integration of Lagrange Interpolating Polynomials
Select { x
0
, . . . , x
n
} in [ a, b ] and integrate the Lagrange polynomial
P
n
( x ) =
∑
n
i =0
f ( x
i
) L
i
( x ) and its truncation error term over [ a, b ]
to obtain
Z